navier–stokes equations

Advances in Mechanics and Mathematics 34

Grzegorz ŁukaszewiczPiotr Kalita

Navier–Stokes EquationsAn Introduction with Applications

Advances in Mechanics and Mathematics

Volume 34

Series Editors:David Y. Gao, Virginia Polytechnic Institute and State UniversityTudor Ratiu, École Polytechnique Fédérale

Advisory Board:Ivar Ekeland, University of British ColumbiaTim Healey, Cornell UniversityKumbakonam Rajagopal, Texas A&M UniversityDavid J. Steigmann, University of California, Berkeley

More information about this series at http://www.springer.com/series/5613

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Grzegorz Łukaszewicz • Piotr Kalita

Navier–Stokes EquationsAn Introduction with Applications

123

Grzegorz ŁukaszewiczFaculty of Mathematics,

Informatics, and MechanicsUniversity of WarsawWarszawa, Poland

Piotr KalitaFaculty of Mathematics and Computer

ScienceJagiellonian University in KrakowKrakow, Poland

ISSN 1571-8689 ISSN 1876-9896 (electronic)Advances in Mechanics and MathematicsISBN 978-3-319-27758-5 ISBN 978-3-319-27760-8 (eBook)DOI 10.1007/978-3-319-27760-8

Library of Congress Control Number: 2015960205

© Springer International Publishing Switzerland 2016This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part ofthe material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting, reproduction on microfilms or in any other physical way, and transmission or informationstorage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodologynow known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoes not imply, even in the absence of a specific statement, that such names are exempt from the relevantprotective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in this bookare believed to be true and accurate at the date of publication. Neither the publisher nor the authors orthe editors give a warranty, express or implied, with respect to the material contained herein or for anyerrors or omissions that may have been made.

Printed on acid-free paper

This Springer imprint is published by Springer NatureThe registered company is Springer International Publishing AG Switzerland

To Renata, Agata, and Jacek, with love(Grzegorz)

To my beloved wife Kasia(Piotr)

Preface

Admittedly, as useful a matter as the motion of fluid and relatedsciences has always been an object of thought. Yet until this dayneither our knowledge of pure mathematics nor our command ofthe mathematical principles of nature have a successfultreatment.

–Daniel Bernoulli

Incompressible Navier–Stokes equations describe the dynamic motion (flow) ofincompressible fluid, the unknowns being the velocity and pressure as functions oflocation (space) and time variables. To solve those equations would mean to predictthe behavior of the fluid under knowledge of its initial and boundary states. Theseequations are one of the most important models of mathematical physics. Althoughthey have been a subject of vivid research for more than 150 years, there are stillmany open problems due to the nature of nonlinearity present in the equations.The nonlinear convective term present in the equations leads to phenomena suchas eddy flows and turbulence. In particular the question of solution regularity forthree-dimensional problem was appointed by Clay Mathematics Institute as one ofthe Millennium Problems, that is, the key problems in modern mathematics. This is,on one hand, due to the fact that the problem remains challenging and fascinatingfor mathematicians and, on the other hand, that the applications of the Navier–Stokes equations range from aerodynamics (drag and lift forces), through designof watercrafts and hydroelectric power plants, to the medical applications of themodels of flow of blood in vessels.

This book is aimed at a broad audience of people interested in the Navier–Stokesequations, from students to engineers and mathematicians involved in the researchon the subject of these equations.

It originated in part from a series of lectures of the first author given over the past15 years at the Faculty of Mathematics, Informatics and Mechanics of the Universityof Warsaw; at summer schools at UNICAMP, Campinas, Brasil; and at UniversitéJean Monnet, Saint-Etienne, France. The lectures were based on the leading bookson the then young theory of infinite dimensional dynamical systems, focused onmathematical physics, in particular, on Temam [220]; Chepyzhov and Vishik [61];Doering and Gibbon [88]; Foias, Manley, Rosa, and Temam [99]; and Robinson[197].

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The lectures at the Mathematics Faculty of the University of Warsaw were alsoattended by students and PhD students from the Faculty of Physics and Facultyof Geophysics, and it became clear that a routine mathematical lecture had to beextended to include additional aspects of hydrodynamics. Some students asked for“more physics and motivation” and “more real applications”; others were mainlyinterested in the mathematics of the Navier–Stokes equations, and yet others wouldlike to see the Navier–Stokes equations in a more general context of evolutionequations and to learn the theory of infinite dimensional dynamical systems on theresearch level. These several aspects of hydrodynamics well suited the tastes andinterests of the lecturer, and also the second author was welcomed to join the projectof the book at a later stage.

In consequence, the audience of the book is threesome:

Group I: Mathematicians, physicists, and engineers who want to learn about theNavier–Stokes equations and mathematical modeling of fluidsGroup II: University teachers who may teach a graduate or PhD course on fluidmechanics basing on this book or higher-level students who start research on theNavier–Stokes equationsGroup III: Researchers interested in the exchange of current knowledge ondynamical systems approach to the Navier–Stokes equations

Although, in principle, all these three groups can find interest in all chapters ofthe book, Chaps. 2–7 are primarily targeted at Group I, Chaps. 3, 4, 7, 8, 11, and 12aimed mainly at Group II, and Chaps. 7–16 for Group III.

For a reader with reasonable background on calculus, functional analysis, andtheory of weak solutions for PDEs, the whole book should be understandable.

The book was planned to be a monograph which could also be used as a textbookto teach a course on fluid mechanics or the Navier–Stokes equations. Typicalcourses could be “Navier–Stokes equations”, “partial differential equations”, “fluidmechanics”, “infinite dimensional dynamics systems.” To this end many chaptersof this book include exercises. Moreover, we did not restrain ourselves to includea number of figures to liven the text and make it more intuitive and less formal.We believe that the figures will be helpful. Special care was undertaken to keep theindividual chapters self-contained as far as possible to allow the reader to read thebook linearly (in linear portions). That demanded several small repetitions here andthere.

To understand the first chapters of this book, just the basic knowledge oncalculus, that can be learned from any calculus textbook, should be enough.

The book is planned to be self-contained, but, to understand its last chapters,some knowledge from a textbook like “Partial Differential Equations” by L.C. Evans(which contains all necessary knowledge on functional analysis and PDEs) wouldbe helpful. Each chapter contains an introduction that explains in simple words thenature of presented results and a section on bibliographical notes that will place itin the context of past and current research.

Preface ix

Several people greatly contributed, knowingly or not knowingly, to the creationof the book. Our thanks go to our colleagues and collaborators: Guy Bayada, MahdiBoukrouche, Thomas Caraballo, José Langa, Pepe Real, James Robinson etc.

The first author is grateful to Guy Bayada who introduced him to the problemsof lubrication theory and flows in narrow films during his visits at INSA, Lyon,and to Mahdi Boukrouche with whom he collaborated for several years on thissubject. Thomas Caraballo, José Langa, and Pepe Real introduced him to the subjectof pullback attractors during his visit at the University of Seville. Thanks for theopportunity to give the summer courses in Campinas and Saint-Etienne go to MarcoRojas-Medar and Mahdi Boukrouche, respectively. Many thanks go also to ChunyouSun, Meihua Yang, and Yongqin Xie for their kind invitation of the first author togive several lectures at the University of Lanzhou, then at Huazhong Universityof Science and Technology in Wuhan and at The University in Changsha, China,in June 2013. Inspirational discussions and exchange of ideas with these Chinesefriends, including also Qingfeng Ma and Yuejuan Wang, contributed to the form ofthe last chapters of the book.

The research group at Jagiellonian University with their leader, StanisławMigórski, greatly motivated the authors as regards contact problems. The secondauthor owes a lot to his colleagues and teachers from Jagiellonian University; hewould like to express his thanks especially to Zdzisław Denkowski and StanisławMigórski. He would also like to thank Robert Schaefer who first introduced to himthe topics of fluid mechanics. He is also grateful for inspiring discussions in the fieldof contact mechanics to his collaborators from the University of Perpignan, MirceaSofonea and Mikaël Barboteu.

We would like to thank Wojciech Pociecha for his help with the preparation ofthe figures.

The work was in part supported by the National Science Center of Poland underthe Maestro Advanced Project no. DEC-2012/06/A/ST1/00262.

Finally, we express our gratitude to the AMMA Series editor, David Y. Gao, andto Marc Strauss and the editors of Springer Publishing House for their care andencouragement during the preparation of the book.

Warszawa, Poland Grzegorz ŁukaszewiczKraków, Poland Piotr KalitaOctober 2015

Contents

1 Introduction and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Equations of Classical Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1 Derivation of the Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 The Stress Tensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4 Navier–Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5 Vorticity Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.6 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.7 Similarity of Flows and Nondimensional Variables . . . . . . . . . . . . . . . . 282.8 Examples of Simple Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.9 Comments and Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1 Theorems from Functional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Sobolev Spaces and Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.3 Some Embedding Theorems and Inequalities. . . . . . . . . . . . . . . . . . . . . . . 483.4 Sobolev Spaces of Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.5 Evolution Spaces and Their Useful Properties . . . . . . . . . . . . . . . . . . . . . . 613.6 Gronwall Type Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.7 Clarke Subdifferential and Its Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.8 Nemytskii Operator for Multifunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.9 Clarke Subdifferential: Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.10 Comments and Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4 Stationary Solutions of the Navier–Stokes Equations . . . . . . . . . . . . . . . . . . 834.1 Basic Stationary Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.1.1 The Stokes Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.1.2 The Nonlinear Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.1.3 Other Topological Methods to Deal with the

Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.2 Comments and Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

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5 Stationary Solutions of the Navier–Stokes Equations with Friction . . 955.1 Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.2 Friction Operator and Its Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.3 Weak Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.4 Existence of Weak Solutions for the Case of Linear

Growth Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.5 Existence of Weak Solutions for the Case of Power

Growth Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.6 Comments and Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6 Stationary Flows in Narrow Films and the Reynolds Equation . . . . . . . 1116.1 Classical Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.2 Weak Formulation and Main Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.3 Scaling and Uniform Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.4 Limit Variational Inequality, Strong Convergence,

and the Limit Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256.5 Remarks on Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1286.6 Strong Convergence of Velocities and the Limit Equation . . . . . . . . . 1346.7 Reynolds Equation and the Limit Boundary Conditions . . . . . . . . . . . 1376.8 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1416.9 Comments and Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

7 Autonomous Two-Dimensional Navier–Stokes Equations . . . . . . . . . . . . . 1437.1 Navier–Stokes Equations with Periodic Boundary Conditions . . . . 1437.2 Existence of the Global Attractor: Case of Periodic

Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1517.3 Convergence to the Stationary Solution: The Simplest Case. . . . . . . 1577.4 Convergence to the Stationary Solution for Large Forces . . . . . . . . . . 1597.5 Average Transfer of Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1637.6 Comments and Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

8 Invariant Measures and Statistical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1698.1 Existence of Invariant Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1698.2 Stationary Statistical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1768.3 Comments and Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

9 Global Attractors and a Lubrication Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 1839.1 Fractal Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1839.2 Abstract Theorem on Finite Dimensionality and an Algorithm. . . . 1859.3 An Application to a Shear Flow in Lubrication Theory . . . . . . . . . . . . 193

9.3.1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1939.3.2 Energy Dissipation Rate Estimate . . . . . . . . . . . . . . . . . . . . . . . . . 1969.3.3 A Version of the Lieb–Thirring Inequality . . . . . . . . . . . . . . . . 2009.3.4 Dimension Estimate of the Global Attractor . . . . . . . . . . . . . . 201

9.4 Comments and Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

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10 Exponential Attractors in Contact Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20710.1 Exponential Attractors and Fractal Dimension . . . . . . . . . . . . . . . . . . . . . 20710.2 Planar Shear Flows with the Tresca Friction Condition . . . . . . . . . . . . 211

10.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21110.2.2 Existence and Uniqueness of a Global in Time Solution . 21810.2.3 Existence of Finite Dimensional Global Attractor . . . . . . . . 22210.2.4 Existence of an Exponential Attractor . . . . . . . . . . . . . . . . . . . . . 229

10.3 Planar Shear Flows with Generalized TrescaType Friction Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23110.3.1 Classical Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . 23110.3.2 Weak Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . 23310.3.3 Existence and Properties of Solutions . . . . . . . . . . . . . . . . . . . . . 23610.3.4 Existence of Finite Dimensional Global Attractor . . . . . . . . 24110.3.5 Existence of an Exponential Attractor . . . . . . . . . . . . . . . . . . . . . 246

10.4 Comments and Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

11 Non-autonomous Navier–Stokes Equations and PullbackAttractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25111.1 Determining Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25111.2 Determining Nodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25611.3 Pullback Attractors for Asymptotically Compact

Non-autonomous Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26011.4 Application to Two-Dimensional Navier–Stokes

Equations in Unbounded Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26811.5 Comments and Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

12 Pullback Attractors and Statistical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 27712.1 Pullback Attractors and Two-Dimensional

Navier–Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27712.2 Construction of the Family of Probability Measures . . . . . . . . . . . . . . . 28112.3 Liouville and Energy Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28512.4 Time-Dependent and Stationary Statistical Solutions . . . . . . . . . . . . . . 28812.5 The Case of an Unbounded Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29112.6 Comments and Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

13 Pullback Attractors and Shear Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29713.1 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29713.2 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29813.3 Existence and Uniqueness of Global in Time Solutions. . . . . . . . . . . . 30113.4 Existence of the Pullback Attractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30513.5 Fractal Dimension of the Pullback Attractor . . . . . . . . . . . . . . . . . . . . . . . . 31013.6 Comments and Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

14 Trajectory Attractors and Feedback Boundary Control inContact Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31714.1 Setting of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31714.2 Weak Formulation of the Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

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14.3 Existence of Global in Time Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32214.4 Existence of Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32914.5 Comments and Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

15 Evolutionary Systems and the Navier–Stokes Equations . . . . . . . . . . . . . . 33715.1 Evolutionary Systems and Their Attractors . . . . . . . . . . . . . . . . . . . . . . . . . 33715.2 Three-Dimensional Navier–Stokes Problem

with Multivalued Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34015.3 Existence of Leray–Hopf Weak Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 34215.4 Existence and Invariance of Weak Global Attractor,

and Weak Tracking Property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35115.5 Comments and Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

16 Attractors for Multivalued Processes in Contact Problems . . . . . . . . . . . . 35916.1 Abstract Theory of Pullback D-Attractors

for Multivalued Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35916.2 Application to a Contact Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36616.3 Comments and Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

1Introduction and Summary

When you put together the science of movements of water,remember to put beneath each proposition its applications, sothat such science may not be without uses.

– Leonardo da Vinci

This chapter provides, for the convenience of the reader, an overview of the wholebook, first of its structure and then of the content of the individual chapters.

The outline of the book structure is as follows.Chapter 2 shows the derivation of the Navier–Stokes equations from the prin-

ciples of physics and discusses their physical and mathematical properties andexamples of the solutions for some particular cases, without going into complicatedmathematics. This part is aimed to fill in a gap between an engineer and mathe-matician and should be understood by anybody with basic knowledge of calculusno more complicated than the Stokes theorem.

In Chap. 3, a necessary mathematical background including these parts of func-tional analysis and theory of Sobolev spaces which are needed to understand modernresearch on the Navier–Stokes equations is presented. Chapters 4–6 comprise threeexamples of stationary problems.

Then we smoothly move to the research level part of the book (Chaps. 7–16)which presents the analysis from the point of view of global attractors of theasymptotic (in time) behavior of the velocities being the solutions of the Navier–Stokes system. Roughly speaking we endeavor to show how the modern theory ofglobal attractors can be used to construct the mathematical objects that enclose theseemingly chaotic and unordered eddy and turbulent flows. We tame these flowsby showing their fine properties like the finite dimensionality of global attractors,which means that the description of unrestful and turbulent states can be done byfinite number of parameters or existence of invariant measures which means that theflow becomes, in statistical sense, stationary.

We deal with non-autonomous problems using the recent and elegant theory ofso-called pullback attractors that allows to cope with flows with changing in timesources, sinks, or boundary data. We also solve problems with multivalued boundaryconditions that allow to model various contact conditions between the fluid andenclosing object, such as the stick/slip frictional boundary behavior.

© Springer International Publishing Switzerland 2016G. Łukaszewicz, P. Kalita, Navier–Stokes Equations, Advancesin Mechanics and Mathematics 34, DOI 10.1007/978-3-319-27760-8_1

1

2 1 Introduction and Summary

The analysis is primarily done for the two-dimensional Navier–Stokes equations,where, as a model, the problem from lubrication theory, that can be reduced to twodimensions is used. The study with various types of boundary conditions, includingmultivalued ones, is presented.

Some of the results presented in this part are based on the previous and alreadypublished work of the authors, but some results, that are the subjects of currentresearch, are yet unpublished elsewhere.

Below we present the content of the book in some more detail.In Chap. 2 we give an overview of the equations of classical hydrodynamics.

We provide their derivation, discuss the associated physical quantities, comment onthe constitutive laws, stress tensor, and thermodynamics, finally we present someelementary properties of the derived system and also some cases where it is possibleto calculate the exact solutions of the following system of the incompressibleNavier–Stokes equations,

@u

@t� ��u C .u � r/u C 1

�rp D f ;

div u D 0;

which are the main subject of the book.In Chap. 3 we introduce the basic preliminary mathematical tools to study the

Navier–Stokes equations, including results from linear and nonlinear functionalanalysis (e.g., the Lax–Milgram lemma, fixed point theorems) as well as the theoryof function spaces (e.g., compactness theorems). We present, in particular, someof the most frequently used in the sequel embedding theorems and inequalities.We discuss the versions of the Gronwall lemma used in the sequel, and providesome necessary background in the theory of Clarke subdifferential and differentialinclusions.

In Chaps. 4–6 we consider stationary problems. Chapter 4 is devoted to thestationary Navier–Stokes equations in a bounded three-dimensional domain Q,

��u C .u � r/u C rp D f in Q;

divu D 0 in Q;

with one of the boundary conditions:

1. Q D Œ0;L�3 in R3 and we assume periodic boundary conditions, or

2. Q is a bounded domain in R3, with smooth boundary, and we assume the

homogeneous boundary condition u D 0 on @Q.

This basic problem serves as an introduction to the mathematical theory ofthe Navier–Stokes equations. We introduce the suitable function spaces in whichwe (usually) seek solutions of the stationary problem, then we present the weakformulation. It allows us to use the theories of linear and nonlinear functionalanalysis (Lax–Milgram lemma and fixed point theorems, respectively) to prove theexistence of solutions.

1 Introduction and Summary 3

To show typical methods used when dealing with nonlinear problems, we presenta number of proofs based on various linearizations and fixed point theorems instandard function spaces. The solutions, due to the nonlinearity of the Navier–Stokesequations are not in general unique, however, under some restriction on the massforce and viscosity coefficient (quite intuitive from the physical point of view), onecan prove their uniqueness.

In Chap. 5 we consider the stationary Navier–Stokes equations with friction inthe three-dimensional bounded domain ˝. The domain boundary @˝ is dividedinto two parts, namely the boundary �D on which we assume the homogeneousDirichlet boundary condition and the contact boundary �C on which we decomposethe velocity into the normal and tangent directions. In the normal direction weassume u � n D 0, i.e., there is no leak through the boundary, while in the tangentdirection we set �T� 2 h.x; u� /, the multivalued relation between the tangentstress and tangent velocity. This relation is the general form of the friction law onthe contact boundary. The existence of solution is shown by the Kakutani–Fan–Glicksberg fixed point theorem and some cut-off techniques.

In Chap. 6 we consider a typical problem for the hydrodynamic equations comingfrom the lubrication theory. In this theory the domain of the flow is usually very thinand the engineers are interested in the distribution of the pressure therein. Becauseof the thinness of the domain, in practice one assumes that the pressure woulddepend only on two and not three independent variables. The pressure distributionis governed then by the Reynolds equation, which depends on the original boundarydata. Our aim is to obtain the Reynolds equation starting from the stationary Stokesequations, considered in the three-dimensional domains ˝", " > 0,

��u C rp D f in ˝";

div u D 0 in ˝";

with a Fourier boundary condition on the top � "F and Tresca boundary conditions

on the bottom part of the boundary �C, respectively. We show how to pass, in aprecise mathematical way, as " ! 0, from the three-dimensional Stokes equationsto a two-dimensional Reynolds equation for the pressure distribution (see Fig. 1.1).

The passage from the three-dimensional problem to a two-dimensional onedepends on several factors and additional scaling assumptions. The existence ofsolutions of the limit equations follows from the existence of solutions of theoriginal three-dimensional problem. Finally, we show the uniqueness of the limitsolution.

In Chaps. 7–10 we consider the nonstationary autonomous Navier–Stokesequations

ut � ��u C .u � r/u C rp D f ; (1.1)

div u D 0; (1.2)

in two-dimensional domains. In these chapters the solutions are global in time andunique.


Fig. 1.1 Schematic view of the problem considered in Chap. 6. Incompressible static Stokesequation is solved on the three-dimensional domain ˝". The domain thickness is given by "h,where h is a function of the point in the two-dimensional domain !. As " ! 0, Reynolds equationon ! is recovered

Chapter 7 is a general introduction to evolutionary two-dimensional Navier–Stokes equations. We prove some basic properties of solutions assuming that theexternal forces do not depend on time and that the domain of the flow is bounded.The boundary conditions are either periodic or homogeneous Dirichlet ones. In thischapter we introduce the notion of the global attractor, one of the main objects tostudy also in Chaps. 8–10.

In Chap. 8 we prove the existence of invariant measures associated with two-dimensional autonomous Navier–Stokes equations. The invariant measures aresupported on the global attractor. Then we introduce the notion of a stationarystatistical solution and prove that every invariant measure is also such a solution.Existence of the invariant measures (stationary statistical solutions) supported onthe global attractor reveals the statistical properties of the potentially chaotic fluidflow after a long time of evolution when the external forces do not depend on time.The non-autonomous case is considered in Chap. 12.

In Chaps. 9–10 we consider system (1.1) and (1.2) in the domain ˝ depicted inFig. 1.2, with homogeneous condition u D 0 on �D, periodic condition u.0; x2/ Du.L; x2/ on �L, and several contact boundary conditions on �C. The motivation forsuch problem setup comes again from problems in contact mechanics, the theory oflubrication and shear flows in narrow films.

One may look at the domain ˝ as a rectification of the ring-like domainconsidered in the theory of lubrication, where it represents a cross section of aninfinite journal bearing. The problem reduces to describing a flow between twocylinders. The outer cylinder is at rest and the inner cylinder rotates providing adriving force to the fluid (lubricant). Since the cylinders are infinitely long it canbe assumed in the first approximation that the flow is two-dimensional. Describeddomain geometry is schematically presented in Fig. 1.3.

The boundary conditions on �C include the following ones.In Chap. 9 we pose

u D Ue1; U 2 R; U > 0 on �C;


Fig. 1.2 Schematic view ofthe flow domain and itsboundaries in Chaps. 9 and 10

Fig. 1.3 Three-dimensional infinite ring-like domain and its rectification considered in Chaps. 9and 10

by which we mean that the boundary �C is moving with a constant velocityU0e1 D .U0; 0/ and the velocity of the fluid at the boundary equals the velocityof the boundary.

We prove the existence of a global attractor and estimate from above its fractaldimension in terms of given data and geometry of the domain of the flow.

In Chap. 10 we consider two problems. We assume that there is no flux across �C

so that the normal component of the velocity on �C satisfies

u � n D 0 on �C;

and that the tangential component of the velocity u� on �C is unknown and satisfiesthe Tresca friction law with a constant and positive maximal friction coefficient k.This means that


jT� .u; p/j � k

jT� .u; p/j < k ) u� D U0e1

jT� .u; p/j D k ) 9� � 0 such that u� D U0e1 � �T� .u; p/

9>>>>>=

>>>>>;

on �C;

(1.3)

where T� is the tangential component of the stress tensor on �C and U0e1 D .U0; 0/,U0 2 R, is the velocity of the lower surface producing the driving force of the flow.

In the second problem the boundary �C is also assumed to be moving withthe constant velocity U0e1 D .U0; 0/ which, together with the mass force, producesthe driving force of the flow. The friction coefficient k is assumed to be related tothe slip rate through the relation k D k.ju� � U0j/, where k W RC ! R

C. If thereis no slip between the fluid and the boundary then the friction is bounded by thethreshold k.0/

u� D U0 ) jT� j � k.0/ on �C; (1.4)

while if there is a slip, then the friction force density (equal to tangential stress) isgiven by the expression

u� ¤ U0 ) �T� D k.ju� � U0j/ u� � U0

ju� � U0j on �C: (1.5)

Note that (1.4) and (1.5) generalize the Tresca law (1.3) where k was assumed tobe a constant. Here k depends of the slip rate, this dependence represents the factthat the kinetic friction is less than the static one, which holds if k is a decreasingfunction.

We prove that for both problems above there exist exponential attractors, inparticular the global attractors of finite fractal dimension.

In Chaps. 11–13 we consider the time asymptotics of solutions of the two-dimen-sional Navier–Stokes equations. First, in Chap. 11 we prove two properties of theequations in a bounded domain, concerning the existence of determining modes andnodes. Then we study the equations in an unbounded domain, in the framework ofthe theory of infinite dimensional non-autonomous dynamical systems, introducingthe notion of the pullback attractor.

Chapter 12 presents a construction of invariant measures and statistical solutionsfor the non-autonomous Navier–Stokes equations in bounded and some unboundeddomains in R

2. More precisely, we construct the family of probability measuresftgt2R and prove the relations t.E/ D �.U.t; �/�1E/ for t; � 2 R, t � � andBorel sets E in H. The support of each measure t is included in the section A.t/of the pullback attractor. We prove also the Liouville and energy equations. Finally,we consider statistical solutions of the Navier–Stokes equations supported on thepullback attractor.


In Chap. 13 we consider the problem of existence and finite dimensionality of thepullback attractor for a class of two-dimensional turbulent boundary driven flows.We generalize here the results from Chap. 9 to the non-autonomous problem. Thenew element in our study with respect to that in Chap. 9 is the allowance of thevelocity of �C to depend on time, i.e.,

u D U.t/e1; U.t/ 2 R on �C:

Our aim is to study the time asymptotics of solutions in the frame of the dynamicalsystems theory. We prove the existence of the pullback attractor and estimate itsfractal dimension. We shall apply the results from Chap. 11, reformulated here inthe language of evolutionary processes.

Chapters 14–16 are devoted to global in time solutions of the Navier–Stokesequations which are not necessary unique. We introduce theories of global attractorsfor multivalued semiflows and multivalued processes to include this situation.We study further examples of contact problems in both autonomous and non-autonomous cases.

In Chap. 14 we consider two-dimensional nonstationary Navier–Stokes shearflows in the domain ˝ as in Fig. 1.2, with nonmonotone and multivalued boundaryconditions on �C. Namely, we assume the following subdifferential boundarycondition

Qp.x; t/ 2 @j.un.x; t// on �C;

where Qp D p C 12juj2 is the Bernoulli (total) pressure, un D u � n, j W R ! R is a

given locally Lipschitz superpotential, and @j is a Clarke subdifferential of j.Our considerations are motivated here by feedback control problems for fluid

flows in domains with semipermeable walls and membranes and by the theory oflubrication. We prove the existence of global in time solutions of the consideredproblem which is governed by a partial differential inclusion, and then we provethe existence of a trajectory attractor and a weak global attractor for the associatedmultivalued semiflow.

In Chap. 15 we study the three-dimensional problem in a bounded domain ˝.The problem domain is the three-dimensional counterpart of the one presented inFig. 1.2. The boundary of ˝ is divided into three parts: the lateral one �L on whichwe assume the periodic boundary conditions, the homogeneous Dirichlet one and,finally, the contact one �C on which we consider a general form of multivaluedfrictional type boundary conditions �T� 2 g.u� /. We prove the existence of theLeray–Hopf weak solutions and, using the framework of evolutionary systems,existence of the weak global attractor.

Finally, in Chap. 16 we consider further non-autonomous and multivalued evo-lution problems, this time in the frame of the theory of pullback attractors formultivalued processes. First we prove an abstract theorem on the existence ofpullback D-attractor and then apply it to study a two-dimensional incompressible


Navier–Stokes flow with a general form of multivalued frictional contact conditionson �C. We assume that there is no flux across �C and hence we have

un.t/ D 0 on �C;

and that the tangential component of the velocity u� on �C is in the followingrelation with the tangential stresses T� ,

�T� .t/ 2 @j.x; t; u� .t// on �C:

In above formula j W �C � R � R ! R is a potential which is locally Lipschitz andnot necessarily convex with respect to the last variable, and @ is the subdifferentialin the sense of Clarke taken with respect to the last variable u� .

The tangent conditions on �C in Chaps. 15 and 16 represent the frictional contactbetween the fluid and the wall, where the friction force depends in a nonmonotoneand even discontinuous way on the slip rate, and are a generalization of theconditions considered in Chap. 10. For this case we prove the existence of theattractor.

Most chapters are devoted to two-dimensional problems. Three-dimensionalproblems are considered only in Chaps. 2, 4, 5, 6, and 15. One reason for that isassociated with the character of the Navier–Stokes equations, namely the fact thatin the two-dimensional problems it is relatively easy to prove the uniqueness of thesolutions which allows us to use the well-developed theory of infinite dimensionaldynamical systems for semigroup and processes, while the uniqueness of the three-dimensional Navier–Stokes equations is in general an open question. We alsoconsider the two- and three-dimensional problems without assuming the solutionuniqueness in the framework of (more recent) theories of trajectory attractors,multivalued semiflows, evolutionary systems, and multivalued processes.

The other reason to focus on the two-dimensional flows concern the simplicity.Our aim was to test first the more elementary two-dimensional models of somereal engineering problems. The word “test” here means not only checking the wellposedness of a particular problem. In Chap. 9 we estimate the attractor dimensionand show how it depends on the shape of the domain (cf. [24, 26], where the upperbounds of the attractor dimension depend also on the geometry of �D). Assumethat the answer to the question on the dependence of the attractor dimension on thegeometry of the boundary is such that in the two-dimensional case the estimate fromabove of the attractor dimension is independent of geometry (for example, on theroughness of the boundary represented by the oscillations of the function h D h.x1/describing �D). Such a result would be contradictory to our intuition, provided theintuitive hypotheses

attractor dimension � level of chaos in the flow � geometry of the flow domain

where “�” means “is related to,” are justified.Such a contradiction with our intuition could be resolved in the following

ways:


1. there is no such contradiction in the “real” three-dimensional case, it appearsonly in the two-dimensional case (but where lies the difference?),

2. the attractor dimension does not represent the level of chaos in the fluid flowdescribed by the (good) model of the Navier–Stokes equations,

3. the Navier–Stokes equations model is not good enough to give the right answerto the problems of chaotic movement of the classical fluids.

The close correspondence between the level of chaos in the fluid flow and thegeometry of the domain is evident as a physical phenomenon (recall observing aflow of water in the river).

To confirm the agreement of the results provided by the modeling with ourphysical intuition or else to confront the above potential possibilities motivated usto study the problems of the existence and properties of the attractor. There are stillmany interesting and important problems close to these considered in the book andwe were able to touch only a few ones. One example is to further study the relationsbetween the (type of) boundary conditions and the attractor dimension.

Finally, we remark that this book is devoted to incompressible flows, for themathematical treatise of compressible ones see, e.g., [157, 187].

2Equations of Classical Hydrodynamics

The neglected borderline between two branches of knowledge isoften that which best repays cultivation, or, to use a metaphor ofMaxwell’s, the greatest benefits may be derived from across-fertilization of the sciences.

– John William Strutt, 3rd Baron Rayleigh

In this chapter we give an overview of the equations of classical hydrodynamics.We provide their derivation, comment on the stress tensor, and thermodynamics,finally we present some elementary properties and also some exact solutions of theNavier–Stokes equations.

2.1 Derivation of the Equations of Motion

Fluid flow may be represented mathematically as a continuous transformation ofthree-dimensional Euclidean space into itself. The transformation is parametrizedby a real parameter t representing time.

Let us introduce a fixed rectangular coordinate system .x1; x2; x3/. We refer tothe coordinate triple .x1; x2; x3/ as the position and denote it by x. Now considera particle P moving with the fluid, and suppose that at time t D 0 it occupies aposition X D .X1;X2;X3/ and that at some other time t, �1 < t < C1, it hasmoved to a position x D .x1; x2; x3/. Then x is determined as a function of X and t

x D x.X; t/ or xi D xi.X; t/ : (2.1)

If X is fixed and t varies, Eq. (2.1) specifies the path of the particle initially at X.On the other hand, for fixed t, (2.1) determines a transformation of a region initiallyoccupied by the fluid into its position at time t.


11

12 2 Equations of Classical Hydrodynamics

We assume that the transformation (2.1) is continuous and invertible, that is, thereexists its inverse

X D X.x; t/; .or Xi D Xi.x; t// :

Also, to be able to differentiate, we assume that the functions xi and Xi aresufficiently smooth.

From the condition that the transformation (2.1) possess a differentiable inverseit follows that its Jacobian

J D J.X; t/ D det

�@xi

@Xj

�

satisfies

0 < J < 1 : (2.2)

The initial coordinates X of the particle will be referred to as the materialcoordinates of the particle. The spatial coordinates x may be referred to as itsposition, or place. The representation of fluid motion as a point transformationviolates the concept of the kinetic theory of fluids, as in this theory the particles aremolecules, and they are in random motion. In the theory of continuum mechanicsthe state of motion at a given point x and at a given time t is described by a numberof functions such as � D �.x; t/, u D u.x; t/, D .x; t/ representing density,velocity, temperature, and other hydrodynamical variables.

Due to the transformation (2.1), each such variable f can also be expressed interms of material coordinates

f .x; t/ D f .x.X; t/; t/ D F.X; t/ : (2.3)

The velocity u at time t of a particle initially at X is given, by definition, as

u.x; t/ D U.X; t/ D d

dtx.X; t/ ; .x D x.X; t// : (2.4)

Above, X is treated as a parameter representing a given fixed particle, and this is thereason that we use the ordinary derivative in (2.4).

Having the velocity field u.x; t/, we can (in principle) determine the transforma-tion (2.1), solving the ordinary differential equation

d

dtx.X; t/ D u.x.X; t/; t/;

with x.X; 0/ D X, where X is a parameter.We shall always write

d

dtF.X; t/ and

@

@tf .x; t/ ;

2.1 Derivation of the Equations of Motion 13

where F and f are related by (2.3). We have thus

d

dtF.X; t/ D d

dtf .x.X; t/; t/ D @f

@xi.x.X; t/; t/

dxi

dtC @f

@t.x.X; t/; t/ ;

so that by (2.4) we obtain a general formula

d

dtF.X; t/ D D

Dtf .x; t/ ; (2.5)

where DDt f .x; t/ � @f

@t .x; t/C u.x; t/ � rf .x; t/ is called the material derivative of f .

Transport Theorem Let ˝.t/ denote an arbitrary volume that is moving with thefluid and let f .x; t/ be a scalar or vector function of position and time. The transporttheorem states that

d

dt

Z

˝.t/f .x; t/ dx (2.6)

DZ

˝.t/

�@f

@t.x; t/C u.x; t/ � rf .x; t/C f .x; t/ div u.x; t/

�

dx :

For the proof consider the transformation

x W ˝.0/ ! ˝.t/; x D x.X; t/ ;

as in (2.1). Then

Z

˝.t/f .x; t/ dx

DZ

˝.0/

f .x.X; t/; t/J.X; t/ dX DZ

˝.0/

F.X; t/J.X; t/ dX ;

so that

d

dt

Z

˝.t/f .x; t/dx D d

dt

Z

˝.0/

F.X; t/J.X; t/ dX (2.7)

DZ

˝.0/

�d

dtF.X; t/J.X; t/C F.X; t/

d

dtJ.X; t/

�

dX :


By (2.5) we have

Z

˝.0/

d

dtF.X; t/J.X; t/ dX

DZ

˝.0/

�@f

@t.x.X; t/; t/C u.x.X; t/; t/ � rf .x.X; t/; t/

�

J.X; t/ dX

DZ

˝.t/

�@f

@t.x; t/C u.x; t/ � rf .x; t/

�

dx :

To prove (2.6) it remains to prove the Euler formula

d

dtJ.X; t/ D div u.x.X; t/; t/J.X; t/ ; (2.8)

the proof of which we leave to the reader as an exercise.The fluid is called incompressible if for any domain ˝.0/ and any t,

volume .˝.t// D volume .˝.0// :

From (2.7) with f .x; t/ � 1 we have

d

dtvolume .˝.t// D d

dt

Z

˝.t/dx D

Z

˝.0/

d

dtJ.X; t/dX ;

hence by (2.8), (2.2), and the arbitrariness of choice of the domain ˝.t/ via ˝.0/ anecessary and sufficient condition for the fluid to be incompressible is

div u.x; t/ D 0 :

Exercise 2.1. Prove that the transport theorem can be written in the form

d

dt

Z

˝.t/f .x; t/ dx D

Z

˝.t/

@f

@t.x; t/ dx C

Z

@˝.t/f .x; t/u.x; t/ � n.x; t/ dS ;

where n.x; t/ is the outward unit normal to @˝.t/ at x 2 @˝.t/.Equation of Continuity Let � D �.x; t/ be the mass per unit volume of a fluid atpoint x and time t. Then the mass of any finite volume ˝ is

m DZ

˝

�.x; t/ dx :


The principle of conservation of mass says that the mass of a fluid in a materialvolume ˝ does not change as ˝ moves with the fluid; that is,

d

dt

Z

˝.t/�.x; t/ dx D 0 :

From the transport theorem (2.6) it follows that

Z

˝.t/

�@�

@tC div .�u/

�

dx D 0 ;

whence

@�

@tC div .�u/ D 0 : (2.9)

Sometimes the principle of conservation of mass is expressed as follows. Let ˝ bea fixed volume. Then

d

dt

Z

˝

�.x; t/ dx D �Z

@˝

�u � n dS ; (2.10)

that is, the rate of change of mass in a fixed volume ˝ is equal to the mass fluxthrough its surface.

We notice also the general formula

d

dt

Z

˝.t/�f dx D

Z

˝.t/�

D

Dtf dx : (2.11)

Exercise 2.2. Derive (2.9) from (2.10).

Exercise 2.3 (Cf. [212]). Show that in material coordinates the equation of conti-nuity is

d

dtf�.X; t/J.X; t/g D 0 ;

or

�.X; t/J.X; t/ D �.X; 0/ :

Exercise 2.4 (Cf. [5]). Show that if �0.X/ is the distribution of density of the fluidat time t D 0 and r.div u/ D 0, then

�.x; t/ D �0.X.x; t// exp

�

�Z t

0

div u.x; t/ dt

�

:


Exercise 2.5. Find �.x; t/ for the motion

ui D xi

1C ait.a1 D 2 ; a2 D 1 ; a3 D 0/;

if �0.X/ is the distribution of density of the fluid at time t D 0.

Exercise 2.6. Prove (2.11).

Principle of Conservation of Linear Momentum We assume that the forcesacting on an element of a continuous medium are of two kinds. External, or body,forces, such as gravitation or electromagnetic forces, can be regarded as reachinginto the medium and acting throughout the volume. If f represents such a force perunit mass, then it acts on an element ˝ as

Z

˝

�f dx :

The internal, or contact, forces are to be regarded as acting on an element of volume˝ through its bounding surface. Let n be the unit outward normal at a point ofthe surface @˝, and tn the force per unit area exerted there by the material volumeoutside @˝. Then the surface force exerted on the volume ˝ can be expressed bythe integral

Z

@˝

tn dS :

The Cauchy principle says that tn depends at any given time only on the positionand the orientation of the surface element dS; in other words,

tn D tn.x; t; n/ :

The principle of conservation of linear momentum says that the rate of change oflinear momentum of a material volume equals the resultant force on the volume

d

dt

Z

˝.t/�udx D

Z

˝.t/�f dx C

Z

@˝.t/tn dS ; (2.12)

where f is assumed to be known.By (2.11), (2.12) yields

Z

˝.t/�

Du

Dtdx D

Z

˝.t/�f dx C

Z

@˝.t/tn dS : (2.13)

From this equation we derive a very important fact, namely, that the vector tn (callednormal stress) can be expressed as a linear function of n, in the form

tn.x; t; n/ D n.x; t/T.x; t/ ; (2.14)


where T D fTijg is a matrix called the stress tensor. This will allow us to pass fromthe integral form (2.13) of the equation of conservation of linear momentum to adifferential one.

Let l3 be the volume of˝ D ˝.t/. Dividing both sides of (2.13) by l2 and lettingthe volume tend to zero we obtain

limj˝j!0

l�2Z

@˝

tn dS D 0 ; (2.15)

that is, the stress forces are in local equilibrium.Let ˝ be a domain containing a fluid, and consider a regular tetrahedron with

vertex at an arbitrary point x 2 @˝, and with three of its faces parallel to thecoordinate planes. Let the slanted face have normal n D .n1; n2; n3/ and area ˙ .The normals to the other faces are �e1, �e2, and �e3, and their areas are n1˙ , n2˙ ,and n3˙ . Applying (2.15) to the family of tetrahedrons obtained by letting ˙ ! 0,we obtain

t.n/C n1t.�e1/C n2t.�e2/C n3t.�e3/ D 0 ; (2.16)

where t.n/ D tn D tn.x; t; n/, t.�h/ D t�h for h 2 fe1; e2; e3g, and ni > 0. Bya continuity argument, (2.16) holds for all ni � 0, and then we prove easily thatt.ei/ D �t.�ei/, i D 1; 2; 3, and that it holds for all n. This means that t.n/ may beexpressed as a linear function of n; that is, we can write it in the form (2.14). Thus,by (2.13) and by the Green theorem we obtain

Z

˝.t/�

Du

Dtdx D

Z

˝.t/.�f C div T/ dx ;

whence, by the arbitrariness of the domain of integration,

�Du

DtD �f C div T ; (2.17)

or

�

0

@@

@tui C

3X

jD1uj@

@xjui

1

A D �fi C Tji;j; i D 1; 2; 3 :

This is the general Cauchy equation of motion in differential form.

Exercise 2.7. Give a physical interpretation of the components of the stress tensor.

Notice that we have not specified T yet, that is, we have not made anyassumptions concerning the nature of forces acting on surface elements. Theseforces depend on the kind of fluid, or, more generally, on the kind of medium underconsideration.


In the simplest model the contact forces act perpendicularly to the surfaceelements. We have then

t.n/ D �p.x/n ;

and call p the pressure. The minus sign is chosen so that when p > 0, the contactforces on a closed surface tend to compress the fluid inside; p represents the pressureexerted from outside on a surface of the element of the fluid.

In particular, all fluids at rest exhibit this stress behavior, namely that an elementof area always experiences a stress normal to itself, and this stress is independent ofthe orientation. Such stress is called hydrostatic.

We call this idealized model a perfect fluid. The equation of motion for perfectfluids is

�

�@u

@tC .u � r/u

�

D �f � rp ;

where

.u � r/ui D3X

jD1uj@

@xjui ; i D 1; 2; 3 :

All real fluids when in motion can exert tangential stresses across surface elements,in which case the tensor T is not diagonal.

The stress tensor may always be written in the form

Tij D �pıij C Pij :

In this case Pij is called the viscous stress tensor.In classical fluid dynamics it is assumed that the stress tensor is symmetric, that is,

Tij D Tji :

This assumption has very important consequences. It may be also considered as atheorem if we assume a specific form of the equation of conservation of angularmomentum. We shall discuss this in Sect. 2.2.

Exercise 2.8 (Cf. [5]). Show that the Cauchy equation of motion can be written as

@

@t.�ui/ D �fi C .Tji � �ujui/;j ;

and interpret it physically.


Exercise 2.9 (Cf. [5]). Show that if F is any function of position and time, thenZ

@˝

FTjinj dS DZ

˝

�

TjiF ;j C�F

�Dui

Dt� fi

��

dx

(theorem of stress means).

Equation of Energy The first law of thermodynamics in classical hydrodynamicsstates that the increase of total energy (we shall consider here only kinetic andinternal energies) in a material volume is the sum of the heat transferred and thework done on the volume. We denote by q the heat flux (then �q � n is the heat fluxinto the volume) and by E the specific internal energy. Then the balance expressedby the first law of thermodynamics is, cf. [5],

d

dt

Z

˝.t/�

�1

2juj2 C E

�

dx (2.18)

DZ

˝.t/�f � u dx C

Z

@˝.t/tn � u dS �

Z

@˝.t/q � n dS :

The first integral on the right-hand side is the rate at which the body force does work,the second integral represents the work done by the stress, and the third integralis the total heat flux into the volume.

We shall write this equation in another form. From the theorem of stress means(Exercise 2.9) we have, with F D ui,

Z

@˝.t/uiTjinj dS D

Z

˝.t/

�

Tjiui;j C �uiDui

Dt� �fiui

�

dx :

Rearranging the terms and using the transport theorem, we obtain

d

dt

Z

˝.t/�1

2juj2 dx D

Z

˝.t/�1

2

D

Dtjuj2 dx (2.19)

DZ

˝.t/�fiui dx �

Z

˝.t/Tjiui;j dx C

Z

@˝.t/ui.tn/i dS :

Thus the rate of change of kinetic energy of a material volume is the sum of threeparts: the rate at which the body forces do work, the rate at which the internalstresses do work, and the rate at which the surface stresses do work.

From (2.18), (2.19), the transport theorem, and the Green theorem we obtain

Z

˝.t/

�

�DE

DtC r � q � T W .ru/

�

dx D 0 ;

where T W .ru/ is the dyadic notation for Tjiui;j, the scalar product of T and ru.


Thus

�DE

DtD �r � q C T W .ru/ :

Conservation Laws of Classical Hydrodynamics Above we obtained the follow-ing system of conservation laws of classical hydrodynamics

D�

DtD ��r � u ; (2.20)

�Du

DtD r � T C �f ; (2.21)

�DE

DtD �r � q C T W .ru/ : (2.22)

They are laws of conservation of mass, momentum, and energy, respectively.If we assume the Fourier law for the conduction of heat,

q D �kr .k � 0/ ; (2.23)

where k is the thermal conductivity of the fluid then the energy equation takes theform

�DE

DtD r � .kr/C T W .ru/ :

2.2 The Stress Tensor

In the classical hydrodynamics the stress tensor T is defined by

Tij D .�p C �uk;k/ıij C .ui;j C uj;i/ : (2.24)

If we define the deformation tensor

Di;j D 1

2.ui;j C uj;i/; (2.25)

then the above formula takes the form

Tij D .�p C �uk;k/ıij C 2Dij : (2.26)

Remark 2.1. Formula (2.26) is a consequence of a number of postulates, comingoriginally from G. Stokes, about the fundamental properties of fluids. Thesepostulates can be formulated as follows (cf. [5, 212]):

(a) The stress tensor T is a continuous function of the deformation tensor D and thelocal thermodynamic state, but independent of other kinematic quantities.

2.2 The Stress Tensor 21

(b) The fluid is homogeneous; that is, T does not depend explicitly on x.(c) The fluid is isotropic; that is, there is no preferred direction.(d) When there is no deformation (D D 0), and the fluid is incompressible

(uk;k D 0), the stress is hydrostatic (T D �pI, I is the unit matrix).

Fluids that satisfy these postulates are called Stokesian. It can be proved (cf. [5, 212])that the most general form of the stress tensor in this case is

T D .�p C ˛/I C ˇD C �D2 ;

where p; ˛; ˇ; � are some functions that depend on the thermodynamic state, ˛; ˇ; �being dependent as well on the invariants of the tensor D.

Moreover, when the dependence of the components of T on the components ofD is postulated to be linear, the stress tensor can be written as

T D .�p C � div u/I C 2D ;

which coincides with (2.24). Such linear Stokesian fluids are called Newtonian.Fluids that are not Newtonian are called non-Newtonian. One important exampleof the latter are the micropolar fluids [92, 159].

The Stress Tensor and the Law of Conservation of Angular MomentumLooking at the form of the equation of conservation of linear momentum

d

dt

Z

˝.t/�u dx D

Z

˝.t/�f dx C

Z

@˝.t/tn dS ;

and recalling the definition of angular momentum in mechanics of mass pointsor rigid particles, it seems natural to assume the following form of the law ofconservation of angular momentum:

d

dt

Z

˝.t/�.x � u/ dx D

Z

˝.t/�.x � f / dx C

Z

@˝.t/x � tn dS : (2.27)

In fact, this form of the law of conservation of angular momentum holds if weassume that all torques arise from macroscopic forces. This is the case in mostcommon fluids, but a fluid with a strongly polar character, e.g., a polyatomic fluid,is capable of transmitting stress torques and being subjected to body torques. Wecall such fluids polar.

Theorem 2.1. For an arbitrary continuous medium satisfying the continuityequation (2.9) and the dynamical equation (2.17) the following statements areequivalent:

(i) the stress tensor is symmetric,(ii) equation (2.27) holds.


Remark 2.2. In classical hydrodynamics the stress tensor is symmetric, and the lawof conservation of angular momentum is defined by Eq. (2.27). In consequence,in classical hydrodynamics the law of conservation of angular momentum can bederived from the law of conservation of mass and the law of conservation of linearmomentum, and as such adds nothing to the description of the fluid.

Proof. Let us assume (ii), and we shall deduce (i). Applying formula (2.11), wehave from (2.27)

d

dt

Z

˝.t/�.x � u/ dx (2.28)

DZ

˝.t/�

D

Dt.x � u/ dx D

Z

˝.t/�

�

x � Du

Dt

�

dx

DZ

˝.t/�.x � f / dx C

Z

@˝.t/x � tn dS :

By the Green theorem,Z

@˝.t/x � tn dS D

Z

˝.t/.x � .r � T/C Tx/ dx ; (2.29)

where r � T is another notation for div T , and Tx is the vector �ijkTjk (�ijk is thealternating tensor of Levi-Civita), so that by (2.28)

Z

˝.t/x �

�

�Du

Dt� �f � r � T

�

dx DZ

˝.t/Tx dx :

The left-hand side vanishes identically by the Cauchy equation; hence the right-handside vanishes for an arbitrary volume, and so Tx D 0. However, the components ofTx are T23 � T32, T31 � T13, T12 � T21, and the vanishing of these implies Tij D Tji,so that T is symmetric.

We leave to the reader the proof that (i) implies (ii). ut

2.3 Field Equations

Substituting the stress tensor (2.24) into the system (2.20)–(2.22) we obtain thesystem of field equations of classical hydrodynamics

D�

DtD ��r � u ; (2.30)

�Du

DtD �rp C .�C /r div u C �u C �f ; (2.31)

2.4 Navier–Stokes Equations 23

�DE

DtD �p div u C �˚ � r � q ; (2.32)

where

�˚ D �.div u/2 C 2D W D (2.33)

is the dissipation function of mechanical energy per mass unit.Let us assume that the fluid is viscous and incompressible, namely, that > 0

and

div u D 0 ; (2.34)

that the specific internal energy of the fluid is proportional to its temperature,

E D cr ; where cr D const > 0 ; (2.35)

and that Fourier’s law (2.23) (with k D const � 0) holds. With (2.34), (2.35), (2.23),and (2.33), system (2.30)–(2.32) becomes

@�

@tC u � r� D 0 ; div u D 0 ; (2.36)

�

�@u

@tC .u � r/u

�

D �rp C �u C �f ; (2.37)

�cr

�@

@tC u � r

�

D 2D W D C k�: (2.38)

2.4 Navier–Stokes Equations

Assuming that the density � of the fluid is uniform and denoting � D

�, D k

�(�

is called the kinematic viscosity coefficient), Eqs. (2.36)–(2.38) reduce to

@u

@tC .u � r/u D �1

�rp C ��u C f ; (2.39)

div u D 0 ; (2.40)

cr

�@

@tC u � r

�

D 2�D W D C �: (2.41)


When the body forces f do not depend on temperature, the first two equations of theabove system,

@u

@tC .u � r/u D �1

�rp C ��u C f ; (2.42)

div u D 0 (2.43)

constitute a closed system of equations with respect to variables u; p, and are calledNavier–Stokes equations of viscous incompressible fluids with uniform density (weshall call them just the Navier–Stokes equations). The mechanical energy of the flowgoverned by (2.42) and (2.43) due to viscous dissipation is lost and appears as heat.This can be seen from Eq. (2.41) in which the term 2�D W D is positive, providedthe flow is not uniform. In real fluids, however, density depends on temperature,so that our system (2.39)–(2.41) may be physically impossible. In fact, due toviscosity and high velocity gradients the temperature rises in view of (2.41), and thisproduces density fluctuations, contrary to our assumption that density is uniform inthe flow domain. Thus, reduced problems often play the role of more or less justifiedapproximations. For more considerations of this kind cf. [109, Chap. 1].

When the body forces depend on temperature, f D f ./, we have to take intoaccount the whole system (2.39)–(2.41). One of the considered in the literaturesystem of equations of heat conducting viscous and incompressible fluid are theso-called Boussinesq equations,

@u

@tC .u � r/u D � 1

�0rp C ��u C 1

�0g˛. � 0/ ; (2.44)

div u D 0 ; (2.45)

@

@tC u � r D

cr� ; (2.46)

where g represents the vertical gravity acceleration, ˛ is the thermal expansioncoefficient, and

cris the thermal diffusion coefficient. Moreover, �0 and 0 are

some reference density and temperature, respectively. In the velocity equation thevertical buoyancy force 1

�0g˛. � 0/ results from changes of density associated

with temperature variations � � �0 D �˛. � 0/. This is the only term in thesystem where changes of density were taken into account. We have also abandonedthe viscous dissipation term in the temperature equation.

2.5 Vorticity Dynamics

Taking the curl of the equation of motion

@u

@tC .u � r/u D �1

�rp C ��u;

2.5 Vorticity Dynamics 25

we obtain

@!

@tC .u � r/! D .! � r/u C ��!; (2.47)

where the vector field ! D r � u is called vorticity of the fluid. It has a simplephysical interpretation. In the case of two-dimensional motion with

u D .u1.x; y/; u2.x; y/; 0/;

the vorticity reduces to

! D .0; 0; !3.x1; x2// D�

0; 0;@u2.x1; x2/

@x1� @u1.x1; x2/

@x2

�

;

where the third component represents twice the angular velocity of a small(infinitesimal) fluid element at point .x1; x2/. The vorticity field is, by definition,divergence free,

div! D 0:

In the case of two-dimensional motions the Eq. (2.47) reduces to

@!

@tC .u � r/! D ��!;

and we can see that the vorticity in the fluid is transported by two agents: convectionand diffusion, just as the temperature in the system (2.44)–(2.46).

For inviscid fluids (� D 0) the vorticity field has very important properties thatallow us to imagine behavior of complicated turbulent flows [83]. In this case,vorticity is a local variable which means that we can isolate a patch of vorticity andobserve how it is transported along the velocity field trajectories with a finite speed.For two-dimensional flows this is evident as then the vorticity equation reduces to

@!

@tC .u � r/! D 0:

For more information, cf. [166].

Exercise 2.10. Vorticity has nothing in common with rotation of the fluid as awhole. Calculate the vorticity of the flows: (a) u.x1; x2; x3/ D .u1.x2/; 0; 0/ and(b) u.r; �; z/ D .0; k=r; 0/ for r > 0.


2.6 Thermodynamics

Equations of State From the point of view of thermodynamics the state of ahomogeneous fluid can be described by some definite relations among a numberof certain state variables, the most important being the volume V .V D 1=�/, theentropy S, the internal energy E, the pressure p, and the absolute temperature ,cf. [212].

In such a description one may start with a relation of the form (cf. [212])

E D E.S;V/ (Gibbs relation) (2.48)

and define p and by

p D �@E

@V; D @E

@S; (2.49)

with p; > 0 by assumption. In this case, taking the total differential in (2.48) andusing (2.49), we obtain

dE D dS � p dV or dE D dS � p1

�2d� : (2.50)

A simple phase system is said to undergo a differentiable process if its state variablesare differentiable functions of time: V D V.t/, S D S.t/, etc. Assuming such adependence one usually assumes, together with (2.50), that

DE

DtD

DS

Dt� p

DV

Dt

or

DE

DtD

DS

Dt� p

1

�2D�

Dt: (2.51)

Relation (2.51) makes it possible to write a definite form of the balance of entropywhen we know the laws of conservation of mass and internal energy. We shall usethis relation in the sequel.

Second Law of Thermodynamics and Constraints on Viscosity CoefficientsConsider the law of conservation of energy (2.32)

�DE

DtD r � .kr/ � p div u C �˚ ; (2.52)

where Fourier’s law is assumed, and �˚ is given by (2.33). We see that the internalenergy increases with the influx of heat transfer, compression, and the viscousdissipation.

2.6 Thermodynamics 27

From the law of conservation of mass

D�

DtC � div u D 0;

we have

div u D �1�

D�

Dt: (2.53)

Substituting (2.53) into (2.52) we can write

�DE

Dt� p

�

D�

DtD r � .kr/C �˚ : (2.54)

Now we shall transform the left-hand side of (2.54). From (2.51) we have

�DS

DtD �

DE

DtC p

�

D�

Dt: (2.55)

Thus

�DS

DtD 1

r � .kr/C 1

�˚ ; (2.56)

so thatZ

˝.t/�

DS

Dtdx D d

dt

Z

˝.t/�S dx (2.57)

DZ

˝.t/

�1

r � .kr/C 1

�˚

�

dx

DZ

˝.t/

�

r ��

k

r�

C k

2jr j2 C 1

�˚

�

dx :

In the end we obtain the following integral form of the balance of entropy

Z

˝.t/�

DS

Dtdx D

Z

@˝.t/

k

@

@ndS C

Z

˝.t/

�k

2jr j2 C 1

�˚

�

dx: (2.58)

Equation (2.58) with ˚ as in (2.33) and with � 0, 3� C 2 � 0 (cf. [212])is consistent with the second law of thermodynamics, which says that the rate ofincrease of entropy is not less than the heat transfer into the material volume. Thefirst integral on the right-hand side of (2.58) is just the heat transferred into thematerial volume divided by the temperature


Z

@˝.t/

k

@

@ndS D

Z

@˝.t/

�q � n

dS ;

and the second integral on the right-hand side of (2.58) is nonnegative.

Remark 2.3. For a perfect fluid, with no heat conductivity and viscosity, Eq. (2.56)becomes

DS

DtD 0 ;

so that the entropy is conserved.

Remark 2.4. If we assume that the Fourier law holds, the second law of thermody-namics demands

k

2jr j2 C 1

�˚ � 0 ; (2.59)

where �˚ is given by (2.33). The inequality (2.59) gives us the restrictions on thecoefficients in the constitutive equation (2.24). Without assuming the Fourier law,the second law of thermodynamics states that

� 1

q � r C �˚ � 0 :

This law will be satisfied if q � r � 0 (i.e., the heat flux is not directed againstthe temperature gradient) and ˚ � 0 (i.e., deformation always absorbs energyconverting it to heat).

Exercise 2.11. Prove that inequality (2.59), with ˚ given in (2.33), yields thefollowing constraints on the coefficients

k � 0; 3�C 2 � 0; � 0 :

Hint. The left-hand side of inequality (2.59) must be nonnegative for an arbitrarychoice of ;i and Dij. It is clear that this implies k � 0. The nonnegativity of theterms containing Dij is equivalent to the condition

3�C 2 � 0 and � 0 :

2.7 Similarity of Flows and Nondimensional Variables

Let us consider two initial boundary value problems

�Du

DtD �rp C �u; div u D 0 (2.60)

2.7 Similarity of Flows and Nondimensional Variables 29

in ˝ D Œ0;L�n, u D u.x; t/, p D p.x; t/, with initial condition u.x; 0/ D u0.x/, and

�0 Du0

Dt0D �rp0 C 0�u0; div u0 D 0 (2.61)

in ˝ 0 D Œ0;L0�n, u0 D u0.x0; t0/, p0 D p0.x0; t0/, with initial condition u0.x0; 0/ Du00.x

0/, respectively, and with periodic boundary conditions, n D 2 or 3.Changing dependent and independent variables in (2.60) as follows:

u?.x?; t?/ D u.x; t/

U; p?.x?; t?/ D p.x; t/ � p0

�U2(2.62)

and

x? D x

L; t? D U

Lt; (2.63)

we obtain the system

Du?

Dt?D �rp? C 1

Re�u?; div u? D 0 (2.64)

in ˝? D Œ0; 1�n, u? D u?.x?; t?/, p? D p?.x?; t?/, with initial condition u?.x?; 0/ Du0.x?/ D u0.Lx?/=U, with periodic boundary condition, and with

Re D �LU

:

In the case of problem (2.60) the symbols �;L;U; denote dimensional charac-teristic quantities of the flow, describing the density of the fluid (�), characteristiclinear dimension of the domain of the flow (L), some characteristic velocity (U),and viscosity (). For U we can take, for example, the square root of the averageinitial kinetic energy in˝, see [88] for more details. According to (2.62) and (2.63),in (2.64) all dependent and independent variables are nondimensional, together withthe Reynolds number Re. The characteristic quantities of the flow in (2.64) are equalto one, with the exception of viscosity which equals 1=Re.

In the same way we can reduce system (2.61) to system (2.64), now with theReynolds number

Re0 D �0L0U0

0 :

Assume now that Re D Re0. Then both systems (2.60) and (2.61) (including theboundary conditions) are represented, in nondimensional variables, by the samesystem (2.64). We call systems (2.60) and (2.61) dynamically similar. Solvingsystem (2.64) we generate solutions to an infinite three-parameter family of systemshaving the same Reynolds number.


The physical interpretation of the Reynolds number is that it describes therelation between the magnitudes of the inertial term .u � r/u and the viscous term��u (� D =�) in the equation of motion, according to

inertial term

viscous termD O

� j.u � r/ujj��uj

�

D O

�U.U=L/

�.U=L2/

�

D O.Re/;

or, saying it differently, the Reynolds number can be interpreted as a ratio of thestrength of inertial forces to viscous forces.

Exercise 2.12. Let u?; p? solve system (2.64). Show that

u.x; t/ D Uu?�

x

L;

U

Lt

�

; p.x; t/ D �U2p?�

x

L;

U

Lt

�

C p0

solve system (2.60).

Exercise 2.13. Let � D �0 and D 0, and let u0; p0 solve system (2.61). Showthat

u.x; t/ D U

U0 u0�

L0

Lx;

L0ULU0 t

�

;

p.x; t/ D U2

U02

�

p0�

L0

Lx;

L0ULU0 t

�

� p00

�

C p0

solve system (2.60). Here arbitrary constants p0; p00 may represent typical values of

pressures in the considered problems for systems (2.60) and (2.61), respectively,cf. (2.62).

Exercise 2.14. Let u0; p0 solve the system of equations

�0 Du0

DtD �rp0 C �u0 ; div u0 D 0 (2.65)

in the whole space. Show that

u.x; t/ D lu0.lx; l2t/; p.x; t/ D l2.p0.lx; l2t/ � p00/C p0;

where l > 0, also solve this system of equations. Thus having one (nontrivial)solution of system (2.65) we can generate in principle an infinite number of differentsolutions of the same system.

Consider the transformations,

.x; t; u; p/ !�

lx; l2t;1

lu;1

l2p

�

; l > 0; (2.66)

2.8 Examples of Simple Exact Solutions 31

or

x0 D lx ;

t0 D l2t ;

u0 D 1

lu ;

p0 D 1

l2p:

Show that transformations (2.66) form a group. This group is called a one parametersymmetry group of scalings of system (2.65).

A solution .u0; p0/ is called self-similar if u.x; t/ D u0.x0; t0/; p.x; t/ D p0.x0; t0/,cf. [17].

Exercise 2.15. Let u0; p0 solve the system of equations

�0 Du0

DtD �rp0 C �u0; div u0 D 0

in the whole space Rn. Show that

u.x; t/ D u0.x � ct; t/C c;

p.x; t/ D p0.x � ct; t/;

where c is any vector in Rn, also solve this system of equations (Galilean

invariance), and

u.x; t/ D Qtu0.Qx; t/;

p.x; t/ D p0.Qx; t/;

where Q is any rotation matrix (Qt D Q�1) also solve this system of equations(rotation symmetry).

2.8 Examples of Simple Exact Solutions

The Flow Due to an Impulsively Moved Plain Boundary Let the domain of theflow be half-space

f.x; y; z/ 2 R3 W y > 0g;

and assume that at times t < 0 the flow is at rest, and that at time t D 0 the planey D 0 is suddenly jerked into motion in the x-direction, with a constant velocity U.


Due to the geometrical symmetry of the problem and under additional “intuitivelyapparent” preconceptions, we assume that the flow is of the form u D .u.y; t/; 0; 0/.Substituting u of the above form into the Navier–Stokes equations (2.42) we obtain

@u

@tD �1

�

@p

@xC �

@2u

@y2; (2.67)

@p

@yD @p

@zD 0: (2.68)

As u depends only on y, from (2.67) together with (2.68) it follows that @p@x depends

only on t. Assume additionally that @p@x D 0 and that the flow is at rest for y D 1.

Then the above system together with the initial and boundary conditions reads

@u

@tD �

@2u

@y2; (2.69)

with

u.y; 0/ D 0 for y > 0 ; (2.70)

u.0; t/ D U for t > 0 ; (2.71)

u.1; t/ D 0 for t > 0: (2.72)

We say that a solution u D u.y; t/ of the initial and boundary value problem (2.69)–(2.72) is invariant or self-similar with respect to the one-parameter symmetry groupof scalings

y0 D ˛y; t0 D ˛2t; u0 D u; ˛ > 0; (2.73)

if the function u0 D u0.y0; t0/ D u.y; t/ is a solution of the problem

@u0

@t0D �

@2u0

@y02 ;

with

u0.y0; 0/ D 0 for y0 > 0 ;

u0.0; t0/ D U for t0 > 0 ;

u.1; t0/ D 0 for t0 > 0:

Observe that the transformation of the independent variables does not change thespace-time domain.


The relation u0.y0; t0/ D u.y; t/ satisfied by the invariant solution implies thatthe number of its independent variables can be reduced by one [17]. Together withthe dimensional homogeneity principle [33] stating that physical phenomena mustbe described by laws that do not depend on the unit of measure applied to thedimensions of the variables that describe the phenomena, we arrive at the followingform of the solution

u.y; t/ D f .�/; � D yp�t: (2.74)

Notice that the new variable �, called similarity variable is nondimensional and isinvariant with respect to the symmetry group of scalings y0 D ˛y, t0 D ˛2t.

Exercise 2.16. Assume that problem (2.69)–(2.72) is invariant with respect to adilation group

y0 D eay; t0 D ebt; u0 D ecu;

for some a; b; c, that is if u.x; t/ is a solution of the problem then u0.x0; t0/ is also asolution of the same problem. Then we get (2.73).

Observe, that if a problem is uniquely solvable and invariant with respect to asymmetry group of scalings x ! x0; t ! t0; u ! u0 then its unique solution isalso invariant.

Substituting function of the form (2.74) to problem (2.69)–(2.72) we obtain

f 00 C 1

2�f 0 D 0;

with

f .0/ D U; f .1/ D 0:

Integrating this equations we obtain

u.y; t/ D f .�/ D U

�

1 � 1p�

Z �

0

e�s24 ds

�

; (2.75)

or, in the nondimensional form

u.y; t/

UD 1 � 1p

�

Z �

0

e�s24 ds:

The example of the velocity profile given by (2.75) is depicted in Fig. 2.1.


00

0,5

1,5

2,5

3,5

1

2

3

=

0,1 0,2 0,3 0,4 0,5 0,6 0,7

u(h)

h

U

y

0,8 0,9 1

vt

Fig. 2.1 Velocity profile for the flow in the half-space due to the impulsively moved boundary. Inthe plot we have U D 1. The variable on the horizontal axis is the velocity u, and on the verticalaxis � D y

p

�t

Exercise 2.17. Check that the pair .u; p/ D .u..y; t/; 0; 0/;C/ where C is aconstant and u is the function given in (2.75) is a solution of the Navier–Stokesequations (2.42) and (2.43) with initial and boundary conditions (2.70)–(2.72).Observe that we have found only one of, perhaps, many other solutions of theproblem. We cannot claim that the solution is unique in u.

From (2.75) it follows that the velocity of the flow is a nonlocal variable, that is,it spreads information through the region of the flow with infinite speed. Althoughat negative times the flow was at rest, at any, however small, positive time t andany, however large, y > 0, we have u.y; t/ > 0. The same observation concerns thevorticity vector (the third and only nonzero component of which is ! D � @u

@y ) asboth functions u and ! satisfy the diffusion equation.

The Poiseuille Flow Consider a stationary flow u D .u.y/; 0; 0/ in the domain

f.x; y; z/ 2 R3 W 0 < y < hg;

and with homogeneous boundary conditions u D 0 at y D 0 and y D h. Substitutingu of the above form to the Navier–Stokes equations we obtain the equations ofmotion

�@2u

@y2D 1

�

@p

@x;

@p

@yD @p

@zD 0 ;

from which it follows that

�@2u

@y2D 1

�

@p

@xD C (2.76)


0,014

0,012

0,008

0,006

0,004

0,002

00 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

0,01 u(y)

y

h = 1

Fig. 2.2 Parabolic velocity profile for the Poiseuille flow. The constants were chosen as h D 1,� D 10, and C D �1. Velocity u.y/ is on the vertical axis, and the space variable y is on thehorizontal axis

for some constant C. Taking into account the boundary conditions we obtain

u.y/ D C

2�y.y � h/ ;

where

C D 1

�

@p

@x:

From the last equation we have p.x/ D C�x C B, for an arbitrary constant B. Letus take B D 0. We can see that there is an infinite number of solutions .u; p/ of theform

u.y/ D C

2�y.y � h/; p.x/ D C�x: (2.77)

We call them the Poiseuille flows. For C D 0 the flow is just at rest, both velocity andpressure equal zero. For C < 0 the fluid flows towards the increasing x-direction,u > 0, driven by the force due to the pressure gradient (as C < 0 the pressuredecreases with increasing x). The example of the velocity profile given by (2.77) fornegative value of C is depicted in Fig. 2.2.


u(y)

h = 1

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1y

2

1,8

1,6

1,4

1,2

0,8

0,6

0,4

0,2

0

1

U

Fig. 2.3 Linear velocity profile for the Couette flow. The constants were chosen as h D 1 andC D 2. Velocity u.y/ is on the vertical axis, and the space variable y is on the horizontal axis

The Couette Flow Let us consider a flow u D .u.y/; 0; 0/ in the same region asin the preceding example, this time, however, with different boundary conditions.We pose u D 0 at y D 0 and u D U at y D h. As the driving force now comesfrom the motion of the upper boundary y D h, we pose C D 0 which correspondsto p D const in the whole domain. Solving Eq. (2.76) in u with C D 0 and with thenew boundary conditions we obtain the solution

u.y/ D U

hy; p D const: (2.78)

of the problem, with a uniform pressure distribution. We call this flow the Couetteflow. The velocity profile given by (2.78) for negative value of C is depicted inFig. 2.3.

2.9 Comments and Bibliographical Notes

Remarks on Modeling Exact solutions serve, among other things, to test a giventheory of hydrodynamics. By theory we mean here a set of governing equations(conservation laws) together with additional conditions (boundary or initial andboundary conditions), that is, a boundary or initial and boundary value problem. Inthe following chapters we shall prove several results about such theories of classicalhydrodynamics.

A particular theory of hydrodynamics is well-set, that is, that it possesses aunique solution depending continuously upon the data (boundary data, externalforces, etc.).

2.9 Comments and Bibliographical Notes 37

As was observed by Birkhoff in [16], “until it has been shown that the boundaryvalue problem is well-set, one cannot conclude that its equations are erroneous.”The latter means “false” according to the following definition, cf. [16]: A theoryof rational hydrodynamics will be called incomplete if its conditions do notuniquely determine the flow, overdetermined if its conditions are mathematicallyincompatible, false if it is well-set but gives grossly incorrect predictions.

Since from a particular theory in its whole generality, it is in most cases quiteimpossible to predict qualitative properties of the flow (it is sometimes easier,however, to compare various flows among themselves), we are forced to “simplify,”whatever that means. In this connection, there are very often exact solutions ofspecial problems that serve to test whether a given theory gives correct predictionsand consequently is not a false theory.

We enumerate a number of such simplifications (which we have already used inSect. 2.8):

1. linearization of governing equations,2. setting a special form of boundary conditions,3. assumption of some symmetry in space (which reduces the number of spatial

dimensions),4. assumption that the problem is time-independent,5. assumption that acting forces are potential or absent.

After suitable simplifications have been done, the original problem reduces tothe one that can be (luckily) solved explicitly by solving, e.g., a linear systemof ordinary differential equations with constant coefficients. Having the exactsolutions, we try to compare them with experiments.

When simplifying, however, one has to be very careful in order not to over-simplify and obtain paradoxes, for example. It is well known (cf. [16]) thatthe various plausible intuitive assumptions we make implicitly when simplifyingmay be fallacious. Following [16], we state the assumptions that are especiallysuggestive:

1. intuition suffices for determining which physical variables require consideration,2. small causes produce small effects, and infinitesimal causes produce infinitesimal

effects,3. symmetric causes produce effects with the same symmetry,4. the flow topology can be guessed by intuition,5. the processes of analysis can be used freely: the functions of rational hydrody-

namics can be freely integrated, differentiated, expanded in series, integrals, etc.,6. mathematical problems suggested by intuitive physical ideas are well-set.

In Chaps. 14–16 we consider problems whose solutions might not be unique.For more information about classical hydrodynamics we refer the reader to

[1, 14, 83, 149].For a history of hydrodynamics and the derivation and development of the

Navier–Stokes equations, in particular, see [81].

3Mathematical Preliminaries

In this chapter we introduce the basic preliminary mathematical tools to study theNavier–Stokes equations, including results from linear and nonlinear functionalanalysis as well as the theory of function spaces. We present, in particular, someof the most frequently used in the sequel embedding theorems and differentialinequalities.

3.1 Theorems from Functional Analysis

Theorems from Linear Functional Analysis The core of this subsection is theLax–Milgram lemma (which we state for separable Hilbert spaces) as well asits proof using the Galerkin method. This method will be very useful in oursubsequent considerations, in the proofs of the existence of solutions of both linearand nonlinear problems.

Theorem 3.1. In a reflexive Banach space every closed ball is weakly compact.

Proof. A proof can be found, e.g., in [106], for separable spaces in [118]. utIn particular, let fxng be a sequence in a reflexive Banach space B such that

kxnkB � M for some M > 0. Then there exists a subsequence fxg of the sequencefxng and an element x in B with kxkB � M such that fxg converges weakly to x inB; that is, for every linear and continuous functional L on B,

lim!1 L.x/ D L.x/ :


39

40 3 Mathematical Preliminaries

Theorem 3.2 (Lax–Milgram Lemma for Separable Spaces). Let H be a separa-ble Hilbert space with a norm k�k, L 2 H0 a linear functional on H, a.u; v/ a bilinearand continuous form on H � H, coercive, that is, such that for some ˛ > 0 and forall u 2 H

a.u; u/ � ˛kuk2 :Then there exists a unique element u 2 H such that

a.u; v/ D L.v/ for all v 2 H : (3.1)

Proof. We use the Galerkin method, to which we shall refer in the followingchapters. It consists of proving the existence of elements um 2 Hm such that

a.um; v/ D L.v/ for all v 2 Hm ; (3.2)

where Hm are finite dimensional subspaces of H such that H1 H2 � � � Hm � � � and

S1mD1 Hn is a dense subset of H, and then by passing to the limit with n to

obtain (3.1).Let !1, !2, !3, : : : be a basis of H and Hm D spanf!1; : : : ; !mg, m D

1; 2; 3; : : : .

1. We shall show that for each positive integer m there exists um 2 Hm such that (3.2)holds. Let

um DmX

kD1�k!k :

Then (3.2) is equivalent to the system of linear equations

mX

kD1�ka.!k; !l/ D L.!l/ ; l D 1; 2; : : : ;m :

This system has a unique solution .�1; : : : ; �m/ for every right-hand side ifand only if the matrix fa.!k; !l/gk;l�m is nonsingular. We shall show that thehomogeneous system

mX

kD1�ka.!k; !l/ D 0 ; l D 1; 2; : : : ;m ;

has a unique solution � D .0; 0; : : : ; 0/. We multiply the lth equation by �k andadd the equations to get

mX

lD1

mX

kD1�l�ka.!k; !l/ D a

mX

kD1�k!k;

mX

lD1�l!l

!

D a.um; um/ D 0 :

3.1 Theorems from Functional Analysis 41

From the coerciveness of the form a.�; �/ we obtain um D 0. As the vectors!1; : : : ; !m are linearly independent, we conclude that �1 D �2 D � � � D �m D 0.Hence the matrix fa.!k; !l/gk;l�m is nonsingular, and for each positive integer mthere exists an approximate solution um 2 Hm.

2. Convergence of the sequence fumg. We have

˛kumk2 � a.um; um/ D L.um/ � kLkH0kumk ;whence

kumk � 1

˛kLkH0

for each positive integer m. From Theorem 3.1 it follows that there exists a sub-sequence fug of the sequence of approximate solutions fumg and an elementu 2 H such that u!u weakly in H. For � j we have

a.u; v/ D L.v/ for all v 2 Hj H ;

so that

lim!1 a.u; v/ D a.u; v/ D L.v/ for v 2

1[

jD1Hj :

AsS1

jD1 Hj is a dense subset of H, we conclude from the continuity of a.�; �/ andL.�/ that (3.1) holds for every v 2 H.

3. Uniqueness of u. Let us suppose that we have two different elements u1 andu2 such that a.u1; v/ D L.v/ and a.u2; v/ D L.v/ for all v 2 H. Thusa.u1 � u2; v/ D 0, and taking v D u1 � u2 we obtain

0 D a.u1 � u2; u1 � u2/ � ˛ku1 � u2k2 ;whence u1 D u2. We have come to a contradiction, which proves the uniqueness.

utRemark 3.1. If L is a linear and continuous functional on the Banach space B, wewill write L 2 B0, the dual space of B, and we will use the notation L.v/ D hL; vifor v 2 B.

Exercise 3.1. Prove that the whole sequence of approximate solutions convergesweakly to u.

As a corollary we obtain the following

Theorem 3.3 (Riesz–Fréchet Theorem for Separable Spaces). Let H be a sep-arable Hilbert space with scalar product .�; �/ and norm k � k and let L 2 H0 bea linear and continuous functional on H. Then there exists a unique element u 2 Hsuch that

.u; v/ D L.v/ for each v 2 H

and kuk D kLkH0 .


Remark 3.2. In the above two theorems the separability of the space H is notessential; for the proofs see [110].

Fixed Point Theorems Fixed point theorems are the basic tool that we shall use.We begin with the Banach contraction principle, the only theorem in this subsectionthat guarantees the uniqueness of the fixed point. The existence of a unique fixedpoint is a strong condition, however, and in most cases we shall make use of theSchauder or the Leray–Schauder theorems. They both follow from the Brouwerfixed point theorem.

Theorem 3.4 (Banach Contraction Principle). Let T be an operator defined onthe Banach space X with values in X. Assume that T is a contraction, i.e., that anumber ˛, 0 � ˛ < 1, exists such that for all pairs of elements u; v 2 X we have

kTu � TvkX � ˛ku � vkX :

Then there exists a unique element u 2 X such that Tu D u.

Proof. Let u0 be an arbitrary element of the space X. Then the sequenceu0; u1; u2; : : :, where for n � 1, un are defined recursively by un D Tun�1, convergesin X to the fixed point of the operator T . utExercise 3.2. Provide the details of the proof and show that

kun � ukX � ˛n�1

1 � ˛ ku1 � u0kX;

for n D 1; 2; : : : .

Theorem 3.5 (Brouwer). Let K be a nonempty, convex, and compact set in Rn. If

T W K!K is a continuous mapping, then it has at least one fixed point, that is, thereexists u0 2 K such that T.u0/ D u0.

Proof. For a proof see, e.g., [118, 158]. utTheorem 3.6 (Schauder). Let K be a closed, nonempty, bounded, and convexset in a Banach space X. Let T be a completely continuous (that is, compact andcontinuous) operator defined on K such that

T.K/ K :

Then there exists at least one element u0 2 K such that T.u0/ D u0.

Remark 3.3. If X D Rn, then the Schauder theorem reduces to the Brouwer

theorem.

Proof. The closure of the set T.K/ is compact. For each n 2 N there existsa sequence x1, x2, : : :, xr.n/ in T.K/ such that

min1�i�r.n/

kx � xikX < " D 1

nfor each x 2 T.K/ :

3.1 Theorems from Functional Analysis 43

Let Xn be the linear hull of the set fx1; : : : ; xr.n/g, that is, the set of all points of theform �1x1 C : : :C �r.n/xr.n/, where �1, : : :, �r.n/ are arbitrary real numbers. We willwrite Xn D span fx1; : : : ; xr.n/g. Set Kn D K \ Xn. The set Kn is convex, bounded,closed, nonempty, and lies in a finite dimensional subspace of X. We define the mapTn W K!Kn by

Tn.x/ D F 1nT.x/ ;

where

F".x/ DPr.n/

iD1 mi.x/xiPr.n/

iD1 mi.x/;

mi.x/ D8<

:

" � kx � xikX when kx � xikX < " ;

0 when kx � xikX � " :

We have F".K/ Kn. The map Tn is continuous, as the functions mi arecontinuous. For an arbitrary x 2 K we have

kTx � TnxkX D��

Pmi.Tx/Tx �P

mi.Tx/xiP

mi.Tx/

��

X

�P

mi.Tx/kTx � xikXP

mi.Tx/<1

n: (3.3)

Thus, we have approximated in a uniform way the compact operator T by a finitedimensional operator Tn.

Since Tn W Kn!Kn satisfies the hypothesis of the Brouwer fixed point theorem,there exist Nxn such that Tn Nxn D Nxn. As T is compact, the sequence fNxng hasa convergent subsequence Nxnk !Nx. From the continuity of T it follows that T Nxnk !T Nx.From (3.3) we conclude that T Nxnk �Tnk Nxnk !0. Thus T Nx D Nx, and T has a fixed pointin K. utExercise 3.3. Prove, by giving simple examples, that all assumptions in theSchauder theorem are necessary.

Theorem 3.7 (Leray–Schauder). Let T be a completely continuous mapping ofa Banach space X into itself, and suppose that there exists a constant M such that

kxkX < M (3.4)

for all x 2 X and � 2 Œ0; 1� satisfying x D �Tx. Then T has a fixed point.


Proof (cf. [110]). We may assume without loss of generality that M D 1. Let usdefine the map T� by

T�x D

8ˆ<

ˆ:

Tx if kTxkX � 1 ;

Tx

kTxkXif kTxkX > 1 :

Then, T� is a continuous mapping of the closed unit ball B.1/ D fx 2 X W kxkX � 1ginto itself. Since the set TB.1/ is precompact, the same is true for T�B.1/.

From the Schauder fixed point theorem it follows that T� has a fixed point x. Weshall show that x is a fixed point of T . In fact, if kTxkX > 1, then x D T�x D �Txwith � D 1

kTxkXand kxkX D kT�xkX D 1, which contradicts (3.4). ut

In Chaps. 5 and 15 we will use the following Kakutani–Fan–Glicksberg fixed pointtheorem (see [3], Corollary 17.55) to prove the existence of weak solutions.

Theorem 3.8 (Kakutani–Fan–Glicksberg). Let S X be nonempty, compact,and convex set, where X is a locally convex Hausdorff topological vector space andlet the multifunction ' W S ! 2S have nonempty convex values and closed graph.Then the set of fixed point of ' (i.e., fx 2 S W x 2 '.x/g) is nonempty and compact.

3.2 Sobolev Spaces and Distributions

We assume that ˝ is a nonempty open subset of Rn, n a positive integer.

Definition 3.1. By Lp.˝/, 1 � p < 1, we denote a linear space of real-valuedfunctions f defined on ˝ such that jf jp is integrable on ˝ with respect to Lebesguemeasure. The functional

f 7! kf kLp.˝/ D�Z

˝

jf jp dx

�1=p

is a norm in Lp.˝/ (provided that we identify functions that are equal to each otheralmost everywhere in ˝).

The space Lp.˝/ is a Banach space. For p D 2 it is a Hilbert space with the scalarproduct

.f ; g/ DZ

˝

fg dx:

We have

Theorem 3.9. Let ffng be a Cauchy sequence in Lp.˝/, that is,

limm;n!1 kfm � fnkLp.˝/ D 0:

3.2 Sobolev Spaces and Distributions 45

Then there exists f 2 Lp.˝/ and a subsequence ff�g of the sequence ffng such that

lim�!1 kf� � f kLp.˝/ D 0

(completeness of Lp.˝/), and f�.x/ ! f .x/ for almost all x 2 ˝. Moreover thereexists h 2 Lp.˝/ with h.x/ � 0 a.e. in ˝ such that jf�.x/j � h.x/ for a.e. x 2 ˝.

Definition 3.2. By L1.˝/ we denote the set of measurable real functions f on ˝for which

ess supfjf .x/j W x 2 ˝g � inffk > 0 W .fx 2 ˝ W jf .x/j > kg/ D 0g < 1:

The linear space L1.˝/ with norm

kf kL1.˝/ D ess supfjf .x/j W x 2 ˝g

is a Banach space.We say that a real-valued function f on ˝ is locally integrable on ˝, and write

f 2 L1loc.˝/, if for every compact K ˝, f is integrable on K.Similarly, we define spaces Lp

loc.˝/, for 1 < p < 1.For f defined on ˝ we define the support of f as the closure of the set

fx 2 ˝ W f .x/ ¤ 0g, and denote it by supp f .Let ˛ D .˛1; : : : ; ˛n/ be a multi-index the coordinates of which are nonnegative

integers. We write j˛j D ˛1 C � � � C ˛n, and

D˛ D @j˛j

@˛1x1 � � � @˛n xn:

We say that f 2 Ck.˝/ (resp. f 2 Ck.˝/) if there exist partial derivatives D˛f forall ˛ with j˛j � k that are continuous in ˝ (resp. ˝).

Moreover,

C1.˝/ D1\

kD1Ck.˝/:

By C10 .˝/ we denote the set of f 2 C1.˝/ for which supp f ˝. If ˝ is

bounded, supp f is a compact subset of ˝.

Lemma 3.1 (cf. [2, 142]). The set C10 .˝/ is dense in Lp.˝/ for 1 � p < 1.

Lemma 3.2 (Du Bois–Reymond, cf. [2, 182]). Let f 2 L1loc.˝/, and let

Z

˝

f' dx D 0

for each ' 2 C10 .˝/. Then f D 0 almost everywhere in ˝.


Definition 3.3 (Generalized Derivative). Let ˛ D .˛1; : : : ; ˛n/ be a multi-indexthe coordinates of which are nonnegative integers. We call a function f ˛ 2 L1loc.˝/

the ˛th generalized (or weak) derivative in˝ of f 2 L1loc.˝/ if for each ' 2 C10 .˝/

Z

˝

fD˛' dx D .�1/j˛jZ

˝

f ˛' dx:

Exercise 3.4. Using Lemma 3.2 show that if f 2 Cj˛j.˝/, then f ˛ D D˛f ; thatis, there exists the generalized derivative f ˛ , and it equals the usual classical partialderivative D˛f .

Exercise 3.5. Prove that generalized partial derivatives are uniquely determined(provided that they exist).

Exercise 3.6. Let ˝ be the interval .0; 1/ on the real line, and let f .x/ D jxj. Provethat f has the generalized first derivative f 0, and f 0.x/ D sgn x almost everywherein ˝.

In the sequel we shall write D˛f instead of f ˛ for generalized derivatives.

Definition 3.4 (Sobolev Space Wm;p.˝/). Let m be a positive integer, 1 � p < 1.By Wm;p.˝/ we denote a linear subspace of elements f in Lp.˝/ for whichgeneralized partial derivatives D˛f exist for all j˛j � m and belong to Lp.˝/, withnorm

kf kWm;p.˝/ D0

@X

j˛j�m

kD˛f kpLp.˝/

1

A

1=p

: (3.5)

Lemma 3.3 (cf. [2, 142]). The space Wm;p.˝/ is a Banach space. For 1 < p < 1Wm;p.˝/ is reflexive.

Exercise 3.7. Prove that the space Wm;p.˝/ is complete.

Hint. Use Theorem 3.9, Lemma 3.2, and Definition 3.4.

Lemma 3.4. For 1 < p < 1 every bounded sequence in Wm;p.˝/ contains aweakly convergent subsequence.

Proof. The space Wm;p.˝/ is reflexive. utDefinition 3.5. By Wm;p

0 .˝/ we denote the closure of the set C10 .˝/ in the norm

of Wm;p.˝/.

The spaces Wm;p.˝/ are also denoted by Wmp .˝/. The spaces Wm;2.˝/ and

Wm;20 .˝/, denoted also by Hm.˝/ and Hm

0 .˝/, respectively, are Hilbert spaces withscalar product

.f ; g/ DX

j˛j�m

Z

˝

D˛fD˛g dx:

3.2 Sobolev Spaces and Distributions 47

Definition 3.6. Let m be a positive integer, 1 < p < 1, and 1=p C 1=q D 1.By W�m;q.˝/ we denote the space of linear continuous functionals on the spaceWm;p0 .˝/.

The spaces W�m;2.˝/ are also denoted by H�m.˝/.Now we shall define distributions and distributional derivatives.Let us introduce in the set C1

0 .˝/ the following notion of convergence.

Definition 3.7. We say that a sequence f'ng C10 .˝/ converges to zero if there

exists a compact K ˝ such that

(i) supp 'n K for each 'n,(ii) lim

n!1D˛'n D 0 for each multi-index ˛, uniformly on ˝.

We denote by D.˝/ the set C10 .˝/ together with the convergence introduced

above.

Definition 3.8. We call the map T W D.˝/ ! R a distribution on ˝ if it is linear,that is,

T.˛' C ˇ / D ˛T.'/C ˇT. /

for all ˛; ˇ 2 R and '; 2 D.˝/, and if

limn!1 T.'n/ D 0

for each sequence f'ng D.˝/ that converges to zero in D.˝/.

We denote by D 0.˝/ the set of all distributions on˝ and write hT; 'i instead T.'/.For f 2 L1loc.˝/, the map on D.˝/ defined by

hTf ; 'i DZ

˝

f' dx

is a distribution. By Lemma 3.2 we can identify it with f , and we write

hf ; 'i DZ

˝

f' dx :

Distributions that can be represented by locally integrable functions we call regular.

An example of a distribution that is not regular is the map

ı.x0/ W D.˝/ 3 ' ! ıx0 .'/ D '.x0/ 2 R;

for some x0 2 ˝, called the Dirac delta.


Definition 3.9 (Distributional Derivative). Let T 2 D 0.˝/. Then the map

@T

@xiW D.˝/ ! R;

i 2 f1; : : : ; ng; defined by@T

@xi; '

D �

T;@'

@xi

for each ' 2 D.˝/, is a distribution. Similarly, for an arbitrary multi-index ˛ wedefine D˛T 2 D 0.˝/ by

hD˛T; 'i D .�1/j˛jhT;D˛'ifor each ' 2 D.˝/.

For T 2 D 0.˝/, we call D˛T the ˛th distributional derivative of T .

Exercise 3.8. Prove that each distribution has all distributional derivatives of anyorder.

Exercise 3.9. Show that f 00.x/ D 2ı.x/ for f from Exercise 3.6 above. One canshow that there is no locally integrable function g on ˝ such that f 00 D g.

Definition 3.10. We say that an open and bounded subset ˝ Rn belongs to the

class Ck;˛ , 0 � ˛ � 1, k a nonnegative integer, if for every point x0 2 @˝ thereexists a ball B centered at x0 and a one-to-one mapping of B on D R

n such that

(i) .B \˝/ RnC D fx D .x1; : : : ; xn/ 2 R

n W xn > 0g ,(ii) .B \ @˝/ @RnC ,

(iii) 2 Ck;˛.B/, �1 2 Ck;˛.D/.

If the boundary of ˝ is smooth enough, say ˝ 2 C0;1, then the Sobolev spacesWm;p.˝/ defined as in Definition 3.4 above can be described alternatively as theclosure of the set Cm.˝/ in the norm (3.5), cf. [142]. From this new definitionof Wm;p.˝/ it is clear that the set Cm.˝/ is dense in Wm;p.˝/ (provided that theboundary of ˝ is smooth enough).

Still another description, equivalent to Definition 3.4, of Sobolev spaces can begiven in terms of distributions. We define the space Wm;p.˝/ as the linear subspaceof those f 2 Lp.˝/ for which all distributional derivatives D˛f , j˛j � m, belong toLp.˝/, with norm (3.5).

3.3 Some Embedding Theorems and Inequalities

We begin with three general results.

Theorem 3.10 ([110, Theorem 7.26], [185]). Let ˝ be an open and boundedsubset of Rn with Lipschitz boundary, that is, ˝ 2 C0;1. Then

3.3 Some Embedding Theorems and Inequalities 49

(i) if kp < n, then the space Wk;p.˝/ is continuously embedded in the spaceLp�

.˝/, p� D np=.n � kp/, and compactly embedded in Lq.˝/ for q < p�;(ii) if 0 � m < k�n=p < mC1, then the space Wk;p.˝/ is continuously embedded

in Cm;˛.˝/, ˛ D k�n=p�m, and compactly embedded in Cm;ˇ.˝/, for ˇ < ˛.

Lemma 3.5 (A Version of the Rellich Theorem, cf. [182]). Let ˝ be an openand bounded subset of Rn. Then the embedding of H1

0.˝/ in L2.˝/ is compact.

Theorem 3.11 ([105, Theorem 10.1], [235, p. 1034]). Let˝ be a bounded domainin R

n with Lipschitz boundary, and let u be any function in Wm;r.˝/ \ Lq.˝/,1 � r; q � 1. For any multi-index j, 0 � jjj < m, and for any number from theinterval jjj=m � � 1, set

1

pD jjj

mC

�1

r� m

n

�

C .1 � /1q:

If m � jjj � n=r is not a nonnegative integer, then

kDjukLp.˝/ � CkukWm;r.˝/kuk1�Lq.˝/: (3.6)

If m � jjj � n=r is a nonnegative integer, then inequality (3.6) holds for D jjj=m.The constant C depends only on ˝, r, q, m, j, .

Now we shall state some special cases of Theorems 3.10 and 3.11, useful in thefollowing chapters.

Let ˝ R3 be as in Theorem 3.10. Then

1. H1.˝/ is continuously embedded in L6.˝/,

kukL6.˝/ � CkukH1.˝/

(cf. Lemma 3.9 below).2. H1.˝/ is compactly embedded in L4.˝/.3. H2.˝/ is continuously embedded in C0;1=2.˝/, and C0;1=2.˝/ is compactly

embedded in C.˝/; in particular, we have

ess supx2˝

ju.x/j � CkukH2.˝/:

Exercise 3.10. Show that W2;3=2.˝/ is continuously embedded in W1;3.˝/ for˝ R

3 as in Theorem 3.10.

Let ˝ be a bounded domain in R3 with Lipschitz boundary. Then

kukL4.˝/ � Ckuk3=4H1.˝/

kuk1=4L2.˝/

[cf. inequality (3.10)].


Lemma 3.6 (Poincaré Inequality). Let˝ be an open and bounded subset of Rn, na positive integer, d D diam˝ D supfjx�yj W x; y 2 ˝g. Then, for each u 2 H1

0.˝/

and i 2 f1; : : : ; ng,

kukL2.˝/ � dp2

��@u

@xi

��

L2.˝/

: (3.7)

Proof. Let u 2 C10 .˝/, and fix i D 1. We have

u.x1; : : : ; xn/ DZ x1

�1@u

@t.t; x2; : : : ; xn/dt:

By the Schwarz inequality,

ju.x1; : : : ; xn/j �Z x1

x�

1

ˇˇˇˇ@u

@t.t; x2; : : : ; xn/

ˇˇˇˇ

2

dt .x1 � x�1 /

�Z C1

�1

ˇˇˇˇ@u

@x1.x/

ˇˇˇˇ

2

dx1 .x1 � x�1 / ;

where x�1 D infft W u.t; x2; : : : ; xn/ D 0; .t; x2; : : : ; xn/ 2 ˝g: Integrating with

respect to x1 we obtain

Z C1

�1ju.x/j2dx1 � d2

2

Z C1

�1

ˇˇˇˇ@u

@x1.x/

ˇˇˇˇ

2

dx1:

Now, integrating with respect to x2; : : : ; xn we obtain inequality (3.7) for i D 1 andu 2 C1

0 .˝/. Let u 2 H10.˝/. There exists a sequence fung C1

0 .˝/ convergingto u in H1

0.˝/. We prove (3.7) for u, passing to the limit in

kunkL2.˝/ � dp2

��@un

@xi

��

L2.˝/

;

for i 2 f1; : : : ; ng. utCorollary 3.1. Let ˝ R

n be an open and bounded set. Then the following twonorms are equivalent in H1

0.˝/:

kuk D0

@kuk2L2.˝/ CX

j˛jD1kD˛uk2L2.˝/

1

A

1=2

;

and

jjjujjj D0

@X

j˛jD1kD˛uk2L2.˝/

1

A

1=2

:


Lemma 3.7 (Ladyzhenskaya Inequality in Two Dimensions). Let˝ R2 be an

open and bounded set. Then for all u 2 H10.˝/

kukL4.˝/ � 21=4kuk1=2L2.˝/

kDuk1=2L2.˝/

: (3.8)

Proof (cf. [146]). We shall prove inequality (3.8) for smooth functions withcompact support in ˝, and the general case will follow immediately from thedensity of these functions in H1

0.˝/. Let u 2 C10 .˝/. For convenience we consider

u as defined on the whole space R2 and equal zero outside of ˝.

We have

u2.x1; x2/ D 2

Z xk

�1uuxk dxk for k D 1; 2 ;

so that

maxxk

u2.x1; x2/ � 2

Z 1

�1juuxk j dxk for k D 1; 2 ; (3.9)

and by the Schwarz inequality,

Z

R2

u4dx �Z C1

�1max

x2u2dx1

Z C1

�1max

x1u2dx2

� 4

Z

R2

juux2 jdxZ

R2

juux1 jdx

� 2

Z

R2

juj2dxZ

R2

jDuj2dx;

which proves the lemma for smooth functions with compact support in ˝. utLemma 3.8 (Ladyzhenskaya Inequality in Three Dimensions). Let ˝ R

3 bean open and bounded set. Then for all u 2 H1

0.˝/

kukL4.˝/ � p2kuk1=4

L2.˝/kDuk3=4

L2.˝/: (3.10)

Proof (cf. [146]). As in the above lemma, it suffices to prove inequality (3.10)for smooth functions with compact support. Let u 2 C1

0 .˝/. We have, by (3.8)and (3.9),

Z

R3

u4dx � 2

Z C1

�1

�Z

R2

u2dx1dx2

Z

R2

.u2x1 C u2x2 /dx1dx2

�

dx3

� 2maxx3

Z

R2

u2dx1dx2

Z

R3

jDuj2dx


� 4

Z

R3

juux3 jdxZ

R3

jDuj2dx

� 4

�Z

R3

juj2dx

�1=2 �Z

R3

jDuj2dx

�3=2

;

which gives inequality (3.10). utLemma 3.9 (cf. [146]). Let ˝ R

3 be an open and bounded set. Then for allu 2 H1

0.˝/,

kukL6.˝/ � 481=6kDukL2.˝/: (3.11)

Proof. Without loss of generality we can assume that u 2 C10 .˝/ and u � 0. We

have

I �Z

R3

u6dx DZ C1

�1

�Z

R2

u3u3dx2dx3

�

dx1

�Z C1

�1

�Z C1

�1max

x2u3dx3

Z C1

�1max

x3u3dx2

�

dx1

� 9

Z C1

�1

�Z

R2

ju2ux2 jdx2dx3

Z

R2

ju2ux3 jdx2dx3

�

dx1

� 9

Z

R1

( Z

R2

u4dx2dx3

�Z

R2

u2x2dx2dx3

�1=2 �Z

R2

u2x3dx2dx3

�1=2)

dx1:

Now we use the Schwarz inequality and proceed as follows:

I � 9maxx1

Z

R2

u4dx2dx3

�Z

R3

u2x2dx

�1=2 �Z

R3

u2x3dx

�1=2

� 36

Z

R3

ju3ux1 jdx

�Z

R3

ju2x2 jdx

�1=2 �Z

R3

ju2x3 jdx

�1=2

� 36p

I3Y

iD1

�Z

R3

ju2xijdx

�1=2

:


In the end,

pI � 36

3Y

iD1kuxikL2.˝/ D 36

(3Y

iD1kuxik2L2.˝/

) 1=2

� 36

8<

:3�3

3X

iD1kuxik2L2.˝/

!39=

;

1=2

(as .a1a2a3/1=3 � .a1 C a2 C a3/=3 for ai � 0), which gives inequality (3.11). utTheorem 3.12 (Hardy Inequality, cf. [112, 142]). Let a; b 2 R, a < b, u 2Lp.a; b/ with p > 1. Then

Z b

a

�1

x � a

Z x

aju.s/jds

�p

dx ��

p

p � 1�p Z b

aju.x/jpdx; (3.12)

and

Z b

a

�1

b � x

Z b

xju.s/jds

�p

dx ��

p

p � 1�p Z b

aju.x/jpdx: (3.13)

Proof. Let un.x/ D 0 for x 2 �a; a C 1

n

�and un.x/ D u.x/ for x 2 �

a C 1n ; b

�. We

prove the first inequality. Integration by parts and Hölder inequality give

Z b

a

�1

x � a

Z x

ajun.s/jds

�p

dx D 1

.b � a/p�1.1 � p/

�Z b

ajun.s/jds

�p

C p

p � 1Z b

a

1

.x � a/p�1

�Z x

ajun.s/jds

�.p�1/jun.x/jdx

� p

p � 1Z b

a

1

.x � a/p�1

�Z x

aju.s/jds

�p�1jun.x/jdx

� p

p � 1�Z b

a

1

.x � a/p

�Z x

ajun.s/jds

�p

dx

�.p�1/=p �Z b

ajun.x/jpdx

�1=p

:

Thus,

Z b

a

�1

x � a

Z x

ajun.s/jds

�p

dx ��

p

p � 1�p Z b

aju.x/jpdx (3.14)

for n 2 N. Applying the Fatou lemma we obtain (3.12). In a similar way we obtainthe second inequality.


Exercise 3.11. Deduce inequality (3.12) from inequality (3.14) and proveinequality (3.13).

Exercise 3.12. Prove that for v 2 W1;p.a; b/ with v.a/ D 0, p > 1, the followinginequality holds:

Z b

a

ˇˇˇˇv.x/

x � a

ˇˇˇˇ

p

dx ��

p

p � 1�p Z b

ajv0.x/jpdx:

3.4 Sobolev Spaces of Periodic Functions

We consider Sobolev spaces Hsp.Q/ of L-periodic functions on the m-dimensional

domain Q D Œ0;L�m. Let C1p .Q/ be the space of restrictions to Q of infinitely

differentiable functions which are L-periodic in each direction, that is, u.x C Lej/ Du.x/, j D 1; : : : ;m (by ej we denote the vectors of the canonical basis, ej D.0; : : : ; 1; : : : ; 0/, where 1 is on the jth coordinate).

Definition 3.11. For an arbitrary nonnegative integer s we define the Sobolev spaceHs

p.Q/ as the completion of C1p .Q/ in the norm

kukHsp.Q/ D

8<

:

X

0�j˛j�s

kD˛uk2L2.Q/

9=

;

1=2

:

We denote, ˛ D .˛1; : : : ; ˛m/ for nonnegative integers ˛1; : : : ; ˛m, j˛j D ˛1 C � � � C˛m, ˛Š D ˛1Š˛2Š � � �˛mŠ. For x D .x1; : : : ; xm/, x˛ D x˛11 x˛22 � � � x˛m

m , D˛ D @j˛j

@x˛ .In this notation the Taylor series in R

m can be written just as

f .x/ DX

j˛j�0

1

˛ŠD˛f .x0/.x � x0/

˛:

Characterization of Sobolev Spaces Hsp.Q/ by Using the Fourier Series Each

function u 2 C1p .Q/ has a representation

u.x/ DX

k2Zm

cke2� ik xL ;

where ck are complex numbers. We consider real functions, for which c�k D ck. Itis known that for functions from C1

p .Q/ the above series converges uniformly.We have

D˛u.x/ D�2� i

L

�j˛j X

k2Zm

ckk˛e2� ik xL :

3.4 Sobolev Spaces of Periodic Functions 55

From the Parseval identity

kD˛uk2L2.Q/ D Lm

�2�

L

�2j˛j X

k2Zm

jckj2k2˛:

We introduce a new norm in Hsp.Q/ by

kukHsf .Q/ D

(X

k2Zm

�1C jkj2s

� jckj2) 1=2

:

Lemma 3.10. The norms k � kHsp

and k � kHsf .Q/ are equivalent, that is

c1kukHsf .Q/ � kukHs

p.Q/ � c2kukHsf .Q/

for all u 2 Hsp.Q/ and some positive constants c1; c2.

Proof. For every integer s there exist positive constants C1, C2 such that for allk 2 R

m

C1 �P

0�j˛j�s k2˛

1C jkj2s� C2: (3.15)

Now,

kuk2Hsp.Q/

DX

0�j˛j�s

kD˛uk2L2.Q/ D LmX

0�j˛j�s

�2�

L

�2j˛j X

k2Zm

jckj2k2˛!

� cX

k2Zm

0

@X

0�j˛j�s

jkj2j˛j1

A jckj2 � cC2X

k2Zm

�1C jkj2s

� jckj2 D cC2kuk2Hsf .Q/:

Setting c2 D cC2 we have the second inequality of the lemma. Using onceagain (3.15) we obtain the first inequality of the lemma. utWe have thus

Lemma 3.11. The space Hsp.Q/ coincides with

(

u 2 L2.Q/ W u.x/ DX

k2Zm

cke2� ik xL ; ck D c�k;

X

k2Zm

jkj2sjckj2 < 1)

:

Lemma 3.12. In the space

PHsp.Q/ D

�

u 2 Hsp.Q/ W

Z

Qu.x/dx D 0

�


(of these u 2 Hsp.Q/ for which c0 D 0 in their series representation) the expression

kuk PHsp.Q/

D X

k2Zm

jkj2sjckj2!1=2

is a norm equivalent to the norm k � kHsp.Q/.

Proof. The proof follows from the Poincaré inequality

kukL2.Q/ � L

2�kDukL2.Q/; (3.16)

which holds for all u in PH1p.Q/. In fact, we have

kDuk2L2.Q/ DX

j˛jD1kD˛uk2L2.Q/ D

X

j˛jD1Lm

�2�

L

�2 X

k2Zmnf0gjckj2jkj2

��2�

L

�2

LmX

k2Zmnf0gjckj2 D

�2�

L

�2

kukL2.Q/;

which ends the proof. utIn the space k � k PHs

p.Q/we introduce the scalar product

.u; v/ PHsp.Q/

DX

k2Zm

jkj2sckdk;

for u.x/ D Pk2Zm cke2� ik x

L and v.x/ D Pk2Zm dke2� ik x

L . We have also

ck D Ou.k/ D 1

Lm

Z

Qu.x/e�2� ik x

L dx:

The Fourier coefficients Ou of u are complex amplitudes of harmonics e2� ik xL

associated to wave numbers 2�kL . The length scale related to the wave number 2�k

L

equals Ljkj . Notice that

Lpmkmax

� L

jkj D Lq

k21 C � � � C k2m

� L

kmin:

Theorem 3.13 (Sobolev Embedding Theorem). Let u 2 Hsp.Q/, s > m

2. Then

u 2 C0p.Q/ and

supx2Q

ju.x/j D kukL1.Q/ � C.s/kukHsp.Q/:


Proof. Let u.x/ D Pk2Zm cke2� ik x

L . Then, for s > m2

we have

kukL1.Q/ �X

k2Zm

jckj �X

k2Zm

1

.1C jkj2s/1=2

�1C jkj2s

�1=2 jckj

�(X

k2Zm

1

1C jkj2s

) 1=2 (X

k2Zm

�1C jkj2s

� jckj2) 1=2

D C.s/kukHsp.Q/:

Moreover, the absolute convergence of the series of coefficientsP

k2Zm jckj impliesthe uniform convergence of the Fourier series of u and, in consequence, thecontinuity of u. utExercise 3.13. Prove that for s > m

2the series

Pk2Zm

11Cjkj2s is convergent.

Exercise 3.14. Prove that if s > m2

D j and u 2 Hsp.Q/ then u is j times continuously

differentiable function in Q (we write u 2 Cj.Q/) and

kukCj.Q/ DX

j˛j�j

supx2Q

jD˛u.x/j � C.s/kukHsp.Q/:

We shall need also the Rellich–Kondrachov theorem on compact embedding.

Theorem 3.14 (Rellich–Kondrachov). The space H1p.Q/ is compactly embedded

in the space L2.Q/ which means that for every bounded in H1p.Q/ sequence of

functions there exists its subsequence that is convergent in L2.Q/.

Proof. Let fung, un.x/ D Pk2Zm cn;ke2� ik x

L , n 2 N be a bounded sequence in H1p.Q/,

that is,X

k2Zm

�1C jkj2� jcn;kj2 � M

for all n and some positive constant M. We have to prove that there exists asubsequence uj which converges to some u? D P

k2Zm c?k e2� ik xL 2 H1

p.Q/ stronglyin L2.Q/. It is known that from every bounded sequence in a Hilbert space one canchoose a weakly convergent subsequence. Assume that uj converges weakly to u?

in H1p.Q/. Then u? satisfies

X

k2Zm

�1C jkj2� jc?k j2 � M:

Exercise 3.15. Let the sequence uj in a Hilbert space H converge weakly to u? 2 H,that is, for every v 2 H, .uj �u?; v/H ! 0 as j ! 1, where .�; �/H denotes the scalarproduct in H. Prove that the norm of u? in H satisfies

ku?kH � lim infj!1 kujkH :


Exercise 3.16. Let uj converge weakly to u? in H1p.Q/. Prove that it is equivalent to

the convergence of all Fourier coefficients, namely, cj;k ! c?k as j ! 1, for everyk 2 Z

m.

We shall prove that the subsequence uj converges to u? in L2.Q/. Observe that

X

k2Zm

�1C jkj2� jcj;k � c?k j2 � 4M;

whence

kuj � u?k2L2.Q/ DX

k2Zm

jcj;k � c?k j2 �X

jkj�K

jcj;k � c?k j2 C 1

K2

X

jkj>K

jcj;k � c?k j2jkj2

�X

jkj�K

jcj;k � c?k j2 C 4M

K2:

For every " > 0 we can take K large enough so that 4MK2

< "2. Then, in view of

Exercise 3.16 we can take j large enough so that the first term on the right-hand sideis smaller than "

2. This proves the convergence uj ! u? in L2.Q/. ut

Exercise 3.17. Prove that HsC1.Q/ is compactly embedded in Hs.Q/.

Laplace Equation in Sobolev Spaces of Periodic Functions Let f 2 L2.Q/.Assume that we look for u 2 Hs

p.Q/ satisfying equation

��u D f : (3.17)

If u 2 Hsp.Q/ is a solution of (3.17) then for an arbitrary constant c, u C c is also

a solution in the same space. To guarantee uniqueness we restrict our attention tosolutions satisfying the additional property

Z

Qu.x/dx D 0; (3.18)

which is equivalent to u 2 PHsp.Q/. But then we have also

Z

Q�u.x/dx D 0: (3.19)

Exercise 3.18. Prove that if u 2 Hsp.Q/ satisfies (3.18) then also (3.19) holds.

It then follows thatZ

Qf .x/dx D 0;


whence f is in PL2.Q/ D PH0p.Q/. For any f 2 L2.Q/ there exists a constant c such that

f C c 2 PL2.Q/. In consequence, to have the solution uniqueness, we shall assumethat f 2 PL2.Q/. Let

f .x/ DX

k2Zm

fke2� ik xL ; f0 D 0

and

u.x/ DX

k2Zm

cke2� ik xL ; u0 D 0: (3.20)

We compute

��u.x/ D�2�

L

�2 X

k2Zm

jkj2uke2� ik xL D

X

k2Zm

fke2� ik xL D f ;

and comparing the coefficients we obtain

uk D L2

4�2fk

jkj2 for k 6D 0: (3.21)

We can prove now the following:

Lemma 3.13. If f 2 PL2.Q/ and u 2 PH1p.Q/ are a generalized solution of the

Laplace equation, that is, u satisfies the integral identityZ

Qru W rv dx D

Z

Qfv dx (3.22)

for all v in PC1p.Q/, then u 2 PH2

p.Q/ and

kuk PH2p .Q/

� Ckf kL2.Q/: (3.23)

Proof. First, observe that in the above integral identity we can take the test functionsfrom the space PH1

p.Q/. Then the existence of a unique generalized solution inPH1

p.Q/ follows easily from the Riesz–Frèchet theorem. This solution is givenby (3.20) where the coefficients are given in (3.21). Moreover, the solution belongsto PH2

p.Q/, as

kuk2PH2p .Q/

DX

k2Zm

jkj4jukj2 DX

k2Zm

jkj4�

L2

4�2fk

jkj2�2

D L4

16�4

X

k2Zm

jfkj2 D L4

16�4kf kL2.Q/:

This ends the proof. ut


Looking on the equation ��u D f as on the abstract operator equation Au D f inthe space PL2.Q/ with operator A W PL2.Q/ D.A/ ! PL2.Q/ we see that

D.A/ D fu 2 PH1p.Q/ W Au 2 PL2.Q/g D PH2

p.Q/:

In fact, from (3.23) it follows that D.A/ PH2p.Q/, on the other hand, for every u in

PH2p.Q/, Au 2 PL2.Q/, whence PH2

p.Q/ D.A/.

Exercise 3.19. Prove that if for all v 2 PH1p.Q/ functions f 2 PHs

p.Q/ and u 2 PH1p.Q/

satisfy (3.22) then u 2 HsC2p .Q/ and (3.23) holds.

Lemma 3.14. The operator A is invertible on PL2.Q/ and A�1 W PL2.Q/ ! PL2.Q/ iscompact, that is, it maps bounded sets in PL2.Q/ to precompact sets in PL2.Q/.Proof. From the above considerations we know that the operator A�1 maps PL2.Q/onto PH2

p.Q/ and that the correspondence is one to one. From Theorem 3.14 it follows

that PH2p.Q/ is compactly embedded in PL2.Q/, whence A�1 is compact. ut

Lemma 3.15. In the considered case of periodic boundary conditions the eigen-functions of the Laplacian operator in PL2.Q/ belong to PC1

p .Q/.

Proof. Let Aw D �w 2 PL2.Q/. From Lemma 3.13 it follows that w 2 PH2p.Q/. Thus

Aw D �w 2 PH2p.Q/. Making use of Exercise 3.19 we conclude that w 2 PH4

p.Q/.

By induction we obtain w 2 PHsp.Q/ for each integer s. Now, it suffices to use the

Sobolev embedding theorem to conclude that w belongs to PC1p .Q/. ut

Exercise 3.20. Prove that if f 2 PC1p .Q/, u 2 PH1

p.Q/, and Au D f then u is infinitely

smooth, namely, u 2 PC1p .Q/.

Theorem 3.15 (cf. [197, Corollary 3.26]). Let H be a Hilbert space and let A be asymmetric linear operator in H whose range is all of H, and suppose further that itsinverse is defined and compact. Then A has an infinite set of real eigenvalues �n withcorresponding eigenfunctions !n: A!n D �n!n. If the eigenvalues are ordered sothat j�nC1j � j�nj one has limn!1 j�nj D 1. Furthermore, the !n can be chosenso that they form an orthonormal basis for H, and in terms of this basis the operatorA can be represented by

Au D1X

jD1�j.u; !j/H!j

on its domain

D.A/ D8<

:u W u D

1X

jD1cj!j;

1X

jD1jcjj2�2j < 1

9=

;:

D.A/ is a Hilbert space with the inner product .u; v/D.A/ D .Au;Av/H andcorresponding norm kukD.A/ D kAukH.

3.5 Evolution Spaces and Their Useful Properties 61

For the operator Au D ��u in PL2.Q/, whence we have

u.x/ DX

k2Zmnf0gcke2� ik x

L DX

k2Zmnf0gck

cos

2�k

x

L

�C i sin

2�k

x

L

��

Dr

Lm

2

X

k2.Zmnf0g/=2

Re.ck/!

.c/k � Im.ck/!

.s/k

�;

where

!.c/k .x/ D

r2

Lmcos

2�k

x

L

�; !

.s/k .x/ D

r2

Lmsin 2�k

x

L

�;

and A!.:/k D 4�2jkj2L2

!.:/k , k!.:/k kL2.Q/ D 1.

For example, for m D 2, the smallest eigenvalue is 4�2

L2jkj2 with jkj D

q

k21 C k22 D 1.It repeats four times, �1 D �2 D �3 D �4 in the sequence

0 < �1 � �2 � �3 � : : : � �n � : : : :

The corresponding eigenfunctions are given as !1 D !.c/.1;0/, !2 D !

.s/

.1;0/, !3 D!.c/.0;1/, !4 D !

.s/

.0;1/.

3.5 Evolution Spaces and Their Useful Properties

The solutions of evolution problems are the functions which depend on bothspace and time variables. Typically, the space variable belongs to a certain domain˝ R

d and the time variable belongs to the time interval I D Œt0; t1�. A typicaltechnique is to deal with these variables differently. Namely, assume that u is afunction of .x; t/ 2 ˝ � I and freeze t 2 I. Then u.�; t/ is only a function of thespace variable. Usually we assume that this function belongs to a certain Banachspace of functions defined on ˝, let us denote this space by V . We write u.t/ 2 V .If we “unfreeze” the time variable now, we can write u W I ! V , so the function of.x; t/ 2 ˝ � I is treated as the Banach space valued function of a time variable only.This chapter is devoted to the recollection of the definition and basic properties ofspaces of such functions.

Definition 3.12. Let V be a separable Banach space. The function u W I ! V isstrongly measurable if there exists a sequence of simple functions un W I ! V suchthat

limn!1 un.t/ D u.t/ for a.e. t 2 I:

A function v W I ! V is simple if v.t/ D PmkD1 �Ak.t/wk for all t 2 I, where Ak are

Lebesgue measurable subsets of I, and wk 2 V .


Definition 3.13. Let V be a separable Banach space. A strongly measurablefunction v W I ! V is Bochner integrable if

Z

Ikv.t/kV dt < 1:

The space of Bochner integrable functions is denoted by L1.II V/. It is possible touse the equivalent definition of this space, which gives rise to the so-called Bochnerintegral.

Theorem 3.16 (cf. [85, Theorem 2, p. 45]). Let v W I ! V be a stronglymeasurable function. Then v 2 L1.II V/ is Bochner integrable if and only if thereexists a sequence of simple functions vn W I ! V such that

limn!1

Z

Ikvn.t/ � v.t/kV dt D 0: (3.24)

Definition 3.14. If the function v W I ! V is simple then its Bochner integral overthe Lebesgue measurable set E I can be defined as

Z

Ev.t/ dt D

mX

iD1wkm.Ak \ E/:

If, in turn, v 2 L1.II V/, then the Bochner integral is defined as

Z

Ev.t/ dt D lim

n!1

Z

Evn.t/ dt;

where vn is the sequence of simple functions from Theorem 3.16.

It is easy to prove that the value of the Bochner integral does not depend on thechoice of the sequence of simple functions which satisfy (3.24). More informationon Bochner integral and its properties can be found in [85, 107, 234]. The definitionsof Bochner integral and the space L1.II V/ do not require the separability of V . Insuch a case, however, the notion of measurability of a Banach space valued functionshould be defined more carefully. For the sake of simplicity, however, in this bookwe always use this notions for separable spaces V .

As a natural extension of L1.II V/ it is possible to define the spaces Lp.II V/ forp 2 Œ1;1� as the spaces of the measurable functions, such that

kf kLp.IIV/ D( �R

I kf .t/kpV dt

� 1p < 1 for p 2 Œ1;1/;

ess supt2I kf .t/kV < 1 for p D 1:

Once we identify any two functions which differ only of a subset of I of Lebesguemeasure zero, the space Lp.II V/ endowed with the norm k � kLp.IIV/ becomes a


Banach space. Moreover, through a natural isometric isomorphism we can makethe following identification Lp.II Lp.˝/m/ D Lp.˝ � I/m valid for p ¤ 1 (notethat L1.˝/m is not a separable space, see Example 1.42 in [204] and Example 23.4in [234]).

We have the following useful results related to the dual space to Lp.II V/.

Proposition 3.1 (cf. [234, Proposition 23.7]). Let V be a separable and reflexiveBanach space and let 1 < p < 1, and 1

p C 1q D 1. Then Lp.II V/ is a reflexive

and separable Banach space. Moreover, the dual space of Lp.II V/ is isometricallyisomorphic with Lq.II V 0/.

Proposition 3.2 (cf. [234, Proposition 23.9]). Let V be a separable and reflexiveBanach space and let 1 < p < 1, and 1

p C 1q D 1. If u 2 Lp.II V/ and v 2 V 0, then

v;

Z

Iu.t/ dt

V0�V

DZ

Ihv; u.t/iV0�V dt:

Moreover from

un ! u strongly in Lp.II V/;

vn ! v weakly in Lq.II V 0/;

it follows thatZ

Ihvn.t/; un.t/iV0�V dt !

Z

Ihv.t/; u.t/iV0�V dt:

A function v 2 L1.II V/ is called a distributional time derivative of a functionu 2 L1.II V/, provided the following equality

Z

Iu.t/�0.t/ dt D �

Z

Iv.t/�.t/ dt for all � 2 C1

0 .I/; (3.25)

holds in V . In such a case we write v D u0 D ut D dudt . Sometimes, the distributional

time derivative of a function belonging to L1.II V/ may not exist as a functionbelonging to the same space, but it may have values in a larger Banach space Z,such that we have the embedding V Z. Then, the function v 2 L1.II Z/ is adistributional time derivative of u 2 L1.II V/, provided (3.25) holds in V . Note that,in such a case, the integral on the right-hand side of (3.25) belongs to V , althoughthe integrand has values only in Z, and hence not necessarily in V .

As it can be easily checked, the distributional derivative, provided it exists as afunction u0 2 L1.II Z/, is unique with respect to equality for a.e. t 2 I. Moreover wehave the following equality in Z valid for all s; t 2 I

u.t/ D u.s/CZ t

su0.r/ dr:


Note that if u 2 L1.II V/ and u0 2 L1.II Z/ with V Z, then, after modification ofthe null set we must have u 2 C.II Z/ and hence the pointwise values of u makesense as the elements of Z in the last equality.

An important role in the study of weak solutions of evolution problems is playedby the so-called evolution triples (or Gelfand triples). An evolution triple consistsof the reflexive Banach space V (we will always assume that V is also separable),the Hilbert space H such that we have the continuous and dense embedding V H,and the dual space V 0. As the space H is always identified with its dual we can writeV H V 0 with all embeddings being continuous and dense. From now on wewill always denote the scalar product on H by .�; �/ and associated norm by j � j. Thenorm on V is denoted by k � k and the duality pairing between V and V 0 by h�; �i: Animportant property of evolution triples is the following formula

hu; vi D .u; v/ whenever u 2 H and v 2 V: (3.26)

The following results which hold in the framework of evolution triples will befrequently used later.

Corollary 3.2. Let V H V 0 be an evolution triple such that V H isadditionally compact and V is Hilbert. Let A W V ! V 0 be a linear operatorwhich is

• bounded (kAukV0 � kAkL .VIV0/kuk for all u 2 V),• symmetric (hAu; vi D hAv; ui for all u; v 2 V),• coercive (hAu; ui � ˛kuk2 for all u 2 V with ˛ > 0).

Then A satisfies the assumptions of Theorem 3.15 and we can construct anorthogonal in V and orthonormal in H basis of eigenfunctions !j of A with thecorresponding eigenvalues such that j�jj ! 1 as j ! 1.

Proof. The Lax–Milgram lemma (cf. Lemma 3.2) implies that for any f 2 H thereexists a unique v 2 V such that Av D f . Hence the range of A (treated as an operatorfrom D.A/ H to H) is whole H and A�1 W H ! H is well defined. Moreover,by (3.26), A, when treated as an operator on H, is symmetric. Obviously A�1 islinear. To prove compactness of A�1 let fn be a bounded sequence in H. Let un 2 Vbe such that Aun D fn. Coercivity of A implies that the sequence un is bounded in Vand hence, by the compactness of the embedding V H, for a subsequence, un ! ustrongly in H for certain u 2 H. The proof of compactness is complete. utTheorem 3.17 (cf. [197, Lemma 7.5]). Let V H V 0, A W V ! V 0, and !j beas in Corollary 3.2. Denote Vn D spanf!1; : : : ; !ng. If we define Pn W V 0 ! Vn byPnv D Pn

iD1hv; !iiwi for v 2 V 0 then we have

if v 2 V then kPnvk � kvk and limn!1 kPnv � vk D 0;

if v 2 H then jPnvj � jvj and limn!1 jPnv � vj D 0;

if v 2 V 0 then kPnvkV0 � kvkV0 and limn!1 kPnv � vkV0 D 0:

We will frequently use the following two results.


Lemma 3.16 (cf. [234, Proposition 23.23], [219, Chap. 3, Lemma 1.2]). LetV H V 0 be an evolution triple, let I D .t1; t2/ be a finite time interval,and let 1 < p < 1 and 1

p C 1q D 1. Then

1. The space W .I/ D fu 2 Lp.II V/ W u0 2 Lq.II V 0/g is a Banach space equippedwith the norm

kukW .I/ D kukLp.IIV/ C ku0kLq.IIV0/:

2. The space W .I/ embeds continuously in C.II H/, i.e., every function from W .I/is almost everywhere equal to a continuous function with values in H, and wehave

maxt2I

ju.t/j � CkukW .I/ for all u 2 W .I/:

3. We have the following integration by parts formula valid for all u; v 2 W .I/,

.u.t1/; v.t1// � .u.t0/; v.t0// DZ

Ihv0.t/; u.t/i C hu0.t/; v.t/i dt:

4. For any u 2 W .I/ the following equality holds in the sense of distributions on I

d

dtju.t/j2 D 2hu0.t/; u.t/i:

Corollary 3.3. Let V;H; p; q be as in the previous Lemma. Let un ! u weakly inLp.II V/ and u0

n ! v weakly in Lq.II V 0/. Then v D u0 and un.t/ ! u.t/ weakly inH for all t 2 I.

Proof. The fact that v D u0 follows, for example, from Proposition 23.19 in [234].We will prove the second assertion. We have

un.s/ D un.t/CZ t

su0

n.r/dr for all s; t 2 I:

Taking the duality with any v 2 V and integrating with respect to t over I we get

.t1 � t0/.un.s/; v/ DZ

I.un.t/; v/ dt C

Z

I

Z t

shu0

n.r/; vi dr dt

DZ

I.un.t/; v/ dt C

Z

Ihu0

n.t/; v�s.t/i dt;

where �s D .t0 � s/�.t0;s/ C .t1 � s/�.s;t1/. The assertion follows by passing to thelimit n ! 1 in the last formula and using the density of the embedding V H. utWe now recall two important results on the triples of spaces V1 V2 V3, wherethe embedding V1 V2 is compact. The first result is known as the Ehrling lemma.


Lemma 3.17 (See [204, Lemma 7.6]). Let V1 V2 V3 be three Banach spaceswith all embeddings continuous and V1 V2 compact. Then for any " > 0 thereexists a constant C."/ such that

kvkV2 � "kvkV1 C C."/kvkV3 for all v 2 V1:

The next result is known as Aubin–Lions compactness theorem.

Theorem 3.18 (cf. [6, 155], [216, Corollary 4], [204, Lemma 7.7], [219, Chap. 3,Theorem 2.1]). Let I be a bounded time interval and let 1 � p; q < 1 andlet X1 X2 X3 be three separable Banach spaces with all embeddings beingcontinuous. If the embedding X1 X2 is compact, then the embedding

fu 2 Lp.II X1/ W u0 2 Lq.II X3/g Lp.II X2/

is also compact.

Fairly general form of the above Lemma is given in [216]. Note that the assumptionof separability is not required there.

A function u W I ! V , where V is a Banach space is said to be weakly continuousprovided the function

I 3 t ! hv; u.t/i 2 R

is continuous for all v 2 V 0. We write u 2 Cw.II V/, or u 2 C.II Vweak/, or,sometimes C.II Vw/. Of course C.II V/ Cw.II V/ with the strict inclusion. Wehave the following result.

Lemma 3.18 (cf. [219, Chap. 3, Lemma 1.4]). Let X Y be two Banach spaceswith a continuous embedding and let I be a finite time interval. If u 2 L1.II X/ andu 2 Cw.II Y/ then u 2 Cw.II X/.

Finally we give a parabolic embedding result.

Lemma 3.19 (see [155, Chap. 1, Lemma 6.7]). Let ˝ R3 be an open and

bounded set with Lipschitz boundary and let I be a bounded time interval. If

u 2 L2.II H1.˝// \ L1.II L2.˝//;

then u 2 L4.II L3.˝//.

Proof. By the Sobolev embedding theorem we have H1.˝/ L6.˝/ with acontinuous embedding, so, using the Hölder inequality we have for a.e. t 2 I andfor a constant c > 0

ku.t/kL3.˝/ � ku.t/k 12

L2.˝/ku.t/k 1

2

L6.˝/� cku.t/k 1

2

L2.˝/ku.t/k 1

2

H1.˝/:

3.6 Gronwall Type Inequalities 67

Thus we haveZ

Iku.t/k4L3.˝/ dt � c4 ess sup

t2Ifku.t/k2L2.˝/g

Z

Iku.t/k2H1.˝/

dt;

and the assertion is proved. ut

3.6 Gronwall Type Inequalities

In this section we deal with Gronwall type inequalities, which are the efficient anduseful tool in the study of evolution problems. These inequalities are used in thederivation of the a priori estimates that are satisfied by the solution of the analyzedproblem.

The first Gronwall type inequality is useful in the study of existence anduniqueness of solutions to initial and boundary value problems for evolutionaryPDEs, as well as in the derivation of dissipative a priori estimates for their solutions.

Lemma 3.20 (Gronwall Inequality). Let t0 be a real number. Let x 2 L1loc.t0;1/

be a function such that x0 2 L1loc.t0;1//, and let h 2 L1loc.t0;1/ and k 2L1loc.Œt0;1//. If

x0.t/ � x.t/k.t/C h.t/ for a:e: t 2 .t0;1/; (3.27)

then

x.t/ � x.t0/eR t

t0k.s/ ds C

Z t

t0

h.s/eR t

s k.r/ dr ds for all t 2 Œt0;1/: (3.28)

If, in particular, k.t/ D C for all t � t0, C being a real constant, then

x.t/ � x.t0/e�Ct0eCt C eCt

Z t

t0

h.s/e�Cs ds for all t 2 Œt0;1/: (3.29)

If, furthermore, h.t/ D D for all t � t0, D being a real constant, then

x.t/ � x.t0/e�Ct0eCt C D

C.e�Ct0eCt � 1/ for all t 2 Œt0;1/: (3.30)

Proof. Multiplying (3.27) by the integrating factor e� R tt0

k.s/ ds we get

x0.t/e� R tt0

k.s/ ds � x.t/k.t/e� R tt0

k.s/ ds � h.t/e� R tt0

k.s/ ds a.e. t > t0;


whence

d

dt

x.t/e� R t

t0k.s/ ds

�� h.t/e� R t

t0k.s/ ds for a.e. t > t0:

Integrating the above inequality from t0 to t we obtain (3.28). Assertions (3.29)and (3.30) follow from (3.28) by a straightforward calculation. utIf, in (3.29) or (3.30), we have C < 0, then the resultant estimate is, afterappropriately large time, independent on the initial condition x.t0/. Such estimatesare called dissipative and they are particularly useful in the study of global attractors.If, however, we only have (3.27) with k.t/ positive on .t0;1/, then the resultantbounds grow to infinity as t ! 1. In such situation, we need the following uniformGronwall inequality.

Lemma 3.21 (Uniform Gronwall Inequality). Let t0 2 R. Let x 2 L1loc.t0;1/ bea function such that x0 2 L1loc.t0;1/, and let h 2 L1loc.t0;1/ and k 2 L1loc.Œt0;1//.Let moreover x.t/ � 0, h.t/ � 0, and k.t/ � 0 for a.e. t > t0. If

x0.t/ � x.t/k.t/C h.t/ for a:e: t 2 .t0;1/; (3.31)

andZ tCr

tk.s/ ds � a1;

Z tCr

th.s/ ds � a2;

Z tCr

tx.s/ ds � a3 for all t 2 Œt0;1/;

where a1; a2; a3; r are positive constants, then

x.t C r/ � a3

rC a2

�ea1 for all t 2 Œt0;1/:

Proof. Let t0 � t � t C r. We multiply (3.31) written for p 2 .t; t C r/ by theintegrating factor e� R p

t k.�/ d� , getting

d

dp

x.p/e� R p

t k.�/ d��

D x0.p/e� R pt k.�/ d� � x.p/k.p/e� R p

t k.�/ d�

� h.p/e� R pt k.�/ d� � h.p/ a.e. p 2 .t; t C r/:

We fix s 2 .t; t C r/ and integrate the last inequality over the interval .s; t C r/.We get

x.t C r/e� R tCrt k.�/ d� � x.s/e� R s

t k.�/ d� �Z tCr

sh.p/ dp � a2;

whence we have the inequality

x.t C r/ � x.s/eR tCr

s k.�/ d� C a2eR tCr

t k.�/ d� � x.s/ea1 C a2ea1 ;

3.6 Gronwall Type Inequalities 69

valid for all s 2 .t; t C r/. We integrate the last inequality with respect to s over theinterval .t; t C r/, which gives us

rx.t C r/ �Z tCr

tx.s/ ds ea1 C ra2e

a1 � a3ea1 C ra2e

a1 ;

which immediately yields the assertion. utIf we need to estimate the norm of the unknown function x by a constant that isarbitrarily small, then the following translated Gronwall inequality becomes useful(see [161]).

Lemma 3.22. Let � > 0 be a constant. Let x 2 L1loc.t0;1/ be such that x0 2L1loc.t0;1/ and x.t/ � 0 for a.e. t � t0. Let moreover h 2 L1loc.t0;1/ be such thath.t/ � 0 for a.e. t � t0. If we have

x0.t/C �x.t/ � h.t/ a:e: t > t0; (3.32)

then for all t � t0 and r > 0 we have

x.t C r/ � 2

re�� r

2

Z tC r2

tx.s/ ds C e��.rCt/

Z tCr

th.s/e�s ds:

In particular, taking r D 2, we get, for all t � t0

x.t C 2/ � e��Z tC1

tx.s/ ds C e��.tC2/

Z tC2

th.s/e�s ds:

Proof. Multiplying (3.32) written for p � t0 by the integrating factor e�p, we get

d

dp.x.p/e�p/ D x0.p/e�p C x.p/�e�p � h.p/e�p a.e. p � t0:

Let t � t0 and r > 0. We fix s 2 �t; t C r2

�and integrate the last inequality over the

interval .s; t C r/ leading to

x.t C r/e�.tCr/ � x.s/e�s �Z tCr

sh.p/e�p dp;

whence we have

x.t C r/ � x.s/e�.s�t�r/ C e��.tCr/Z tCr

sh.p/e�p dp

� x.s/e�.s�t�r/ C e��.tCr/Z tCr

th.p/e�p dp:


We integrate the last inequality with respect to s over the interval .t; t C r2/, which

gives

r

2x.t C r/ �

Z tC r2

tx.s/e�.s�t�r/ ds C r

2e��.tCr/

Z tCr

th.s/e�s ds;

and the assertion follows immediately. utWe shall use also the following lemma.

Lemma 3.23 (Generalized Gronwall Inequality). Let ˛ D ˛.t/ and ˇ D ˇ.t/be locally integrable real-valued functions on Œ0;1/ that satisfy the followingconditions for some T > 0:

lim inft!1

1

T

Z tCT

t˛.�/d� > 0; (3.33)

lim supt!1

1

T

Z tCT

t˛�.�/d� < 1; (3.34)

limt!1

1

T

Z tCT

tˇC.�/d� D 0; (3.35)

where ˛�.t/ D maxf�˛.t/; 0g, ˇC.t/ D maxfˇ.t/; 0g. Suppose that � D �.t/ isan absolutely continuous nonnegative function on Œ0;1/ that satisfies the followinginequality almost everywhere on .0;1/,

d�.t/

dtC ˛.t/�.t/ � ˇ.t/: (3.36)

Then �.t/ ! 0 as t ! 1.

For the proof, see [99].

3.7 Clarke Subdifferential and Its Properties

We start this section from the recollection of the definition and properties ofthe Clarke subdifferential, a generalization of Gâteaux derivative to functionalswhich are only locally Lipschitz. The Clarke subdifferential at a given pointassumes multiple values, so it is a multifunction. Then, we give an example of amultivalued partial differential equation, where the semilinear term has the formof a Clarke subdifferential. Finally, we show several examples how to computeClarke subdifferentials and we discuss the usefulness of the Clarke subdifferentialto express various boundary conditions in mechanics. Throughout this chapter X isalways assumed to be a Banach space.

3.7 Clarke Subdifferential and Its Properties 71

Definition 3.15. A functional j W X ! R is locally Lipschitz if for every x 2 Xthere exists an open neighborhood O 2 N .x/ and a constant kO such that for allz; y 2 O jj.z/ � j.y/j � kOkz � ykX .

The following two definitions were introduced by Clarke (see [71, 72]).

Definition 3.16. Let j W X ! R be locally Lipschitz and let x; h 2 X. A generalizeddirectional derivative in the sense of Clarke of the functional j at the point x 2 X andin the direction h 2 X, denoted by j0.xI h/ is defined as

j0.xI h/ D lim supy!x;�!0C

j.y C �h/ � j.y/

�:

Definition 3.17. Let j W X ! R be locally Lipschitz. The generalized subdifferen-tial in the sense of Clarke of the functional j at the point x 2 X is the set @j.x/ X0defined as

@j.x/ D f� 2 X0 W h�; hiX0�X � j0.xI h/ for all h 2 Xg: (3.37)

We recall some results on the properties of the Clarke subdifferential and Clarkegeneralized directional derivative.

Proposition 3.3 (cf. [72, Proposition 2.1.1]). If j W X ! R is locally Lipschitz,then j0 W X � X ! R is upper semicontinuous, i.e.,

�n ! x; hn ! h ) lim supn!1

j0.xnI hn/ � j0.xI h/ for all x; h 2 X:

Moreover, j0.xI �/ is Lipschitz on X with the Lipschitz constant equal to the localLipschitz constant kO of j at x given in Definition 3.15.

Proposition 3.4 (cf. [72, Proposition 2.1.2]). If j W X ! R is locally Lipschitz,then for every x 2 X, the set @j.x/ is nonempty, weakly-* compact, and convex.Moreover, for every � 2 @j.x/ we have k�kX � kO (where kO is the local Lipschitzconstant of j at x given in Definition 3.15) and we have

j0.xI h/ D max�[email protected]/

h�; hiX0�X for all x; h 2 X:

Proposition 3.5 (cf. [72, Proposition 2.1.5]). If j W X ! R is locally Lipschitz, thenthe graph of a multifunction X 3 X ! @j.x/ 2 2X0

is a strong-weak-* sequentiallyclosed set.

For a functional defined on Rd one can calculate its Clarke subdifferential from the

following characterization.


Theorem 3.19 (see [72, Theorem 2.5.1]). Let j W Rd ! R be locally Lipschitz then

@j.s/ D convf limn!1 rj.sn/ W sn ! s; sn … S [ Njg for s 2 R

d; (3.38)

where S is any Lebesgue null set, and Nj is the Lebesgue null set on which j is notdifferentiable.

We recall the Lebourg mean value theorem (cf. [72, Theorem 2.3.7])

Theorem 3.20. Let X be a Banach space and j W X ! R be a locally Lipschitzfunctional. For any x; y 2 X there exists 2 .0; 1/ and � 2 @j.x C .1 � /y/ suchthat h�; y � xiX0�X D j.y/ � j.x/.

We are in position to prove the approximation theorem for Clarke subdifferentials.

Theorem 3.21. Let j W Rd ! R be a locally Lipschitz functional such that the

growth condition j�j � C.1 C jsj/ holds for all s 2 Rd and all � 2 @j.s/ with

C > 0. Let moreover �n W Rd ! R be a sequence of mollifier kernels given by

�n.r/ D nd�.nr/ for r 2 Rd, where � W Rd ! R is a function such that � 2 C1.Rd/,

�.s/ � 0 on Rd,RRd �.s/ ds D 1, and supp � Œ�1; 1�d. Define

jn.s/ DZ

Rd�n.r/j.s � r/ dr:

Then

(A) jn 2 C1.Rd/,(B) jrjn.s/j � D.1C jsj/ for all s 2 R

d with D > 0 independent of n,(C) if .˝;˙;/ is a finite and complete measure space and vn 2 L2.˝/d, is a

sequence such that vn ! v strongly in L2.˝/d and rjn.vn/ ! � weakly inL2.˝/d, then �.x/ 2 @j.v.x// -a.e. in ˝.

Proof. The assertion .A/ follows, for example, from [93], Theorem 6, Appendix C.Let us prove .B/. Let h 2 R

d. We have

rjn.s/ �h D lim�!0

jn.s C �h/ � jn.s/

�D lim

�!0

Z

supp �n

�n.r/j.s C �h � r/ � j.s � r/

�dr:

Using the Lebourg mean value theorem we get

ˇˇˇˇj.s C �h � r/ � j.s � r/

�

ˇˇˇˇ � jz.s; �h; r/jjhj � C.1C jsj C jrj C j�jjhj/jhj;

where z.s; �h; r/ 2 @j.s � r C �h/ with 2 .0; 1/ possibly changing from point topoint. We get

jrjn.s/ � hj � C lim�!0

Z

supp �n

�n.r/.1C jsj C jrj C j�jjhj/jhj dr:

3.7 Clarke Subdifferential and Its Properties 73

As supp �n �� 1n ;

1n

�dwe have

jrjn.s/j � C lim�!0

Z

supp �n

�n.r/

�

1C jsj C 1

nC j�j

�

dr � C.2C jsj/;

and the assertion .B/ follows. We pass to the proof of .C/. As vn ! v in L2.˝/d

we also have, for a subsequence, vn.x/ ! v.x/ and jvn.x/j � h.x/ for -a.e. x 2 ˝

with h 2 L2.˝/. The further argument will be done for this subsequence. Takew 2 L1.˝/d. From the weak convergence rjn.vn/ ! � it follows that

Z

˝

�.x/ � w.x/ d.x/ D limn!1

Z

˝

rjn.vn.x// � w.x/ d.x/

�Z

˝

lim supn!1

rjn.vn.x// � w.x/ d.x/;

where the use of the Fatou lemma is justified by the growth condition .B/ and thebound jvn.x/j � h.x/. We estimate the integrand for -a.e. x 2 ˝ as follows

lim supn!1

rjn.vn.x// � w.x/ D lim supn!1

lim�!0C

jn.vn.x/C �w.x// � jn.vn.x//

�

D lim supn!1

lim�!0C

Z

supp �n

�n.s/j.vn.x/C �w.x/ � s/ � j.vn.x/ � s/

�ds

� lim supn!1�!0C

supr2supp �n

j.vn.x/C �w.x/ � r/ � j.vn.x/ � r/

�

D lim supn!1�!0C

j.vn.x/C �w.x/ � rn;�/ � j.vn.x/ � rn;�/

�;

where rn;� 2 supp �n is such that the supremum over r of the difference quotientis achieved. Observe that as n ! 1 we must have vn.x/ � rn;� ! v.x/. From thedefinition of the Clarke directional derivative we get

lim supn!1

rjn.vn.x// � w.x/ � lim supz!v.x/�!0C

j.z C �w.x// � j.z/

�D j0.v.x/I w.x//;

whereasZ

˝

�.x/ � w.x/ d.x/ �Z

˝

j0.v.x/I w.x// d.x/: (3.39)

We will show that �.x/ � z � j0.v.x/I z/ -a.e. in ˝ for all z 2 Rd and in

consequence we will have the assertion �.x/ 2 @j.v.x// -a.e. in ˝. First we prove


that j0.v.x/I w.x// � �.x/ � w.x/ � 0 for -a.e. x 2 ˝ for all w 2 L1.˝/d. Indeed,suppose it is not the case. Then for a certain Nw 2 L1.˝/d and set N ˝ of positivemeasure we have the inequality j0.v.x/I Nw.x// � �.x/ � Nw.x/ < 0. Take Ow D Nw on Nand Ow D 0 on ˝ n N. We have Ow 2 L1.˝/d andZ

˝

j0.v.x/I Ow.x// � �.x/ � Ow.x/ d.x/ DZ

Nj0.v.x/I Nw.x// � �.x/ � Nw.x/ d.x/ < 0;

a contradiction with (3.39). Let fwng1nD1 be a countable dense set in R

d. By takingconstant functions in place of w we get j0.v.x/I wn/ � �.x/ �wn outside a certain nullset Mn. If follows that j0.v.x/I wn/ � �.x/ � wn for all n outside a null sets

S1nD1 Mn.

By Proposition 3.3 and density of fwng1nD1 in R

d it follows that �.x/ � z � j0.v.x/I z/for all z 2 R

d on a set of full measure and the proof is complete. ut

3.8 Nemytskii Operator for Multifunctions

In this section we prove a result, that will be useful in passing to the limit in theterms containing the Clarke subdifferential. Before we state the theorem, however,we need to recall a useful result on pointwise behavior of weakly convergencesequences (see Proposition 3.6 in [177]). To formulate it we will need the notionof Kuratowski–Painlevé upper limit of a sequence of sets An R

d, given by

K- lim supn!1

An D f x 2 Rd W x D lim

k!1

xnk ; xnk 2 Ank ; 0 < n1 < n2 < : : : < nk < : : : g:

Theorem 3.22. Let .�;˙;/ be a � -finite measure space, and let d be a positivenatural number, and 1 � p < 1. Let un ! u weakly in Lp.�/d and un./ 2 G./for -a.e. 2 � and all n 2 N, where G./ R

d is a nonempty, compact set for-a.e. 2 �. Then

u.x/ 2 conv

�

K- lim supn!1

fun.x/g�

:

Theorem 3.23. Let .�;˙;/ be a � -finite measure space and let W D Lp.�/d

with a positive natural number d and 1 < p < 1. Assume that the multifunctionh W � � R

d ! 2Rd

satisfies the assumptions

.h1/ for every s 2 Rd the multifunction ! h.; s/ has a measurable selection,

i.e., there exists a measurable function f W ˝ ! Rd such that f ./ 2 h.; s/ for

-a.e. 2 �,.h2/ the set h.; s/ is nonempty, compact, and convex in R

d, for all s 2 Rd and

for -a.e. 2 �,.h3/ graph of s ! h.; s/ is a closed set in R

2d for -a.e. 2 �,.h4/ we have j�j � c1./ C c2jsjp�1 for all s 2 R

d, -a.e. 2 � and all � 2h.; s/, with c2 > 0 and c1 2 Lq.�/, where 1

q C 1p D 1.

3.8 Nemytskii Operator for Multifunctions 75

Then the multivalued operator HW W ! 2W0

defined by

H.u/ D f � 2 W 0 W �./ 2 h.; u.// -a.e. on � g;

satisfies

.H1/ H has nonempty and convex values,

.H2/ the graph of H is sequentially closed in .W; strong/ � .W 0;weak/ topology,

.H3/ k�kW � kc1kLq.�/ C c2kukp�1W for all w 2 W and � 2 H.u/.

Proof. Let u 2 W. First we will prove .H3/. Let � 2 H.u/. We have

k�kW0 D supkwkW D1

Z

�

�./ � w./ d � supkwkW D1

Z

�

.c1./C c2ju./jp�1/jw./j d:

Hence we have

k�kW0 � supkwkW D1

.kc1kLq.�/kwkW C c2kukp�1W kwkW/ D kc1kLq.�/ C c2kukp�1

W ;

(3.40)

and .H3/ is proved. We pass to the proof of .H1/. Let u 2 W. Consider amultifunction � 3 ! h.; u.// R

d. We will show that this multifunction hasa measurable selection. Let vn W � ! R

d be a sequence of simple (i.e., piecewiseconstant and measurable) functions such that jvn./j � ju./j and vn./ ! u./a.e. 2 �. By .h1/ it follows that there exists the sequence of measurable functions�n W � ! R

d such that �n./ 2 h.; vn.// a.e. 2 �. By (3.40) it follows that

k�nkW0 � kc1kLq.�/ C c2kvnkp�1W � kc1kLq.�/ C c2kukp�1

W ;

and hence, for a subsequence, not renumbered, we have

�n ! � weakly in W 0 for certain � 2 W 0: (3.41)

Since for a.e. 2 � we have

j�n./j � c1./C c2jvn./j � c1./C c2ju./j;we can define G./ D fz 2 R

d W jzj � c1./ C c2ju./jg and we can useTheorem 3.22 concluding that

�./ 2 conv

�

K- lim supn!1

f�n./g�

a.e. 2 �:

Now z 2 K- lim supn!1f�n./g, whenever �nk./ ! z. Keeping in mind thatvnk./ ! u./ and �nk./ 2 h.; vnk.// and we can use .h3/ to conclude thatz 2 h.; u.// for a.e. 2 �. Hence we have

�./ 2 conv h.; u.// D h.; u.// a.e. 2 �; (3.42)


� being the required measurable selection. Moreover, (3.41) implies that � 2 W 0and the proof of non-emptiness of H.u/ is complete. Convexity of H.u/ followsstraightforwardly from the convexity of h.; s/ in .h2/, and the proof of .H1/ iscomplete.

Finally, we will establish .H2/. Let un ! u in W and �n ! � weakly in W 0 with�n 2 H.un/. Passing to a subsequence, we may assume that un./ ! u./ for a.e. 2 � and jun./j � f ./ for a.e. 2 � with f 2 Lp.�/. Since we already knowthat � 2 W we must prove that �./ 2 h.; u.// for a.e. 2 �. The proof of thisassertion exactly follows the lines of the proof of (3.42) in .H1/, which completesthe proof of the theorem. utRemark 3.4. The assertion .H2/ of the above Lemma also follows directly by theapplication of the convergence theorem of Aubin and Cellina (see Theorem 7.2.1 in[8] or Theorem 1.4.1 in [7]).

We will use the notation

H.u/ D Sqh.�;u.�//;

as H.u/ is the set of all selections of the class Lq.�/d of the multifunction� 3 ! h.; u.// R

d. The operator H is known as the Nemytskii operatorfor a multifunction h.

Now we present a version of above theorem valid if the multifunction h hasthe form of a Clarke subdifferential. In such a case the above theorem simplifies.Note that for a functional of more than one variable, say '.; s/, by @'.; s/ we willalways understand the Clarke subdifferential taken with respect to the last variable s.

Theorem 3.24. Let .�;˙;/ be a finite and complete measure space and denoteW D Lp.�/d with d 2 N and 1 < p < 1. Assume that the functional' W � � R

d ! R satisfies the assumptions

.'1/ for every s 2 Rd the function � 3 ! '.; s/ is measurable,

.'2/ for -almost all 2 � the function Rd 3 s ! '.; s/ is locally Lipschitz,

.'3/ j�j � c1./C c2jsjp�1 for all s 2 Rd, -a.e. 2 � and all � 2 @'.; s/, with

c2 > 0 and c1 2 Lq.�/, where 1q C 1

p D 1.

Then the multivalued operator HW W ! 2W0

defined by

H.u/ D Sq@'.�;u.�// D f � 2 W 0 W �./ 2 @'.; u.// -a.e. on � g

satisfies


.H2/ the graph of H is sequentially closed in .W; strong/ � .W 0;weak/ topology,

.H3/ we have k�kW � kc1kLq.�/ C c2kukp�1W for all w 2 W and � 2 H.u/.

3.9 Clarke Subdifferential: Examples 77

Proof. We will use Theorem 3.23. The hypothesis .h4/ follows from .'3/ while thehypotheses .h2/ and .h3/ follow from the properties of the Clarke subdifferential(Propositions 3.4 and 3.5). It remains to prove .h1/. To this end consider the integralfunctional

I.u/ DZ

�

'.; u.// d for u 2 Lp.�/d:

We use Theorem 5.6.39 in [84] (see also Theorem 2.7.5 in [72], where, however,c1 is assumed to be a constant and not a function of ), whence it follows that I islocally Lipschitz and

@I.u/ Sq@'.�;u.�//:

Let s 2 Rd and let us./ D s for 2 �. As the measure is finite, us 2 Lp.�/d.

Since, by Proposition 3.4, @I.us/ is nonempty, then there exists � 2 Lq.�/d suchthat �./ 2 @'.; s/ for -a.e. 2 � and the proof is complete. ut

3.9 Clarke Subdifferential: Examples

As an example of the application of the Clarke subdifferential in PDEs let usconsider the following example. Let ˝ be a bounded domain in R

d with Lipschitzboundary and let g 2 L1

loc.R/. Consider the following initial and boundary valueproblem

@u.x; t/

@t��u.x; t/C g.u.x; t// D f .x; t/ on ˝ � .0;T/;

u.x; t/ D 0 on @˝ � .0;T/;u.x; 0/ D u0.x/ on ˝:

The question arises how to understand the weak solution of the above problemwhen g is discontinuous. It turns out that the function g can be replaced with itsmultivalued regularization constructed in the following way. First we define thefunctional

j.s/ DZ s

0

g.s/ ds for s 2 R:

As g is locally bounded, j must be locally Lipschitz and it makes sense to [email protected]/. We replace the above initial and boundary value problem by the followingone


@u.x; t/

@t��u.x; t/C �.x; t/ D f .x; t/ on ˝ � .0;T/;

�.x; t/ 2 @j.u.x; t// on ˝ � .0;T/;u.x; t/ D 0 on @˝ � .0;T/;

u.x; 0/ D u0.x/ on ˝:

This problem is called a multivalued partial differential equation, a partial differen-tial inclusion, or a subdifferential inclusion and it turns out, that under appropriateassumptions of f ; u0 and the growth condition on @j it has a weak solutionunderstood in the following sense.

Find u 2 L2.0;TI H10.˝// with u0 2 L2.0;TI H�1.˝// and � 2 L2.0;TI L2.˝//

such that u.0/ D u0, �.t/ 2 [email protected]//, and for a.e. t 2 .0;T/ and all v 2 H10.˝/,

hu0.t/; viH�1.˝/�H10 .˝/

C .ru.t/;rv/L2.˝/d C .�.t/ � f .t/; v/L2.˝/ D 0;

Recall that for v 2 L2.˝/ by [email protected]/ we mean the set of all functions h 2 L2.˝/ such

that h.x/ 2 @j.v.x// a.e. in ˝. If we identify L2.0;TI L2.˝// D L2.˝ � .0;T//,we can also write [email protected]// D [email protected].�;t//.

We can associate with the above problem the following problem known ashemivariational inequality.

Find u 2 L2.0;TI H10.˝// with u0 2 L2.0;TI H�1.˝// such that u.0/ D u0, and

for a.e. t 2 .0;T/ and all v 2 H10.˝/,

hu0.t/; viH�1.˝/�H10 .˝/

C .ru.t/;rv/L2.˝/d

CZ

˝

j0.u.x; t/I v.x// dx � .f .t/; v/L2.˝/:

It follows directly from the definition (3.37) that every solution of above partialdifferential inclusion is also a solution of the hemivariational inequality. Theconverse also holds true, but this has been shown only recently by Carl (see [49]):his proof is more involved and uses the notion of upper and lower solutions.

The Clarke subdifferential is very easy to compute in finite dimensions using thecharacterization (3.38), when the number of points in which the functional is notcontinuously differentiable is a null Lebesgue set.

Example 1. Let us consider the example of the functional ' W R ! R

'.s/ D

8ˆ<

ˆ:

�s � 2 for s < �1;s for s 2 Œ�1; 1�;�s C 2 for s > 1:

3.9 Clarke Subdifferential: Examples 79

This functional is of class C1 on a neighborhood of every s 62 f�1; 1g, and for thepoints f�1; 1g we take convex hull of left and right derivatives

@'.s/ D

8ˆˆˆ<

ˆˆˆ:

f�1g for s < �1;Œ�1; 1� for s D �1;f1g for s 2 .�1; 1/;Œ�1; 1� for s D 1;

f�1g for s > 1:

The potential and its subdifferential are both depicted in Fig. 3.1.

1,5

0,5

−0,5−2 −1,5 −1 −0,5 0,5 1,5 2S10

−1,5

−1

0

1

ϕ(s)1,5

0,5

−0,5

−2 −1,5 −1 −0,5 0,5 1,5 2S10

−1,5

−1

0

1

∂ϕ(s)

Fig. 3.1 The example of a simple locally Lipschitz function, which is nondifferentiable in twopoints, and its Clarke subdifferential

Example 2. Let W Œ0;1/ ! R be a continuous function such that .0/ > 0. Lets0 2 R

d. Define the functional ' W Rd ! R by the formula

'.s/ DZ js�s0j

0

.r/ dr: (3.43)

It is easy to verify, using the mean value theorem for integrals, that ' is locallyLipschitz. It is also easy to calculate the gradient of ' for all s ¤ s0, whichestablishes the fact that ' is of class C1 on the neighborhood of every s ¤ s0.The value of the Clarke subdifferential at s0 is obtained by taking the convex hull ofthe sphere obtained from limits of the gradients of all sequences converging to s0.We get

@'.s/ D8<

:

n.js � s0j/ s�s0js�s0j

ofor s ¤ s0;

fy 2 Rd W jyj � .0/g D B.0; .0// for s D s0:

Graphs of ' and @' for d D 1, s0 D 1, and .r/ D 1 � r are depicted in Fig. 3.2.


0,6

0,5

0,4

0,3

0,2

0,1

0

1,5

0,5

−0,5

−1,5

−1

0

1

−1 −0,5 0,5 1,5 2,50 1 2 3

−1 −0,5 0,5 1,5 2,50 1 2 3

S

S

ϕ(s) ∂ϕ(s)

Fig. 3.2 The second example of a nondifferentiable nonconvex potential and its Clarkesubdifferential

Example 3. Note that it is not required in Example 2 for to be a continuousfunction. For example, we can consider the following function

.r/ D(1 for r 2 Œ0;M/;2 for r 2 ŒM;1/;

where 1;M > 0, and 2 2 R. Then, ' calculated by (3.43) is given by

'.s/ D(1js � s0j for js � s0j 2 Œ0;M/;1M C 2.js � s0j � M/ for js � s0j 2 ŒM;1/:

For s ¤ s0 and js � s0j ¤ M the functional ' is continuously differentiable. Usingthis fact and the characterization (3.38) we can calculate its Clarke subdifferential as

@'.s/ D

8ˆˆ<

ˆˆ:

B.0; 1/ for s D s0;n1

s�s0js�s0jo

for js � s0j 2 .0;M/;˚�M .s � s0/ W � 2 Œ1; 2�

�for js � s0j D M;

n2

s�s0js�s0jo

for js � s0j 2 .M;1/:

The plot of ' and its Clarke subdifferential for d D 1, s0 D 0, 1 D 2, 2 D 1, andM D 1 is presented in Fig. 3.3.

3,5 2,5

1,5

0,5

2

−0,5

−1,5

−2,5

−1

2

1

0

2,5

−1,5−2 −0,5 0,5 1,50 1 2−1

−1,5−2 −0,5 0,5 1,50 1 2−11,5

0,5

0

1

3

2

j(s) ∂j(s)

S

S

Fig. 3.3 The third example of a locally Lipschitz potential and its Clarke subdifferential


Finally, we present a brief comparison of the Clarke subdifferential with theconvex subdifferential. If the functional ' W X ! R [ f1g, where X is a Banachspace, is convex, then its convex subdifferential is defined as

@conv'.s/ D(

; if s … dom';

f� 2 X0 W h�; v � siX0�X � '.v/ � '.s/ 8 v 2 dom'g otherwise;

where dom' is the set of all points in which ' in not equal to 1. If a functional' W X ! R is both convex and locally Lipschitz then both subdifferentials mustcoincide (see, e.g., Proposition 2.2.7 in [72]). There are, however, examples whereonly one of the two notions makes sense. For the functionals which are locallyLipschitz and nonconvex we can speak only of the Clarke subdifferential, while forconvex functionals which assume infinite values (and such functionals are useful, forexample, in constrained optimization or in unilateral problems in solid mechanics)only the convex subdifferential makes sense.


The preliminary material provided in this chapter (and much more) can be found,for example, in the following monographs devoted to the Navier–Stokes equations:[13, 75, 88, 99, 108, 146, 156, 157, 218, 219, 221]. More information aboutdifferential inclusions and theory of multifunctions can be found in [7, 8]. Forthe theory of Clarke subdifferential and hemivariational inequalities we refer thereader to [50, 71, 72, 114, 173, 177, 184]. The applications of this theory in contactmechanics are described in [177, 217].

4Stationary Solutions of the Navier–StokesEquations

Now I think hydrodynamics is to be the root of all physicalscience, and is at present second to none in the beauty of itsmathematics.

– William Thomson, 1st Baron Kelvin

In this chapter we introduce some basic notions from the theory of the Navier–Stokes equations: the function spaces H, V , and V 0, the Stokes operator A with itsdomain D.A/ in H, and the bilinear form B. We apply the Galerkin method andfixed point theorems to prove the existence of solutions of the nonlinear stationaryproblem, and we consider problems of uniqueness and regularity of solutions.

4.1 Basic Stationary Problem

Let Q D Œ0;L�3 and define the spaces

H D fu D .u1; u2; u3/ 2 PL2p.Q/3 W div u D 0g;

and

V D fu D .u1; u2; u3/ 2 PH1p.Q/

3 W div u D 0g;

with the norms

kukH D juj DsZ

Qju.x/j2dx; juj2 D .u; u/;

and

kukV D kuk DsZ

Qjru.x/j2dx; kuk2 D .ru;ru/;


83

84 4 Stationary Solutions of the Navier–Stokes Equations

respectively. V and H are Hilbert spaces and V H. If we identify H with its dualH0 in view of the Riesz–Fréchet representation theorem, we have V H V 0 withcontinuous embeddings and with each space dense in the following one.

4.1.1 The Stokes Operator

We define the Stokes operator A W V ! V 0 by

hAu; vi DZ

Qru � rv dx for all v 2 V: (4.1)

Using this definition we can write the weak formulation of the Stokes problem:Given f 2 V 0, find u 2 V and p 2 PL2.Q/ such that

� ��u C rp D f ; div u D 0; (4.2)

in the weak sense, that is,

�.ru;rv/ � .p; div v/ D hf ; vi for all v 2 PH1p.Q/

3; (4.3)

in an equivalent abstract form

�hAu; vi D hf ; vi for all v 2 V: (4.4)

We shall prove that for f 2 H, the Stokes operator coincides with the minusLaplacian, Au D ��u. To this end we use the explicit spectral representation offunctions in PL2.Q/3. Assume in the beginning that f 2 PL2.Q/3. Then

f .x/ DX

k2Z3

fke2� ik xL ;

X

k2Z3

jfkj2 < 1; f�k D Nfk; f0 D 0; (4.5)

where fk D .f 1k ; f2k ; f

3k /. Let u 2 V and p 2 PL2.Q/ satisfy (4.3). Assuming that

u.x/ DX

k2Z3

uke2� ik xL ; p.x/ D

X

k2Z3

pke2� ik xL (4.6)

with u0 D 0, k � uk D 0, p0 D 0, u�k D Nuk, p�k D Npk we obtain,

�4�2jkj2

L2uk C 2� ik

Lpk D fk:

Multiplying this equation by k and using the relation k � uk D 0 we obtain

pk D L

2� i

fk � k

jkj2 for k 6D 0;

4.1 Basic Stationary Problem 85

and then we compute

uk D 1

�

L2

4�2jkj2�

fk � kfk � k

jkj2�

for k 6D 0:

Now, if f 2 H, then fk � k D 0 and the equation for uk reduces to

uk D 1

�

L2

4�2jkj2 fk for k 6D 0:

Thus, the weak solution u of the Stokes problem (4.4) (with � D 1) coincideswith the weak solution of the problem ��u D f for f 2 H. From this we seeimmediately that u 2 PH2

p.Q/3. Moreover, from the above explicit formulas for the

Fourier coefficients uk and pk we have the following regularity theorem.

Theorem 4.1. If f 2 PL2p.Q/3, then u 2 PH2p.Q/

3, p 2 PH1p.Q/, and

kuk2PH20 .Q/

3 C kpk2PH10 .Q/

� L2

4�2

�L2

�2�2C 1

�

kf k2PL2.Q/3 :

Proof. We have

kuk2PH20 .Q/

3 DX

k2Z3

jkj4jukj2 DX

k2Z3

jkj4 1�2

L4

16�4jkj4ˇˇˇˇfk � k

fk � k

jkj2ˇˇˇˇ

2

� 2X

k2Z3

jkj4 1�2

L4

16�4jkj4 jfkj2 C 2X

k2Z3

jkj4 1�2

L4

16�4jkj4 jfkj2

D L4

4�4�2kf k2PL2.Q/3 ;

and

kpk2PH1p .Q/

DX

k2Z3

jkj2jpkj2 DX

k2Z3

jkj2 L2

4�2

ˇˇˇˇfk � k

jkj2ˇˇˇˇ

2

�X

k2Z3

L2

4�2jfkj2 D L2

4�2kf k2PL2.Q/3 ;

which ends the proof. utRemark 4.1. Observe that setting f , u, and p of the form (4.5) and (4.6), respectively,directly to Eqs. (4.2) we would get the same spectral representations of thesolution u, p.

Thus, in particular,

D.A/ D fu 2 V W Au 2 Hg D V \ PH2p.Q/

3:


4.1.2 The Nonlinear Problem

Let us consider the stationary problem:

��u C .u � r/u C rp D f in Q ;

div u D 0 in Q ;


1. Q D Œ0;L�3 in R3 and we assume periodic boundary conditions, or


homogeneous Dirichlet boundary condition u D 0 on @Q.

In the second case we define

H D fu D .u1; u2; u3/ 2 L2.Q/3 W div u D 0; un D u � nj@Q D 0g;

and

V D fu D .u1; u2; u3/ 2 H10.Q/

3 W div u D 0g ;

with the norms

kukH D juj DsZ

Qju.x/j2 dx; juj2 D .u; u/;

and

kukV D kuk DsZ

Qjru.x/j2 dx; kuk2 D .ru;ru/:

The Stokes operator A is defined in the same way as in the periodic case (cf. (4.1))and we have

D.A/ D fu 2 V W Au 2 Hg D V \ H2.Q/3:

For both cases, let us denote b.u; v;w/ D ..u � rv/;w/ for u; v;w 2 V and define anonlinear operator B W V ! V 0 by .Bu; v/ D b.u; u; v/ for all v 2 V . We shall alsowrite ..u; v// instead of .ru;rv/.Exercise 4.1. Prove that for all u; v;w 2 V it is

b.u; v;w/ D �b.u;w; v/;

and thus b.u; v; v/ D 0.


Exercise 4.2. Prove that for all u; v;w 2 V ,

jb.u; v;w/j � C.Q/kukkvkkwk:

Hint. Use the Hölder inequality and the continuous embedding of L4.Q/ in V .Let H and V be the corresponding function spaces. Then the weak formulation

of the above nonlinear stationary problem is as follows.

Problem 4.1. For f 2 H (or f 2 V 0) find u 2 V such that

�..u; v//C b.u; u; v/ D hf ; vi for all v 2 V

or, equivalently,

�Au C Bu D f in H or V 0:

Remark 4.2. Concerning regularity of solutions one can prove that if f 2 H then allsolutions u belong to D.A/.

We shall prove the following existence theorem.

Theorem 4.2. Let Q D Œ0;L�3 (for the periodic problem) or let Q be a boundeddomain in R

3 with smooth boundary (for the Dirichlet problem). Then

(i) For every f 2 V 0 and � > 0 there exists at least one solution of Problem 4.1.(ii) If �2 � c1.Q/kf kV0 , then the solution of Problem 4.1 is unique.

Proof. ad .i/. To prove existence of solutions we use the Galerkin method. For everym 2 N let

um.x/ DmX

jD1�j;m!j.x/; �j;m 2 R;

be the approximate solution, where !j, j 2 N are the eigenfunctions of the Stokesoperator corresponding to the considered problem. This means that

�..um; v//C b.um; um; v/ D hf ; vi for all v 2 Vm ; (4.7)

where Vm D spanf!1; : : : ; !mg. Such solution exists in view of the Brouwer fixedpoint theorem.

Exercise 4.3. Prove that the operator A W H D.A/ ! H is a self-adjointpositive operator in H and is an isomorphism from D.A/ onto H, and its inverseA�1 W H ! H is continuous, compact, and self-adjoint so there exists a sequence�j, j 2 N, 0 < �1 � �2 � �3 � : : :, �j ! 1, and a family of elements !j 2 D.A/orthonormal in H such that A!j D �j!j for j 2 N, cf. Theorem 3.15.

Exercise 4.4. Prove existence of um using the Brouwer fixed point theorem.


Solution to Exercise 4.4. Consider the ball B D fu 2 Vm W kuk � 1�kf kV0g in Vm and

the map ˚ W Ou ! u where Ou 2 B and u is the unique solution of the linear problem

�..u; v//C b.Ou; u; v/ D hf ; vi for all v 2 Vm:

Setting v D u in this identity we obtain that u 2 B. We shall show that ˚ iscontinuous in Vm. To this end, let

�..w; v//C b. Ow; u; v/ D hf ; vi for all v 2 Vm:

Subtracting the equation for w from that for u, setting v D u � w, and using theestimate for the solution v, we obtain

ku � wk � C.Q/

�2kf kV0kOu � Owk ;

which proves the continuity of ˚ . As Vm is a finite dimensional space and ˚ is acontinuous map from a convex and compact set B to B, we conclude existence ofum, solution of (4.7) in view of the Brouwer fixed point theorem.From (4.7) we conclude

kumk � kf kV0

�:

Thus, for a subsequence,

um ! u weakly in V;

and

um ! u strongly in H;

as V is compactly embedded in H. Passing with m to the limit in Eq. (4.7) we obtainequation

�..u; v//C b.u; u; v/ D hf ; vi

for all v being a finite linear combination of elements of the basis !k, k 2 N. Sincesuch elements v are dense in V , the existence part of the theorem has been proved.

Proof of .ii/. To prove uniqueness of solutions for large viscosity coefficientswith respect to mass forces, �2 > c1.Q/kf kV0 , let us assume that there are twodistinct solutions u1 and u2, that is,

�..u1; v//C b.u1; u1; v/ D hf ; vi;

and

�..u2; v//C b.u2; u2; v/ D hf ; vi


for all v 2 V . Setting v D u1 � u2 and subtracting the second equation from the firstone we obtain

�ku1 � u2k2 D �b.u1 � u2; u2; u1 � u2/ � c1.Q/ku1 � u2k2ku2k

� c1.Q/

�kf kV0ku1 � u2k2;

as ku2k � 1�kf kV0 , whence

�

� � c1.Q/

�kf kV0

�

ku1 � u2k2 � 0 ;

which gives a contradiction. Thus u1 D u2. utExercise 4.5. Prove that Theorem 4.2 is valid also in the two-dimensional case,when Q R

2 is a square (in the periodic case) or a bounded domain with smoothboundary (in the case of the Dirichlet boundary conditions).

In the end we shall show how one can proceed differently to prove existence of theGalerkin approximate solutions um satisfying (4.7).

Theorem 4.3. Let X be a finite dimensional Hilbert space, with a scalar productŒ�; �� and the associated norm Œ��, and let P W X ! X be a continuous map such that

ŒP.�/; �� > 0 for Œ�� D k > 0

for some k. Then there exists � 2 X with Œ�� k for which P.�/ D 0.

Proof. Let B D B.0; k/ be a closed ball in X, centered at zero and with radius k.Assume that P is different from zero in this ball. Then the map

� ! S.�/ D � kP.�/

ŒP.�/�; S W B ! B (4.8)

is continuous in X. From the Brouwer fixed point theorem it follows that S has afixed point in B, that is,

�0 D � kP.�0/

ŒP.�0/�(4.9)

for some �0 2 B. We have Œ�0� D k. Multiplying (4.9) by �0 we get

Œ�0�2 D �kŒP.�0/; �0�

ŒP.�0/�;

which contradicts ŒP.�/; �� > 0 for Œ�� D k. Thus P.�/ D 0 for some � 2 B. ut


Now, let X D Vm with the norm induced from V . Let us defineP D Pm W Vm ! Vm by the relation,

ŒPm.u/; v� D ..Pm.u/; v// D �..u; v//C b.u; u; v/ � hf ; vi (4.10)

for all u; v 2 Vm. The map is well defined in view of the Riesz–Fréchet representa-tion theorem. In fact, for every u in Vm the map v ! �..u; v//C b.u; u; v/ � hf ; videfines a linear and continuous functional on Vm. Thus, there exists a unique Pm.u/in Vm satisfying the above relation.

The map Pm is continuous in Vm and

ŒPm.u/; u� D �kuk2 C b.u; u; u/ � hf ; ui D �kuk2 � hf ; ui� �kuk2 � kf kV0kuk D kukf�kuk � kf kV0g:

Whence, for k > kf kV0

�and kuk D k we have ŒPm.u/; u� > 0. In view of our auxiliary

theorem, there exists um in Vm such that Pm.um/ D 0, that is, (4.7) is satisfied. utExercise 4.6. Prove that the maps S defined in (4.8) and Pm defined in (4.10) arecontinuous.

4.1.3 Other Topological Methods to Deal with the Nonlinearity

In this subsection we shall use the Schauder and, alternatively, the Leray–Schauderfixed points theorems to prove directly the existence of a solution of the nonlinearproblem, thus bypassing the Galerkin approximate solutions. When the viscosity islarge enough with respect to external forces, we shall use the Banach contractionprinciple to prove the existence of the unique solution of the nonlinear problem.

Exercise 4.7. Let f 2 V 0, Q R3 as above. Prove that there exists u 2 V such that


by using

(a) the Schauder fixed point theorem,(b) the Leray–Schauder fixed point theorem,(c) assuming that the viscosity is large enough with respect to external forces, use

the Banach contraction principle to prove the unique solution u.

Solution to Exercise 4.7. (a) Let us define a map T in V by

�..Tu; v//C b.u;Tu; v/ D hf ; vi for all v 2 V :


Using the Lax–Milgram lemma we conclude that the map T is well defined. Settingv D Tu we get

�kTuk2 D hf ;Tui � kf kV0kTuk ;

whence

kTuk � 1

�kf kV0 � r:

Define K D fv 2 V W kvk � rg. Then T.K/ K. Let us consider the mapT W K ! K. We shall show that T is continuous and compact in the topology of V .Let u1, u2 be in K. We have,

�..Tu1; v//C b.u1;Tu1; v/ D hf ; vi for all v 2 V;

and

�..Tu2; v//C b.u2;Tu2; v/ D hf ; vi for all v 2 V:

Subtracting the second equation from the first one we get

�..Tu1 � Tu2; v//C b.u1 � u2;Tu2; v/ C b.u1;Tu1 � Tu2; v/ D 0

for all v in V . Set v D Tu1 � Tu2 and we get, since b.u1;Tu1 � Tu2;Tu1 � Tu2/ D 0,

kTu1 � Tu2k � C

�kTu2kju1 � u2jL4.Q/3 � C1

�rku1 � u2k:

The continuity follows immediately. To prove compactness observe that anysequence .uk/ K contains a subsequence which converges strongly in the L4

topology. Thus, compactness follows easily from the first of the above inequalities.(b) Let us define the map u ! Tu in V by

�..Tu; v//C b.u; u; v/ D hf ; vi for all v 2 V : (4.11)

Using the Riesz–Fréchet representation theorem we conclude that the map T is welldefined.

Assume that �Tu D u, where � 2 Œ0; 1�. We shall prove that all such u are insome ball. If � D 0, then u D 0. Now, for � > 0, Tu D 1

�u. Setting v D u in (4.11)

we obtain �kuk2 � �kf kV0kuk, whence kuk � 1�kf kV0 � M.

Now, we shall prove continuity and compactness of T in V . Let um; uk be in V .Then, subtracting the equations for Tum and Tuk, and setting v D Tum � Tuk weobtain


�kTum � Tukk2 D b.um;Tum � Tuk; um � uk/C b.um � uk;Tum � Tuk; uk/

� jumjL4.Q/3kTum � Tukkjum � ukjL4.Q/3 Cjum � ukjL4.Q/3kTum � TukkjukjL4.Q/3 ;

whence,

kTum � Tukk � 1

�.jumjL4.Q/3 C jukjL4.Q/3 /jum � ukjL4.Q/3

� C

�.jumjL4.Q/3 C jukjL4.Q/3 /kum � ukk:

To prove the continuity of T in V observe that if um ! uk in V then from the aboveestimate, kTum � Tukk ! 0. To prove the compactness of T in V , let fumg be abounded sequence in V . Then, it contains a convergent subsequence fug in L4.Q/3.Thus, from the above estimate it follows that the sequence fTug is convergent.

Using the Leray–Schauder fixed point theorem we conclude that there exists u inV such that Tu D u. This u is of course a solution of our stationary Navier–Stokesproblem.(c) Let us define the map T W V ! V by

�..Tu; v//C b.u;Tu; v/ D hf ; vi for all v 2 V: (4.12)

The map is well defined. We shall prove that it is contracting in a ball B.0; r/ in Vprovided the viscosity is large enough with respect to external forces. Let

�..Tu1; v//C b.u1;Tu1; v/ D hf ; vi for all v 2 V:

We get

�..Tu � Tu1; v//C b.u;Tu � Tu1; v/C b.u � u1;Tu1; v/ D 0

for all v 2 V , whence, with v D Tu � Tu1,

kTu � Tu1k � C

�kTu1kku � u1k:

From (4.12) it follows that for every u 2 V , kTuk � 1�kf kV0 . Therefore,

kTu � Tu1k � C

�2kf kV0ku � u1k:

Consider the closed ball B D B.0; r/ in V with r D 1�kf kV0 . Assume moreover

that C�2

kf kV0 < 1. Then the map T has the following properties: T W B ! Band is contracting on B. By the Banach contraction principle, there exists a uniqueu 2 B.0; r/ such that Tu D u. This u is the unique solution of the stationary Navier–Stokes problem.


Exercise 4.8. Prove the Ladyzhenskaya inequality (3.8) for all functions u 2PH1

p.Q/, Q D Œ0;L�2, knowing that it holds for all functions u 2 H10.˝/, where

˝ is a bounded set in R2.

Exercise 4.9. Using the methods from Exercise 4.7 prove existence of solutions ofthe stationary Navier–Stokes problems in the two-dimensional case, for periodicand homogeneous boundary conditions, respectively. You may find useful theLadyzhenskaya inequality (3.8).


There are many other important and interesting problems concerning stationarysolutions of the Navier–Stokes equations. We mention problems of flows withnonhomogeneous Dirichlet boundary conditions or with other types of boundaryconditions, flows in nonsmooth, unbounded, or exterior domains. Still other prob-lems concern behavior of solutions when the viscosity tends to zero, bifurcation ofsolutions, and the structure of the set of stationary solutions.

In Chaps. 5 and 6 we shall consider stationary flows appearing in applications inthe theory of lubrication.

For thorough treatment of the basic mathematical problems of the Navier–Stokesequations we refer the reader to monographs: [75, 88, 99, 108, 146, 156, 218, 219],and, for the compressible case, to [157, 187].

5Stationary Solutions of the Navier–StokesEquations with Friction

In this chapter we consider the three-dimensional stationary Navier–Stokesequations with multivalued friction law boundary conditions on a part of the domainboundary. We formulate two existence theorems for the formulated problem. Thefirst one uses the Kakutani–Fan–Glicksberg fixed point theorem, and the secondone, with the relaxed assumptions, is based on the cut-off argument.

5.1 Problem Formulation

Let ˝ R3 be a bounded domain with Lipschitz boundary as schematically

presented in Fig. 5.1. The boundary @˝ is divided into two disjoint parts @˝ D�D [ �C. Assume that the viscosity � is a positive constant and the mass forcedensity f does not depend on time. Consider the stationary system of the Navier–Stokes equations

� ��u C .u � r/u C rp D f in ˝; (5.1)

div u D 0 in ˝; (5.2)

with the following boundary conditions.On �D we assume the Dirichlet boundary condition u D 0. Moreover we let

un D 0 on �C; (5.3)

�T� 2 h.u� / on �C; (5.4)

where un D u � n is the outward normal component of the velocity vector, its tangentcomponent is given by u� D u�unn on �C, and T� D Tn� .Tn �n/n is the tangentialpart of the stress vector Tn


95

96 5 Stationary Solutions of the Navier–Stokes Equations with Friction

T D �pI C 2�D.u/; (5.5)

[cf. (2.24)–(2.25)], and h W �C � R3 ! 2R

3is a friction multifunction.

Fig. 5.1 The domain of the studied problem. The set ˝ is a space between two cylinders. TheDirichlet boundary is the wall of the outer cylinder, and two rings bounding ˝ at top and bottom.The contact boundary �C is the wall of the inner, rotating, cylinder. On �D we impose thehomogeneous Dirichlet boundary condition, and on �C we impose the frictional contact conditionwhich allows for the occurrence of either stick, or slip between the liquid and the rotating cylinder

Exercise 5.1. Prove that T� D 2�D.u/� on �C, where D.u/� D D.u/n � D.u/n � n.

5.2 Friction Operator and Its Properties

We will prove the existence of a weak solution for two cases. In the first case, inSect. 5.4 we will assume the linear growth condition on the friction multifunctionh (see .h4/ below) with the restriction on the growth constant (see .h5/ below) andwe will use the Kakutani–Fan–Glicksberg theorem to obtain the solution. Next, inSect. 5.5 we will relax the conditions on h, namely instead of the linear growthcondition we will assume the weaker growth condition with exponent p � 1 where2 � p < 4 (see .h6/ below) This possibility comes from the fact that in three spacedimensions the trace operator is compact from the space H1.˝/ to Lp.@˝/ with1 � p < 4. Moreover we will need no restriction on the constant in the growthcondition. This restriction is clearly nonphysical, i.e., it gives a limitation on themagnitude of the boundary friction force no matter if this force is “dissipative”(i.e., directed opposite to the velocity) or “excitatory” (i.e., directed the same as thevelocity). We will replace .h5/with its one sided counterpart denoted by .h7/ below.This condition enforces the limitation on the friction force only if it has the samedirection as the velocity. The proof of existence for the relaxed assumptions will usethe cut-off argument.

5.2 Friction Operator and Its Properties 97

We assume the following properties of the friction multifunction h:

.h1/ for every s 2 R3 the multifunction x ! h.x; s/ has a measurable selection,

i.e., there exists gs W �C ! R3, a measurable function, such that gs.x/ 2 h.x; s/

for all x 2 �C,.h2/ the set h.x; s/ is nonempty, compact, and convex in R

3, for all s 2 R3 and

for a.e. x 2 �C,.h3/ the graph of s ! h.x; s/ is closed in R

6 for a.e. x 2 �C,.h4/ for all s 2 R

3, a.e. x 2 �C and all � 2 h.x; s/, we have j�j � c1.x/ C c2jsj,with some c2 � 0 and c1 2 L2.�C/.

Denote W D L2.�C/3 and define a multivalued operator HW W ! 2W by

H.u/ D f � 2 W W �.x/ 2 h.x; u.x// a.e. on �C g: (5.6)

The following Lemma is a straightforward consequence of Theorem 3.23.

Lemma 5.1. Assume the multifunction h satisfies .h1/–.h4/. Then the operator Hgiven by (5.6) satisfies


.H2/ the graph of H is sequentially closed in .W; strong/ � .W;weak/ topology,

.H3/ for all w 2 W and � 2 H.u/, k�kW � kc1kL2.�C/ C c2 kukW.

For the first existence result we will need the additional bound on the constant c2in the growth condition .h4/, dependent on the viscosity coefficient and the norm ofthe trace operator k�k,

.h5/ c2k�k2 < 2�.For the second existence result, the assumptions .h4/ and .h5/ will be replacedby the following, weaker ones:

.h6/ there exists 2 � p < 4 such that for all s 2 R3 we have

max�2h.x;s/

j�j � c3.x/C c4jsjp�1 for a.e. x 2 �C;

with some c3 2 L2.�C/ and c4 � 0,and

.h7/ For all s 2 Rd, some d1 2

h0; 2�

k�k2�

and d2 2 L2.�C/ we have

min�2h.x;s/

� � s � �d1jsj2 � d2.x/ for a.e. x 2 �C:

When .h4/ is replaced by .h6/, the multivalued friction operator is defined onLp.�C/

3 in place of L2.�C/3. We will denote U D Lp.�C/

3. Then U0 D Lq.�C/3

with 1p C 1

q D 1 and q 2 � 43; 2�. Define the multifunction G W U ! 2U0

by

G.u/ D f� 2 U0 W �.x/ 2 h.x; u.x// for a.e. x 2 �Cg: (5.7)

Analogously to Lemma 5.1 we have the following result which is a straightforwardconsequence of Theorem 3.23.


Lemma 5.2. Assume the multifunction h satisfies .h1/–.h4/. Then the operator Ggiven by (5.7) satisfies

.G1/ G has nonempty and convex values,

.G2/ the graph of G is sequentially closed in .U; strong/ � .U0;weak/ topology,

.G3/ for all w 2 W and � 2 G.u/, k�kU0 � kc3kLq.�C/ C c4 kukU.

5.3 Weak Formulation

Let

QV D fu 2 C1.˝/3 W div u D 0; u D 0 on �D; un D 0 on �Cg

and

V D closure of QV in H1.˝/3 norm; W D L2.�C/3:

If v 2 V then we have the following Korn inequality (see, for instance, [194] for aproof of this inequality in a general setting)

kvk2H1.˝/3� cK

Z

˝

jD.u/j2 dx: (5.8)

Note that the Korn inequality implies that k � k, where

kuk DsZ

˝

jD.u/j2 dx (5.9)

is a norm on the space V equivalent to the standard H1.˝/3 norm.We denote by � W V ! W the trace operator and by k�k its norm k�kL .VIW/.We can pass to the derivation of the weak formulation of the studied problem. Tothis end we assume that u, p, and T are smooth enough and satisfy (5.1)–(5.4). Wemultiply (5.1) by v 2 QV and integrate over ˝, which gives

� �Z

˝

ui;jjvi dx CZ

˝

ujui;jvi dx CZ

˝

p;ivi dx DZ

˝

fivi dx: (5.10)

Integrating by parts the term with pressure we obtain

Z

˝

p;ivi dx D �Z

˝

pvi;i dx CZ

@˝

pvn dS:

5.3 Weak Formulation 99

Using the fact that div v D 0 and vn D 0 on the whole @˝, it follows that

Z

˝

p;ivi dx D 0: (5.11)

As div u D 0, we have, for smooth u, that uj;ij D 0 and hence

ui;jjvi D .ui;jj C uj;ij/vi D .ui;j C uj;i/;jvi :

From the following Green formula valid for a smooth matrix valued function F(F W ˝ ! M3�3) and v 2 V

Z

˝

Fij;jvi C Fijvi;j dx DZ

@˝

Fijvinj dS; (5.12)

we have

��Z

˝

ui;jjvi dx D �2�Z

˝

D.u/ij;jvi dx

D 2�

Z

˝

D.u/ijvi;j dx � 2�Z

@˝

D.u/nvn C D.u/� � v� dS

D 2�

Z

˝

D.u/ijvi;j dx �Z

�C

T� � v� dS:

Finally, by the symmetry of D.u/

Z

˝

D.u/ijvi;j dx D 1

2

Z

˝

D.u/ijvi;j dx C 1

2

Z

˝

D.u/jivj;i dx

D 1

2

Z

˝

D.u/ijvi;j dx C 1

2

Z

˝

D.u/ijvj;i dx

DZ

˝

D.u/ijD.v/ij dx: (5.13)

In this way we have proved that

� �Z

˝

ui;jjvi dx D 2�

Z

˝

D.u/ijD.v/ij dx �Z

�C

T� � v� dS: (5.14)

Exercise 5.2. Prove the Green formula (5.12).

Using (5.11) and (5.14) in (5.10) we can justify the following weak formulationof the considered problem.


Problem 5.1. Let f 2 L2.˝/3. Find u 2 V and � 2 H.u� / such that for every v 2 V

2�

Z

˝

D.u/ijD.v/ij dx CZ

�C

� � v� dS CZ

˝

.u � r/u � v dx DZ

˝

f � v dx: (5.15)

Remark 5.1. If �C is a piecewise flat surface (i.e., �C D SKkD1 � k

C , where n D conston each � k

C ), then for u; v 2 V

2

Z

˝

D.u/ijD.v/ij dx DZ

˝

ru � rv dx; (5.16)

and (5.15) could be replaced by

�

Z

˝

ru � rv dx CZ

�C

� � v� d� CZ

˝

.u � r/u � v dx DZ

˝

f � v dx:

Let us prove (5.16). To this end first assume that u 2 V and v 2 QV . On each � kC

we can construct an orthonormal basis n; �1; �2, where both vectors �1; �2 lie in thesurface � k

C . Take v 2 QV . On � kC we have rvn D @vn

@n nC @vn@�1�1C @vn

@�1�1 D @vn

@n n, where

the last equality follows from the fact that vn D 0 on � kC . Now, we have

Z

˝

uj;ivi;j dx D �Z

˝

ujvi;ij dx CZ

@˝

ujvi;jni dS:

As div v D 0, and u D 0 on �D, we have

Z

˝

uj;ivi;j dx DZ

�C

ujvi;jni dS DZ

�C

u � @vn

@nn dS D

Z

�C

.unn C u� / � @vn

@nn dS D 0:

The last equality follows from the fact that un D 0 and u� is orthogonal to n on �C.Using (5.13) we have

Z

˝

D.u/ijD.v/ij dx DZ

˝

D.u/ijvi;j dx D 1

2

�Z

˝

ui;jvi;j dx CZ

˝

uj;ivi;j dx

�

D 1

2

Z

˝

ui;jvi;j dx;

and the assertion follows by the density of QV in V .

We introduce the linear operator A W V ! V 0 and the bilinear operator B W V � V !V 0 given by

hAu; vi D 2

Z

˝

D.u/ijD.v/ij dx for u; v 2 V; (5.17)

5.3 Weak Formulation 101

and

hB.u;w/; vi DZ

˝

.u � r/w � v dx for u;w; v 2 V;

respectively, and the functional F 2 V 0 given by

hF; vi DZ

˝

f � v dx for v 2 V:

Note that, for u; v;w 2 V the integral in the definition of B is well defined, asfrom the Sobolev embedding theorem it follows that H1.˝/ L4.˝/. Using thisnotation, Problem 5.1 can be equivalently reformulated as follows.

Problem 5.2. Find u 2 V and � 2 H.u� / such that for every v 2 V we have

h�Au C B.u; u/; vi C .�; v� /W D hF; vi:

We end this section by proving several auxiliary properties of the operators A and B.

Lemma 5.3. The operator A W V ! V 0 defined by (5.17) is bilinear, continuous,and coercive. Moreover, we have

hAu; ui D 2kuk2 for u 2 V:

Proof. The assertion follows in a straightforward way from the definition of A andthe definition of the norm in V , cf. (5.9). utLemma 5.4. For u; v;w 2 V we have

hB.u;w/; vi � cBkukkwkkvk; (5.18)

hB.u; v/; vi D 0; (5.19)

hB.u;w/; vi D �hB.u; v/;wi: (5.20)

Proof. We first prove (5.18). By the Hölder inequality

hB.u;w/; vi � kukL4.˝/3krwkL2.˝/3�3kvkL4.˝/3 :

From the Sobolev embedding theorem it follows that kzkL4.˝/3 � CkzkH1.˝/3 for allz 2 H1.˝/3 with a constant C > 0. Hence, we have

hB.u;w/; vi � C2kukH1.˝/3kwkH1.˝/3kvkH1.˝/3 ;

and (5.18) follows by the Korn inequality (5.8).


For the proof of (5.19), let u; v 2 QV . We have

hB.u; v/; vi DZ

˝

ujvi;jvi dx D 1

2

Z

˝

uj

0

@3X

jD1v2i

1

A

;j

dx

D �12

Z

˝

uj;j

3X

iD1v2i

!

dx C 1

2

Z

@˝

ujnj

3X

iD1v2i dS D 0;

where the last equality follows from the fact that div u D 0 in ˝ and un D u � n D 0

on @˝. Now as QV is dense in V , for u; v 2 V we can construct sequences um; vm 2 QVsuch that kv � vmk ! 0 and ku � umk ! 0 as m ! 1. Hence

hB.u; v/; vi D hB.u � um C um; v � vm C vm/; v � vm C vmiD hB.u � um; v/; vi C hB.um; v � vm/; vi

C hB.um; vm/; v � vmi C hB.um; vm/; vmi:We use the fact that hB.um; vm/; vmi D 0 and (5.18) to get

jhB.u; v/; vij � cB.ku � umkkvk2 C kv � vmkkumk.kvk C kvmk//:The sequences um; vm are bounded in V as they converge in this space, and thesequences ku � umk and kv � vmk converge to zero, whence the assertion is proved.Finally, (5.20) is a straightforward consequence of (5.19). For the proof it sufficesto substitute v C w in place of v in (5.19). utLemma 5.5. The mapping V2 3 .w; u/ ! B.w; u/ 2 V 0 is sequentially weaklycontinuous, i.e.

wk ! w and uk ! u weakly in V (5.21)

) hB.wk; uk/; vi ! hB.w; u/; vi for all v 2 V:

Proof. First, take v 2 QV and let uk ! u;wk ! w weakly in V . We will prove that

hB.wk; v/; uki ! hB.w; v/; ui: (5.22)

As the embedding V L2.˝/3 is compact we have uk ! u and wk ! w stronglyin L2.˝/3. Moreover,

hB.wk; v/; uki � hB.w; v/; ui D hB.wk; v/; uk � ui C hB.wk � w; v/; ui;whence

jhB.wk; v/; uki � hB.w; v/; uij �kvkC1.˝/3.kwkkL2.˝/3kuk � ukL2.˝/3 C kwk � wkL2.˝/3kukL2.˝/3/;

5.4 Existence of Weak Solutions for the Case of Linear Growth Condition 103

whereas (5.22) is proved, and by (5.20)

hB.wk; uk/; vi ! hB.w; u/; vi: (5.23)

Now, let v 2 V and let vm 2 QV be a sequence such that kvm � vk ! 0 as m ! 1.We have

hB.wk; uk/; vi D hB.wk; uk/; vmi C hB.wk; uk/; v � vmi:From the fact that the sequences wk; uk are bounded in V it follows that

limm!1 lim sup

k!1jhB.wk; uk/; v � vmij � lim

m!1 lim supk!1

cBkwkkkukkkv � vmk D 0;

which yields (5.21) and the proof is complete. ut

5.4 Existence of Weak Solutions for the Case of LinearGrowth Condition

Define

S D(

.�;w/ 2 W � V W k�kW � 2�kc1kL2.�C/ C c2k�kkFkV0

2� � c2k�k2 ;

kwk � k�kkc1kL2.�C/ C kFkV0

2� � c2k�k2)

: (5.24)

We formulate the main theorem of this section.

Theorem 5.1. Assume .h1/–.h5/. Then Problem 5.1 has a solution .�; u/ 2 S.Moreover, any solution of Problem 5.1 must belong to S.

The proof of the above theorem will be done through several lemmas. First, weformulate the following auxiliary problem.

Problem 5.3. Fix � 2 W and w 2 V . Find u 2 V such that for every v 2 V we have

h�Au C B.w; u/; vi C .�; v� /W D hF; vi: (5.25)

Lemma 5.6. For a given .�;w/ 2 W � V the auxiliary Problem 5.3 has exactly onesolution u 2 V.

Proof. Define the mapping M W V ! V 0 by M.u/ D �Au C B.w; u/. Obviouslythis mapping is linear. By Lemma 5.3 and (5.18), M is bounded, and hence alsocontinuous. Moreover, by Lemma 5.3 and (5.13)

hMu; ui D �hAu; ui D �2kuk2;


whence M is also coercive. Consider the functional G W V ! R defined as G.v/ DhF; vi � .�; v� /W . Then G is linear and

jG.v/j � kFkV0kvk C k�kWkv�kW � .kFkV0 C k�kWk�k/kvk;

whence G is bounded and, in consequence, it belongs to V 0. By the Lax–Milgramlemma, it follows that Problem 5.3 has exactly one solution u 2 V . utIn view of the unique solvability of Problem 5.3, we can define the mapping�1 W W � V ! V which assigns to the pair .�;w/ 2 W � V the unique solutionu 2 V of (5.25). We can write

h�A�1.�;w/C B.w; �1.�;w//; vi C .�; v� /W D hF; vi: (5.26)

Lemma 5.7. For every .�;w/ 2 W � V we have

k�1.�;w/k � k�k2�

k�kW C kFkV0

2�:

Proof. Taking the test function v D �1.�;w/ in (5.26) we get

2�k�1.�;w/k2 C .�;�1.�;w/� /W D hF; �1.�;w/i:

It follows that

2�k�1.�;w/k2 � k�kWk�kk�1.�;w/k C kFkV0k�1.�;w/k;

whence we have the assertion. utLemma 5.8. If �k ! � weakly in W, wk ! w weakly in V, and �1.�k;wk/ ! �

weakly in V, then � D �1.�;w/, i.e., the graph of �1 is a sequentially closed set in.W;weak/ � .V;weak/ � .V;weak/ topology.

Proof. Let �k ! � weakly in W, wk ! w weakly in V , and �1.�k;wk/ ! � weaklyin V . We have

h�A�1.�k;wk/C B.wk; �1.�k;wk//; vi C .�k; v� /W D hF; vi;

for v 2 V . We pass to the limit in all terms in the last equation. As A is linear andcontinuous, it is also weakly sequentially continuous, and we have

h�A�1.�k;wk/; vi ! h�A�; vi:

Lemma 5.5 implies that

hB.wk; �1.�k;wk//; vi ! hB.w; �/; vi;

5.4 Existence of Weak Solutions for the Case of Linear Growth Condition 105

and weak convergence �k ! � in W implies that

.�k; v� /W ! .�; v� /W :

Summarizing, we get

h�A� C B.w; �/; vi C .�; v� /W D hF; vi;whereas � D �1.�;w/ and the proof is complete. ut

We define the multifunction � W W � V ! 2W�V by the formula

�.�;w/ D H.w� / � f�1.�;w/g:

We will show that� satisfies all assumptions of the Kakutani–Fan–Glicksberg fixedpoint theorem, cf. Theorem 3.8, and, in consequence, � has a fixed point .�;w/.We observe that every fixed point of � must be a solution of Problem 5.1. Indeed,assume that .�;w/ is a fixed point of �. Then � 2 H.w� / and w D �1.�;w/. Fromthe definition of �1 it follows that

h�Aw C B.w;w/; vi C .�; v� /W D hF; vi for all v 2 V:

Clearly, .�;w/ solves Problem 5.1. Hence, be showing the existence of a fixed pointof � we prove the existence of a solution for Problem 5.1.

Lemma 5.9. Assume .h1/–.h5/. Then �.S/ S and if x 2 �.x/, then x 2 S.

Proof. First we prove the assertion that �.S/ S. Let .�; u/ 2 �.�;w/, where.�;w/ 2 S. Then � 2 H.w� / and u D �1.�;w/. From Lemma 5.7 it follows that

kuk � k�k2�

k�kW C kFkV0

2�: (5.27)

By Lemma 5.1 it is possible to use the growth condition .H3/. We get

k�kW � kc1kL2.�C/ C c2k�kkwk: (5.28)

As .�;w/ 2 S, we can use the bounds given in the definition (5.24). Theestimate (5.27) implies that

kuk � k�k2�

2�kc1kL2.�C/ C c2k�kkFkV0

2� � c2k�k2 C kFkV0

2�D k�kkc1kL2.�C/ C kFkV0

2� � c2k�k2 ;

and the estimate (5.28) implies that

k�kW � kc1kL2.�C/Cc2k�kk�kkc1kL2.�C/ C kFkV0

2� � c2k�k2 D 2�kc1kL2.�C/ C c2k�kkFkV0

2� � c2k�k2 :

We have proved that u and � satisfy the estimates given in definition (5.24) of theset S, whence .�; u/ 2 S. The proof of the first assertion is complete.


For the proof of the second assertion assume that .�; u/ 2 �.�; u/. We mustshow that .�; u/ 2 S, i.e., .�; u/ satisfy the bounds given in definition (5.24). ByLemma 5.7 and the growth condition .H3/ in Lemma 5.1 we have

kuk � k�k2�

k�kW C kFkV0

2�; k�kW � kc1kL2.�C/ C c2k�kkuk:

Combining the above two inequalities in two different ways we obtain

kuk � k�k2�

.kc1kL2.�C/ C c2k�kkuk/C kFkV0

2�;

k�kW � kc1kL2.�C/ C c2k�k�k�k2�

k�kW C kFkV0

2�

�

:

A straightforward calculation proves that .�; u/ satisfy the bounds given in thedefinition (5.24), whereas indeed .�; u/ 2 S. The proof is complete. utThe space W � V endowed with the .W;weak/ � .V;weak/ topology is a locallyconvex Hausdorff topological vector space, and the set S is nonempty, compact, andconvex in this space. Assuming .h1/–.h4/, the values of� are nonempty and convex(see .H1/ of Lemma 5.1).

Lemma 5.10. Assume .h1/–.h5/. Then the graph of � is sequentially closed in.W;weak/ � .V;weak/ � .W;weak/ � .V;weak/.

Proof. Let �k ! � weakly in W and wk ! w weakly in V . Let moreover .�k; uk/ besuch that �k ! � weakly in W, uk ! u weakly in V and .�k; uk/ 2 �.�k;wk/. Thelast assertion means that �k 2 H.wk� / and uk 2 �1.�k;wk/. Lemma 5.8 implies thatu 2 �1.�;w/. Since the trace operator � W V ! W is compact, we have wk� ! w�strongly in W. It remains to use .H2/ of Lemma 5.1 to conclude that � 2 H.w� /,whence .�; u/ 2 �.�;w/. The proof is complete. utLemma 5.11. Assume .h1/–.h5/. Then the graph of �jS is a closed set in topology.W;weak/ � .V;weak/ � .W;weak/ � .V;weak/.

Proof. Lemma 5.10 implies that the graph of �jS is sequentially closed in topology.W;weak/ � .V;weak/ � .W;weak/ � .V;weak/. The graph of �jS is containedin S � S, a set which is compact, and, by the Eberlein–Šmulyan theorem alsosequentially compact in .W;weak/� .V;weak/� .W;weak/� .V;weak/ topology.Thus the graph of �jS must be sequentially compact, and by the Eberlein–Šmulyantheorem, it is also compact and hence closed. utWe have all the ingredients to prove that main theorem of this section.

Proof (of Theorem 5.1). The assertion of the theorem is a straightforward appli-cation of the Kakutani–Fan–Glicksberg fixed point theorem and Lemmata 5.9and 5.11. ut

5.5 Existence of Weak Solutions for the Case of Power Growth Condition 107

5.5 Existence of Weak Solutions for the Case of PowerGrowth Condition

As in this section we no longer impose the linear growth condition .h4/, but wereplace it with the more general power growth condition .h6/, we cannot interpretanymore the term

R�C� � v� dS in (5.15) as the scalar product in W. Instead, we

represent this term using the duality between U and U0, and the multivalued operatorG given by (5.7). We are in position to formulate the following problem.

Problem 5.4. Find u 2 V and � 2 G.u� / such that for every v 2 V we have

h�Au C B.u; u/; vi C h�; v� iU0�U D hF; vi: (5.29)

The main result of this section is the following existence theorem for the aboveproblem.

Theorem 5.2. Assume .h1/–.h3/ and .h6/, .h7/. Then Problem 5.4 has a solution.�; u/ 2 W � V.

Proof. The proof consists of two steps. In the first step we will define the cut-offfriction multifunction hm and the associated cut-off Problem 5.5. In the second stepwe pass to infinity with the cut-off constant and we recover the solution of theoriginal problem.

Step 1. Let us first choose a natural number m and define

hm.x; s/ D8<

:

h.x; s/ if jsj � m;

h

x; msjsj�

if jsj > m:

We will show that hm satisfies .h1/–.h5/. Naturally, hm satisfies .h1/ and .h2/. Wewill prove that hm satisfies .h3/, i.e., the graph of s ! hm.x; s/ is closed in R

6 fora.e. x 2 �C. To this end, let sk ! s and �k ! � with �k 2 hm.x; sk/. Define

rk D(

sk if jskj � m;mskjskj if jskj > m:

From the fact that �k 2 hm.x; sk/ and the definition of hm it follows that �k 2 h.x; rk/.We will consider two cases. If jsj � m, then we have rk ! s and we use .h3/ forh whence it follows that � 2 h.x; s/ D hm.x; s/ for a.e. x 2 �C. In the second casewe have jsj > m. Then we have rk ! ms

jsj and we can use .h3/ for h to deduce

that � 2 h

x; msjsj�

D hm.x; s/ for a.e. x 2 �C. The proof that hm satisfies .h3/

is complete. We will now demonstrate that hm satisfies .h4/ and .h5/. Indeed, lets 2 R

3 and � 2 hm.x; s/. By the definition of hm and .h6/ for a.e. x 2 �C we havej�j � c3.x/C c4mp�1, whence .h4/ holds with c1 D c3.�/C c4mp�1 2 L2.�C/ andc2 D 0, which implies that hm satisfies also .h5/.


We have proved that hm satisfies .h1/–.h5/. We can define the multivalued operatorHm W W ! 2W by the formula

Hm.u/ D f� 2 W W �.x/ 2 hm.x; u.x// a.e. on �Cg:We formulate the following cut-off version of Problem 5.4, where the cut-offparameter is denoted by m.

Problem 5.5. Find um 2 V and �m 2 Hm.um� / such that for every v 2 V we have

h�Aum C B.um; um/; vi C .�m; v� /W D hF; vi: (5.30)

As hm satisfies .h1/–.h5/ we can use Theorem 5.1 to deduce that for each m thereexists .�m; um/ 2 W � V , a solution of Problem 5.5.

Step 2. We pass with the cut-off constant m to infinity and we recover the solutionof Problem 5.4. Taking v D um in (5.30) we get

2�kumk2 C .�m; um� /W � kFkV0kumk: (5.31)

Now we use .h7/ to obtain

.�m; um� /W DZ

�C

�m � um� dS

DZ

fjum� j�mg�m � um� dS C

Z

fjum� j>mgjum� j

m�m � mum�

jum� j dS

� �Z

fjum� j�mgd1jum� j2 C d2 dS �

Z

fjum� j>mgjum� j

m.d1m

2 C d2/ dS

� �d1kum�k2W � kd2kL1.�C/ � 1

mkd2kL2.�C/kum�kW :

Using the last inequality in (5.31) we get, by the trace theorem

.2� � d1k�k2/kumk2 � kFkV0kumk C kd2kL1.�C/ C 1

mk�kkd2kL2.�C/kumk:

It follows that the sequence um is bounded in V and thus, for a subsequence, denotedby the same index, um ! u weakly in V . From the fact that p 2 Œ2; 4/ it follows thatthe trace operator �p W V ! U is compact (see, for example, Theorem 2.79 in [50])and um� ! u� strongly in U.

Using .h6/ we estimate

k�mkqU0

�Z

fjum� j�mg.c3 C jum� jp�1/q dS C

Z

fjum� j>mg

c3 Cˇˇˇˇmum�

jum� jˇˇˇˇ

p�1!q

dS

�Z

�C

2q�1.cq3 C jum� jp/ dS:


Since c3 2 L2.�C/ Lq.�C/ and the sequence um� which converges in U stronglyis norm bounded in this space, it follows that �m is bounded in U0 and hence, for asubsequence, not renumbered, �m ! � weakly in U0. We can rewrite (5.30) as

h�Aum C B.um; um/; vi C h�m; v� iU0�U D hF; vi;

and we can pass to the limit to obtain (5.29). We only need to show that � 2 G.u� /.Since um� ! u� strongly in U, then, for a subsequence, denoted by the same index,um� .x/ ! u� .x/ with jum� .x/j � g.x/ for a.e. x 2 �C, where g 2 Lp.�C/. Fix " > 0.By the Luzin theorem there exists a closed set M" �C, with m2.�C n M"/ < " andg continuous on M". Define m" D maxx2M" g.x/. For all natural m � m" we havejum.x/j � g.x/ � m" � m a.e. on M" and hence �m.x/ 2 h.x; um� .x// a.e. on M". Wecan use .H2/ of Theorem 3.23 for the space Z D Lp.M"/

3 and the map

Z 3 u ! f� 2 Z0 W �.x/ 2 h.x; u.x// a.e. on M"g 2 2Z0

(indeed um� ! u� strongly in Z and �m ! � weakly in Z0), whereas a.e. on M" wehave �.x/ 2 h.x; u� .x//. Since " > 0was arbitrary, it follows that �.x/ 2 h.x; u� .x//a.e. on �C and the assertion is proved. This completes the proof of Theorem 5.2. ut


In this chapter the multivalued term has the form of an arbitrary multifunction. Inmost typical applications this multifunction is a Clarke subdifferential of a certainlocally Lipschitz functional, usually nondifferentiable in a discrete or Lebesguenull set of points. In such a case, the multivalued PDE is equivalent to a certainhemivariational inequality, as it has been proved by Carl [49] (the proof in [49] isdone for an evolution problem but it works well for the static case, too). There aremany techniques to prove existence of weak solutions for the multivalued ellipticpartial differential equations or hemivariational inequalities. The idea based onthe use of surjectivity results for pseudomonotone operators has been extensivelyexploited in [184]. The application of this technique to stationary incompressibleNavier–Stokes problem with the multivalued feedback control boundary conditionin the normal direction involving the Bernoulli pressure was presented in [175]. Theapproach based on the Galerkin method and the finite element method was studied in[114], and the techniques using upper and lower solutions were thoroughly analyzedin [50]. Yet another interesting approach to study the existence of weak solutionsusing the Kuratowski–Knaster–Mazurkiewicz (KKM) principle was proposed in[78]. The approach based on the Kakutani–Fan–Glicksberg fixed point theorem is,to our knowledge, new.


Fig. 5.2 Two examples of a frictional contact condition. The vertical line in both plots representsthe stick effect, where the tangential velocity of the fluid u� is equal to the given velocity U0. Insuch a case the friction condition is actually a (nonhomogeneous) Dirichlet condition. If the stickdoes not occur, i.e., u� ¤ U0 we have a slip effect. In such a case, the tangential stress T� is givenas a function of u� . The right plot depicts the possibility of taking into account the dependence ofthe friction coefficient of the slip rate. It is a priori unknown which part of the contact boundaryis a stick zone, and which part is the slip zone. Of course, in the case of evolution problems, bothzones can change with time

The book [177] is devoted to the applications of multivalued PDEs in thecontinuum mechanics and presents many examples of contact conditions in thesolid mechanics leading to multivalued PDEs and hemivariational inequalities. Anexample of the law that can be used in the model presented in this chapter is theTresca friction law

h.x; u� / D8<

:

nk�

u��U0ju��U0jo

for u� ¤ U0;

B.0; k� / otherwise:

The interpretation of this law is the following: on the contact boundary thefoundation is moving with the given velocity U0 2 R

3. If there is no slip, i.e.,u� D U0 then the magnitude of the friction force cannot exceed the frictionbound k� . When the friction force is large enough, its absolute value equal to k� ,then the slip appears and we have u� ¤ U0. The direction of the friction force isalways opposite to the vector u� � U0. The Tresca law is presented in the left plot inFig. 5.2.

Obviously, this example satisfies all the assumptions .h1/–.h5/, as we have .h4/with c2 D 0. It is possible to generalize the Tresca law, making the friction bound k�dependent on ju� � U0j. In such a case we speak of slip rate dependent friction. Inparticular the situation when k� is decreasing with increasing ju� � U0j reflects thefact that the kinetic friction is less than the static friction. This, generalized, versionof the Tresca law is presented in the right plot in Fig. 5.2.

The dependence of the k� on ju� � U0j can also be discontinuous, to accountfor flattening of asperities on the surface. Similar laws were used, for example, in[12, 213] for static problems in elasticity, see Fig. 4.6 in [12] and Fig. 2a in [213] forexamples of particular nonmonotone friction laws.

6Stationary Flows in Narrow Filmsand the Reynolds Equation

In this chapter we study a typical problem from the theory of lubrication, namely, theStokes flow in a thin three-dimensional domain ˝", " > 0. We assume the Fourierboundary condition (only the friction part) at the top surface and a nonlinear Trescainterface condition at the bottom one.

When " ! 0 the domains ˝" become in the limit a two-dimensional domain !,the mutual bottom of the domains ˝". Let .u"; p"/ be the solution of the consideredStokes problem in ˝". An important problem in the theory of lubrication, whichdeals with flows in thin films, is to describe the behavior of solutions .u"; p"/ as" ! 0 in terms of some limit functions .u?; p?/. It happens that the pressure p?

depends only on two variables which define the surface ! and satisfies the Reynoldsequation, a fundamental equation in this theory. We can say that the Reynoldsequation describes a distribution of the pressure “in an infinitely thin domain.”Moreover, the boundary conditions imposed on the solutions of the Stokes equationin ˝" tend in an appropriate sense to some boundary conditions for the Reynoldsequation. Our aim is to provide a strict mathematical analysis of this asymptoticproblem.

6.1 Classical Formulation of the Problem

We consider a boundary value problem for the Stokes equations in ˝",

� ��u C rp D 0 in ˝"; (6.1)

div u D 0 in ˝"; (6.2)


111

112 6 Stationary Flows in Narrow Films and the Reynolds Equation

The domain ˝" is as follows. Let ! R2 be an open bounded set with sufficiently

smooth boundary. Then

˝" D f.x1; x2; x3/ 2 R3 W .x1; x2/ 2 !; 0 < x3 � "h.x1; x2/g:

The boundary of ˝" consists of three parts: the bottom !, the lateral part � "L , and

the top surface � "F , @˝" D ! [ � "

L [ � "F . Our boundary conditions are:

• At the bottom ! we assume

un D u � n D 0 on !: (6.3)

The tangential velocity on ! is unknown and satisfies the Tresca boundarycondition with the friction coefficient k",

jT� j D k" H) 9� � 0 u� D s � �T� ;jT� j < k" H) u� D s;

(6.4)

where j:j denotes the Euclidean norm in R2, and s is the velocity of the lower

surface !, n D .n1; n2; n3/ is the outward unit normal to ˝", and

un D u � n; u� D u � unn:

Boundary condition (6.3)–(6.4) is a form of the well-known Coulomb solid–solid interface law [91] applied here for the fluid–solid interface. Another variant,taking into account also the normal stresses, was considered in [21].

• On the lateral boundary � "L we assume

u D G" on � "L : (6.5)

• At the top surface � "F we assume the Fourier boundary condition

T� C l"u D 0 on � "F ; (6.6)

and

u � n D 0 on � "F ; � "

F given by x3 D h".x1; x2/ D "h.x1; x2/: (6.7)

Two-dimensional cross section of the domain of the studied problem is presentedin Fig. 6.1.

We have to specify the data G" in (6.5), l" in (6.6), and h" in (6.7), depending on ".We will specify this dependence in Sect. 6.3.

6.1 Classical Formulation of the Problem 113

We assume that G" 2 .H1.˝"//3 with div G" D 0 in ˝", G" � n D 0 on � "F [ !.

We assume also that h satisfies

0 < h.x/ � hM; h 2 C2.!/; (6.8)

and that l" > 0. In (6.4) and (6.6) T� is the tangential part of the stress vector Tn onthe boundary @˝", where the stress tensor T is given by

Tij D �p ıij C 2�Dij.u/ where Dij.u/ D 1

2

�@ui

@xjC @uj

@xi

�

for 1 � i; j � 3;

Fig. 6.1 Two-dimensionalcross section of the domain ofthe studied problem. For" D 1 we denote � "

F D �F

and � "L D �L. On �F we

assume the Fourier condition,on �L, the (nonhomogeneous)Dirichlet condition, and on !,the Tresca condition

namely

T� D Tn � Tnn; (6.9)

where Tn D .Tn/ � n is the normal part of the stress tensor on the boundary @˝".

Remark 6.1. In [15, 21] a similar problem was considered, it differed from theabove in a boundary condition at the top surface; namely, in these papers it wasassumed that

u D 0 at � "F : (6.10)

Condition (6.10) helps much in obtaining the solution. With (6.6) and (6.7) inplace of (6.10) the problem is more difficult due to the lack of the usual formsof the Poincaré and Korn inequalities. What is important is to obtain other forms ofthese inequalities, in which the constants do not explode as " tends to zero.

The plan of the rest of the chapter is as follows. In Sect. 6.2 we provide aweak formulation of the problem, prove suitable forms of the Korn and Poincaréinequalities, and obtain key estimates. In Sect. 6.3, taking into account a smallparameter, we introduce a scaling, and using estimates from Sect. 6.2 prove aconvergence theorem for the rescaled functions. In Sect. 6.4 we establish the limitvariational inequality, give precise characterization of the sets to which its solution


and admissible test functions belong, and prove the strong convergence of thevelocity fields. In Sect. 6.5 the Reynolds equation and the limit boundary conditionsare obtained. In the last section we prove the uniqueness of solutions of the limitproblem.

6.2 Weak Formulation and Main Estimates

We start from remarks on the notation in this chapter. We will denote by x D .x1; x2/variables belonging to the set !. Surface element in integrals will be denoted by dS,or, in case of integrals over ! by dx or dx1 dx2. The real variable in the verticaldirection, orthogonal to ! will be denoted by y or x3 and the corresponding lengthelement in integrals by dy or dx3. Volume element in volume integrals will bedenoted by dx dy or dx dx3. Let us define

K D f' 2 .H1.˝"//3 W ' � G" D 0 on � "L ; ' � n D 0 on � "

F [ !g;(6.11)

where

G" 2 .H1.˝"//3 with div G" D 0 in ˝"; G" � n D 0 on � "1 [ !;

(6.12)

Kd D fv 2 K W div v D 0 in ˝"g;and

L20.˝"/ D fp 2 L2.˝"/ W

Z

˝"

p dx dx3 D 0g:

Our aim is to justify the following weak formulation of problem (6.1)–(6.7).

Definition 6.1. The weak solution of problem (6.1)–(6.7) is a pair .u; p/ such that

u 2 Kd; (6.13)

p 2 L20.˝"/;

a.u; ' � u/ �Z

˝"

p div' dx dx3 CZ

� "F

l"u.' � u/ dS

CZ

!

k".j' � sj � ju � sj/ dx � 0 for all ' 2 K; (6.14)

where

a.u; v/ D 2�

Z

˝"

Dij.u/Dij.v/ dx dx3: (6.15)

6.2 Weak Formulation and Main Estimates 115

Remark 6.2. In (6.14), k" comes from the Tresca condition. It is a positive constantand its dependence on " shall be defined in Sect. 6.3.

Observe that from condition (6.13) it follows that:

u D G" on � "L .cf: (6.5)/;

u � n D 0 on � "F [ ! .cf: (6.3) and (6.7)/:

Tresca and slip conditions are included in (6.14). To obtain (6.14) we observe that

��ui C p;i D ��2�.ui;j C uj;i/C pıij�

;j � �Tij;j;

as uj;ij D .uj;j/;i D 0 from div u D 0.We multiply the equations of motion

�Tij;j D 0 .i D 1; 2; 3/

by 'i � ui, integrate by parts in ˝" and add for i D 1; 2; 3, to get

Z

˝"

Tij@.'i � ui/

@xjdx dx3 �

Z

@˝"

Tn � .' � u/ dS D 0: (6.16)

NowZ

˝"

Tij@.'i � ui/

@xjdx dx3 D

Z

˝"

TijDij.'i � ui/ dx dx3;

since

TijDij.v/ D 1

2Tij.vi;j C vj;i/ D 1

2Tijvi;j C 1

2Tjivj;i D Tijvi;j

by symmetry.From (6.16) and (6.15),

Z

˝"

Tij@.'i � ui/

@xjdx dx3 D

Z

˝"

.2�Dij.u/ � p ıij/Dij.' � u/ dx dx3

D a.u; ' � u/ �Z

˝"

p div .' � u/ dx dx3

D a.u; ' � u/ �Z

˝"

p div' dx dx3; (6.17)

as div u D 0 in ˝".


The second term in (6.16) is divided into three parts, as @˝" D � "F [ ! [ � "

L :

We have, by (6.9) and (6.6),

�Z

� "F

Tijnj.'i � ui/ dS D �Z

� "F

ŒT� C Tnn� � ..' � u/� C .' � u/nn/ dS

D �Z

� "F

.�l"u/ � .' � u/� dS

D l"Z

� "F

u � .' � u/ dS;

as .' � u/n D n � .' � u/ D 0 on � "F by (6.11) and (6.12).

Integration over ! and the Tresca condition yieldZ

!

k".j' � sj � ju � sj/ dx � �Z

!

Tn � .' � u/ dx; (6.18)

where

s D .s1; s2; 0/ D G"j!: (6.19)

Observe that G" � n D 0 on ! by (6.12) so that G"3 D 0:

The third integral is

�Z

� "L

Tn � .' � u/ dS D 0; 1

as ' D u D G" on � "L by (6.11) and (6.13).

In this way, from (6.16)–(6.18) we get

a.u; ' � u/ �Z

˝"

p div' dx dx3 C l"Z

� "F

u.' � u/ dS (6.20)

C k"Z

!

.j' � sj � ju � sj/ dx � 0:

Remark 6.3. Observe that in a.:; :/ there is the Newtonian viscosity �. In the sequelwe set � D 1.

We use (6.20) to obtain the basic estimate of the velocity given in Lemma 6.4below.

Lemma 6.1. We have

1

2a.u; u/C l"

2

Z

� "F

juj2 dS C k"Z

!

ju � sj dx � l"

2

Z

� "F

jG"j2 dS C2

Z

˝"

jrG"j2 dx dx3:

(6.21)


Proof. We set ' D G" in (6.20) to obtain,

a.u; u/C l"Z

� "F

juj2 dS C k"Z

!

ju � sj dx � a.u;G"/C l"Z

� "F

uG" dS; (6.22)

as div G" D 0, and G" D s on ! by (6.19). Moreover, as � D 1,

a.u;G"/ D 2

Z

˝"

Dij.u/Dij.G"/ dx dx3

� 2

Z

˝"

.Dij.u/Dij.u//12 .4Dij.G

"/Dij.G"//

12 dx dx3

�Z

˝"

Dij.u/Dij.u/ dx dx3 C 4

Z

˝"

Dij.G"/Dij.G

"// dx dx3

D 1

2a.u; u/C 2a.G";G"/ � 1

2a.u; u/C 2

Z

˝"

jrG"j2 dx dx3: (6.23)

From (6.22) and (6.23) we obtain immediately (6.21). utThe desired estimate of the velocity will follow from (6.21) and a form of the Korninequality, suitable for our problem.

Lemma 6.2 (A Form of the Korn Inequality). We have

Z

˝"

jr.u � G"/j2 dx dx3 � a.u � G"; u � G"/C C.� "F /

Z

� "F

ju � G"j2 dS; (6.24)

where

C.� "F / D 2kD2h

"kC.!/

1C kD1h

"k2C.!/�: (6.25)

Observe that we use here h 2 C2.!/, cf. (6.8), and that C.� "F / is of order " if h" D "h

as in (6.7).

Proof. Assume that v D u � G" is a smooth function. The result will follow fromdensity of smooth functions in Kd. We have

a.v; v/ D 1

2

Z

˝"

�@vi

@xkC @vk

@xi

�2

dx dx3

DZ

˝"

�@vi

@xk

@vi

@xkC @vi

@xk

@vk

@xi

�

dx dx3

DZ

˝"

jrvj2 dx dx3 CZ

˝"

@vi

@xk

@vk

@xidx dx3


DZ

˝"

jrvj2 dx dx3 �Z

˝"

@2vi

@xk@xivk dx dx3 C

Z

@˝"

@vi

@xkvkni dS

DZ

˝"

jrvj2 dx dx3 CZ

˝"

@vi

@xi

@vk

@xkdx dx3

�Z

@˝"

@vi

@xivknk dS C

Z

@˝"

@vi

@xkvkni dS:

If div v D 0, then we have

Z

˝"

jrvj2 dx dx3 D a.v; v/CZ

@˝"

@vi

@xivknk dS �

Z

@˝"

@vi

@xkvkni dS: (6.26)

For v D u�G" we have .u�G"/ �n D 0 on @˝" as u 2 Kd and G" is as in (6.12) thusthe first surface integral on the right-hand side equals zero, and the second surfaceintegral reduces to that over � "

F [!; as v D u � G" D 0 on � "L . We have to estimate

the integral

Z

� "F [[email protected] � G"

i /

@xk.uk � G"

k/ni dS DZ

� "F [!.uk � G"

k/@.ui � G"

i /

@xkni dS: (6.27)

Since .u � G"/ � n D 0 on � "F [ !, then @.u�G"/�n

@xkD 0, i.e.,

3X

iD1

@.ui � G"i /

@xkni D �

3X

iD1.ui � G"

i /ni;k: (6.28)

But ni;k D 0 on !, so that from (6.27) and (6.28) we have

ˇˇˇˇˇ

Z

� "F

.uk � G"k/@.ui � G"

i /

@xkni dS

ˇˇˇˇˇ

� QC.� "F /

Z

� "F

ju � G"j2 dS; (6.29)

where

QC.� "F / � 2max

q2� "Fjni;k.q/j:

As � "F is given by x3 D h".x1; x2/,

n.q/ D � @h"

@x1.x1; x2/;� @h"

@x2.x1; x2/; 1

�

p1C jrh".x/j2 D n.x1; x2/: (6.30)


We compute

n3;i.x1; x2/ D @n3@xi

.x1; x2/ D �.1C jrh"j2/� 32

�@h"

@x1

@2h"

@xi@x1C @h"

@x2

@2h"

@xi@x2

�

;

whenceˇˇˇˇ@n3@xi

ˇˇˇˇ � 2jD1h

"jjD2h"j � jD2h

"j.1C jD1h"j2/;

and similarly

jnj;i.x1; x2/j Dˇˇˇˇ@nj

@xi

ˇˇˇˇ � jD2h

"j.1C jD1h"j2/:

Thus QC.� "F / � C.� "

F /, where C.� "F / is defined in (6.25). In the end, from (6.26)

and (6.29) we obtain (6.24). utNow, having inequality (6.24), we can estimate

R˝" jruj2 dx dx3. We have

C.� "F /

Z

� "F

ju � G"j2 dS � 2C.� "F /

Z

� "F

juj2 dS CZ

� "1

jG"j2 dS

!

; (6.31)

and, by (6.23),

a.u � G"; u � G"/ D a.u; u/ C a.G";G"/ � 2a.u;G"/

� 2a.u; u/C 5

Z

˝"

jrG"j2 dx dx3: (6.32)

MoreoverZ

˝"

jruj2 dx dx3 � 2

Z

˝"

jr.u � G"/j2 dx dx3 C 2

Z

˝"

jrG"j2 dx dx3: (6.33)

In the end, from (6.24), along with (6.31)–(6.33) we have

Lemma 6.3.Z

˝"

jruj2 dx dx3 � 4 a.u; u/ C 12

Z

˝"

jrG"j2 dx dx3 (6.34)

C 4C.� "F /

Z

� "F

juj2 dS CZ

� "F

jG"j2 dS

!

:

The first term on the right-hand side as well as the surface integralR� "F

juj2 dS isestimated in (6.21). From (6.21) and (6.34) we have


Lemma 6.4.Z

˝"

jruj2 dx dx3 ��

28C 16C.� "F /

l"

�Z

˝"

jrG"j2 dx dx3 (6.35)

C 4�l" C 2C.� "

F /�Z

� "F

jG"j2 dS:

Remark 6.4. If we set "l" D Ol 2 .0;1/, then l" ! 1 as " ! 0, and

28C 16C.� "F /

l"D O.1/ as " ! 0;

4.l" C 2C.� "F // D O

�1

"

�

as " ! 0:

Later on we shall show that "R˝" jruj2 dx dx3 D O.1/ as " ! 0. This will come

from estimates of the integrals on the right-hand side of (6.35), by means of theH1.˝/3 norm of a given function OG in H1.˝/3, to which the family of functions G"

is suitably related (here and in the sequel we put ˝ D ˝1).To be able to estimate

R˝

juj2 dx dy we need a suitable form of the Poincaréinequality.

Lemma 6.5 (A Form of the Poincaré Inequality). Set h"M D max! jh".x1; x2/j:Then

Z

˝"

juj2 dx dx3 � 2h"M

Z

� "F

juj2 dS C 2.h"M/2

Z

˝"

ˇˇˇˇ@u

@x3

ˇˇˇˇ

2

dx dx3: (6.36)

Proof. Let .x1; x2; x3/ D .x; t/, x D .x1; x2/ for short. Then

u.x; t/ D u.x; h".x// �Z h".x/

t

@u

@z.x; z/ dz;

so that

ju.x; t/j � 2ju.x; h".x//j2 C 2h".x/Z h".x/

0

ˇˇˇˇ@u

@z.x; z/

ˇˇˇˇ

2

dz:

We integrate over t 2 Œ0; h".x/� to obtain

Z h".x/

0

ju.x; x3/j2dx3 � 2h".x/ju.x; h".x//j2 C 2.h".x//2Z h".x/

0

ˇˇˇˇ@u

@z.x; x3/

ˇˇˇˇ

2

dx3;

and, after integration over !, we obtain (6.36). ut

6.3 Scaling and Uniform Estimates 121

We have now all basic estimates. In Sect. 6.3 we shall state them once again butin new variables: Ou instead of u", etc., defined in ˝1.

We have the following existence theorem in ˝".

Theorem 6.1. For the data as in Sects. 6.1 and 6.2, there exists a unique weaksolution .u; p/ D .u"; p"/ of problem (6.1)–(6.7),

u" 2 Kd and p" 2 L20.˝"/:

Proof. For the proof, which requires some more advanced variational methods tobe presented in this place, we refer the reader to [15] where the functional j is heregiven by

j.'/ DZ

!

k"j'.x; 0/ � sj dx C l"Z

� "F

u' dS: ut

6.3 Scaling and Uniform Estimates

Let ˝ D ˝1 and define, for a velocity u in ˝",

y D x3"; Oui.x; y/ D ui.x; x3/; i D 1; 2 and Ou3.x; y/ D 1

"u3.x; x3/;

Ou.x; y/ D .Ou1.x; y/; Ou2.x; y/; Ou3.x; y// being a new velocity in ˝.For the pressure p in ˝" we define Op.x; y/ D "2p.x; x3/: Let

Ol D "l" and Ok D "k";

where Ol and Ok are some positive numbers, l" stands in (6.6), and k" appears in theTresca condition (6.4).

Now, for a given function OG.x; y/ D . OG1.x; y/; OG2.x; y/; OG3.x; y// in H1.˝/3 suchthat OG � n D 0 on �F [ ! .�F D � 1

F / and

@ OG1

@x1C @ OG2

@x2C @ OG3

@yD 0 .div OG D 0/;

we define a family of functions G" in ˝", by

OGi.x; y/ D G"i .x; x3/; i D 1; 2; OG3.x; y/ D 1

"G"3.x; x3/:


By OK we define the set

OK Dn' 2 .H1.˝//3 W ' � OG D 0 on �L; ' � n D 0 on �F [ !

o;

where �L D � 1L . Observe that div OG D 0 in ˝ implies div G" D 0 in ˝".

It is clear that Ou D OG on �L implies u D G" on � "L , and Ou � n D 0 on ! implies

u � n D 0 on !. Moreover, from (6.30) it follows that also Ou � n D 0 on �F impliesu � n D 0 on � "

F .We have proved in fact that

u 2 Kd is equivalent to Ou 2 OK; div Ou D 0:

To compare solutions .u"; p"/ for various " (these solutions exist by Theorem 6.1) wetransform them as above to the same domain ˝. The transformed solutions .Ou"; Op"/are solutions of a suitably transformed problem from Definition 6.1. We shall writethe problem precisely later on.Now, we obtain estimates for Ou D Ou" directly from those obtained for u D u" inSect. 6.2.

We shall write estimates (6.35) and (6.36) in new variables.

Lemma 6.6. We haveZ

˝"

jrG"j2 dx dx3 � 1

"

Z

˝

jr OGj2 dx dy for all " 2 .0; 1�: (6.37)


Lemma 6.7. We haveZ

� "F

jG"j2 dS � C0.˝/Z

˝

j OGj2 C jr OGj2

�dx dy for all " 2 .0; 1�; (6.38)

where C0.˝/ is independent of ".

Proof. For i D 1; 2 we have

Z

� "F

jG"i j2 dS D

Z

!

jGi.x; h".x//j2

p1C jrh".x/j2dx

�Z

!

j OGi.x; h.x//j2p1C jrh".x/j2dx D

Z

�F

j OGij2 dS; (6.39)

6.3 Scaling and Uniform Estimates 123

(by definition Gi.x; h".x// D OGi.x; h.x//), and

Z

� "F

jG"3j2 dS D

Z

!

jG"3.x; h

".x//j2p1C jrh".x/j2dx

� "2Z

!

j OG3.x; h.x//j2p1C jrh.x/j2

p1C jrh".x/j2

p1C jrh.x/j2 dx

� "2 maxx2!

p1C jrh".x/j2

Z

�F

j OG3j2 dS: (6.40)

From (6.39) and (6.40),

Z

� "F

jG"j2 dS �Z

�F

j OGj2 dS: (6.41)

AsZ

�F

j OGj2 dS � C0.˝/Z

˝

j OGj2 C jr OGj2

�dx dy; (6.42)

where C0.˝/ depends only on ˝, (6.41) and (6.42) give (6.38). utUsing inequalities (6.37) and (6.38) in (6.35) and taking into account Remark 6.4we see that

"

Z

˝"

jruj2 dx dx3 � C.˝; h; Ol/Z

˝

j OGj2 C jr OGj2

�dx dy � QC0 for all " 2 .0; 1�:

(6.43)

Writing (6.43) in terms of Ou, we obtain

"22X

i;jD1

��@Oui

@xj

��

2

L2.˝/

C2X

iD1

��@Oui

@y

��

2

L2.˝/

C "2��@Ou3@y

��

2

L2.˝/

C "42X

iD1

��@Ou3@xi

��

2

L2.˝/

� QC0;(6.44)

and also, as in [15, 21],

��@Op@xi

��

H�1.˝/

� C1 and

��@Op@y

��

H�1.˝/

� "C2: (6.45)

Exercise 6.2. Prove estimates (6.45). First, for 2 H10.˝

"/ set ' D u ˙

in (6.14), where u D u", p D p", to get

a.u"; / �Z

˝"

p"div dx dx3 D 0:


Further estimates follow from the Poincaré inequality (6.36) written in newvariables,

Z

˝

jOuij2 dx dy � 2hM

Z

�F

jOuij2 dS C 2h2M

Z

˝

ˇˇˇˇ@Oui

@y

ˇˇˇˇ

2

dx dy for i D 1; 2; 3:

(6.46)

Observe that changing the variables, we obtain the same form of the Poincaréinequality.


To estimate the integrals on the left-hand side, we have to estimate the surfaceintegrals on the right-hand side. We have,

Z

�F

jOuij2dS DZ

!

jOui.x; h.x//j2p1C jrh.x/j2dx

DZ

!

jOui.x; h".x//j2

p1C jrh".x/j2

p1C jrh.x/j2

p1C jrh".x/j2 dx

� maxx2!

p1C jrh.x/j2

Z

� "F

juij2 dS for i D 1; 2; (6.47)

andZ

�F

jOu3j2 dS DZ

!

jOu3.x; h.x//j2p1C jrh.x/j2dx

� 1

"2maxx2!

p1C jrh.x/j2

Z

� "F

ju3j2 dS: (6.48)

From (6.21) using (6.37) and (6.38) we have

Z

� "F

juj2 dS �Z

� "F

jG"j2 dS C 4

l3

Z

˝"

jrG"j2 dx dx3

� C0.˝/Z

˝

.j OGj2 C jr OGj2/ dx dy C 4

OlZ

˝

jr OGj2 dx dy

� C.˝; Ol; OG/: (6.49)

Thus, from the Poincaré inequality (6.46) with (6.44) and (6.47)–(6.49) we obtainin the end

kOuik2L2.˝/ � C3; (6.50)

"2kOu3k2L2.˝/ � C4: (6.51)

6.4 Limit Variational Inequality, Strong Convergence, and the Limit Equation 125

Let

Vy D�

v 2 L2.˝/ W @v

@y2 L2.˝/

�

be the Banach space with norm

kvkVy D

kvk2L2.˝/ C��@v

@y

��

2

L2.˝/

! 12

:

Theorem 6.2. There exist u?i 2 Vy for i D 1; 2, p? 2 L20.˝/, and a subsequence" ! 0 such that

Oui ! u?i weakly in Vy for i D 1; 2; (6.52)

"@Oui

@xj! 0 weakly in L2.˝/ for i; j D 1; 2; (6.53)

"@Ou3@y

! 0 weakly in L2.˝/; (6.54)

"2@Ou3@xi

! 0 weakly in L2.˝/ for i D 1; 2; (6.55)

Op ! p? weakly in L2.˝/; p? depends only on x; (6.56)

"Ou3 ! 0 weakly in L2.˝/: (6.57)

Proof. From (6.44) and (6.50) we have (6.52), and from (6.44) and (6.52) we

get (6.53). By (6.53) andP2

jD1@Ouj

@xjD � @Ou3

@y , (6.54) holds. From (6.51) and (6.44)

we deduce (6.55). From (6.45) we obtain (6.56), and from (6.51), div Ou D 0, and asuitable choice of the test function we obtain also (6.57). ut

6.4 Limit Variational Inequality, Strong Convergence,and the Limit Equation

To obtain the variational inequality for limit functions u?, p? in ˝ we change vari-ables in the inequality defining weak solutions in ˝" and then using Theorem 6.2,pass to zero with ".First, we write (6.20) in new variables Ou; Op in ˝. In the place of the test function 'in ˝" we put the new test function O' in ˝, which is connected with ' in the same


way as Ou is connected with u. For ' 2 K we have O' 2 OK, so that summing over Kwith ' we sum over OK with O'. We have, after multiplying (6.20) by ",

1

2

2X

i;jD1

Z

˝

�

"@Oui

@xjC "

@Ouj

@xi

��

"@

@xj. O'i � Oui/C "

@

@xi. O'j � Ouj/

�

dxdy

C 1

2

2X

iD1

Z

˝

�@Oui

@yC "2

@Ou3@xi

��@

@y. O'i � Oui/C "2

@

@xi. O'3 � Ou3/

�

dxdy

C 2

Z

˝

"@Ou3@y"@

@y. O'3 � Ou3/dxdy �

Z

˝

Op�@ O'1@x1

C @ O'2@x2

C @ O'3@y

�

dxdy

C2X

iD1

Z

!

OlOui.x; h.x//Œ O'i.x; h.x// � Oui.x; h.x//�p1C jrh".x/j2dx

CZ

!

Ok.j O' � sj � jOu � sj/dx � 0: (6.58)

If we take all nonlinear terms in (6.58) to the right-hand side and take lim inf"!0

on both sides (in fact on the left-hand side the simple lim"!0 exists), we obtain thelimit variational inequality,

2X

iD112

Z

˝

@u?i@y

@. O'i � u?i /

@ydxdy �

Z

˝

p?.x/

�@ O'1@x1

C @ O'2@x2

�

dxdy

�Z

!

p?.x/

�

O'1.x; h.x// @h

@x1.x/C O'2.x; h.x// @h

@x2.x/

�

dx

C2X

iD1OlZ

!

u?i .x; h.x//� O'i.x; h.x// � u?i .x; h.x//

�dx

CZ

!

Ok .j O' � sj � ju? � sj/ dx � 0: (6.59)

We shall confine ourselves to present the main points of the proof.

1. In the above inequality (6.59) the third components of u? and O' do not occur, sothat we can write N' D . O'1; O'2/, and (6.59) holds for all N' that are projections ofO' 2 OK on the first two components. Let

˘. OK/ D f N' D . O'1; O'2/ 2 .H1.˝//2 W 9 O'3 2 H1.˝/; O' D . O'1; O'2; O'3/ 2 OKg:

6.4 Limit Variational Inequality, Strong Convergence, and the Limit Equation 127

The second line in (6.59) is written in terms of N', and equals

Z

˝

p?.x/@ O'3@y

dydx DZ

!

Z h.x/

0

p?.x/@ O'3@y

dydx DZ

!

p?.x/ O'3.x; h.x//dx;

as O'3.x; 0/ D 0 and O'3 D 0, on �L. Since O'1n1 C O'2n2 C O'3n3 D 0, and

O'3 D �1n3. O'1n1 C O'2n2/ D

p1C jrh.x/j2 . O'1n1 C O'2n2/ D O'1 @h

@x1C O'2 @h

@x2;

(cf. (6.30)), we have

Z

˝

p?.x/@ O'3@y

dydx DZ

!

p?.x/

2X

iD1O'i.x; h.x//

@h

@xi.x/

!

dx:

2. Integrals over �F. We know that, cf. (6.21), (6.37), and (6.38),

l"Z

� "F

u2 dS D O

�1

"

�

;

and from (6.47) and (6.48) we get

Z

�F

jOuij2dS � C.h/Z

� "F

juij2dS for i D 1; 2;

Z

�F

jOu3j2dS � 1

"2C.h/

Z

� "F

ju3j2dS;

whence from (6.49),Z

�F

jOuij2dS � C.h/C.˝; Ol; OG/ andZ

�F

jOu3j2dS � 1

"2C.h/C.˝; Ol; OG/:

(6.60)

From (6.60) we conclude that Oui ! v?i weakly in L2.�F/, and, since

"l"Z

� "F

ui.'i � ui/dS

D OlZ

!

Oui.x; h.x// Œ O'i.x; h.x// � Oui.x; h.x//�p1C jrh".x/j2dx;

we get the terms

OlZ

!

u?i .x; h.x//� O'i.x; h.x// � u?i .x; h.x//

�dx

on the left-hand side of (6.59), asp1C jrh".x/j2 ! 1 strongly in L2.!/

with " ! 0.


The third component (for i D 3) is nonpositive in the limit and we can omit it.In fact,

"l"Z

� "F

u3.'3 � u3/dS � OlZ

� "F

u3'3dS

D OlZ

!

u3.x; h".x//'3.x; h

".x//p1C jrh".x/j2dx

D OlZ

!

"Ou3.x; h.x//" O'3.x; h.x//p1C jrh".x/j2dx;

and from (6.60) f"Ou3g is bounded in L2.�F/ and

" O'3.x; h.x//p1C jrh".x/j2 ! 0 in L2.!/;

so that

lim"!0

OlZ

� "F

u3.'3 � u3/ dS � 0:

6.5 Remarks on Function Spaces

In this subsection we give precise characterization (e.g., in terms of densitytheorems) of the admissible test functions in the limit inequality (6.59), crucialfor further considerations (strong convergence of the velocity field, limit equations,uniqueness).

Consider the limit variational inequality (6.59). The function space to which thetest functions belong is

˘. OK/ WD f N' D . O'1; O'2/ 2 .H1.˝//2 W 9 O'3 2 H1.˝/; O' D . O'1; O'2; O'3/ 2 OK;that is; O' 2 .H1.˝//3; O' � OG D 0 on �L; O' � n D 0 on �F [ !g:

Here, OG 2 OK, div OG D @ OG1@x1

C @ OG2@x2

C @ OG3@y D 0 in ˝. Denote NG D . OG1; OG2/ for

OG D . OG1; OG2; OG3/ 2 OK: We would like to characterize the space ˘. OK/ in terms ofO'1; O'2 only. Instead of working directly with ˘. OK/ it will be easier to consider thespace ˘0. OK/ WD f N' 2 .H1.˝//2 W N' C NG 2 ˘. OK/g: Thus,

˘0. OK/ D f N' D . O'1; O'2/ 2 .H1.˝//2 W there exists O'3 2 H1.˝/ such that

O' D . O'1; O'2; O'3/ D 0 on �L; O' � n D 0 on �F [ !g: (6.61)

6.5 Remarks on Function Spaces 129

We shall characterize ˘0. OK/ in terms of . O'1; O'2/. Let N' D . O'1; O'2/ be an arbitraryelement of .H1.˝//2 such that O'1 D O'2 D 0 on �L. We shall prove that one canalways find O'3 2 H1.˝/ such that . O'1; O'2; O'3/ D O' satisfies condition in (6.61), thatis, we shall prove that ˘0. OK/ D �0. OK/, where

�0. OK/ D f N' 2 .H1.˝//2 W O'1 D O'2 D 0 on �Lg:

We can see that if N' 2 ˘0. OK/ then N' 2 �0. OK/, that is, ˘0. OK/ �0. OK/: We shallprove that �0. OK/ ˘0. OK/. Let N' 2 �0. OK/. We have to define N'3 2 H1.˝/ suchthat O' D . O'1; O'2; O'3/ D 0 on �L, O' � n D 0 on �F [ !. The first condition is justO'3 D 0 on �L, the second is O'3 D 0 on ! and

O'1.x; h.x// @h

@x1.x/C O'2.x; h.x// @h

@x2.x/ D O'3.x; h.x// on �F:

Thus, we have to show existence of O'3 D 2 H1.˝/ such that D 0 on �L [ !,

D O'1.x; h.x// @h

@x1.x/C O'2.x; h.x// @h

@x2.x/ D 1 on �F:

As N' D . O'1; O'2/ 2 .H1.˝//2, 1 is well defined on �F and belongs to H12 .�F/

because h 2 C2.!/. Since O'1 D O'2 D 0 on �L, we see that 1 D 0 on @˝. Thus,there is no singularity at @!, and the function

b D�0 on �L [ !; on �F;

belongs to H12 .@˝/. Then, to prove the existence of 2 H1.˝/ such that D b

on @˝ we find as the solution of the Laplace problem

� D 0 in ˝; D b on @˝:

This proves that �0. OK/ D ˘0. OK/, and we characterized ˘0. OK/ in terms of thefunctions O'1 and O'2 only (and not O'3—the third component). Since ˘. OK/ D˘0. OK/C NG, we have

˘. OK/ D f N' D . O'1; O'2/ 2 .H1.˝//2 W N' D NG on �Lg:

Lemma 6.8. Let N' D . O'1; O'2/ 2 ˘. OK/ be such that the following condition, whichwill be denoted as condition (D’) holds

Z

˝

�

O'1.x; y/ @@x1

.x/C O'2.x; y/ @@x2

.x/

�

dxdy D 0 for all 2 C10.!/: (6.62)


ThenZ

˝

p?.x/

�@ O'1@x1

C @ O'2@x2

C @ O'3@y

�

dxdy D 0; (6.63)

which is equivalent toZ

˝

p?.x/

�@ O'1@x1

C @ O'2@x2

�

dxdy

CZ

!

p?.x/

�

O'1.x; h.x// @h

@x1C O'2.x; h.x// @h

@x2

�

dx D 0: (6.64)

Proof. Let m 2 C10.!/, m ! p? in L2.!/. Then

Z

˝

m.x/

�@ O'1@x1

C @ O'2@x2

C @ O'3@y

�

dxdy

D �Z

˝

�

O'1 @m

@x1C O'2 @m

@x2

�

dxdy CZ

@˝

m.x/

3X

iD1O'ini

!

dS;

as @m@y D 0: Now, the surface integral equals zero since m 2 C1

0.!/ and O' � n D 0

on �F [ !. The first integral on the right-hand side equals zero by condition (D’).Thus, the integral on the left-hand side equals zero. We pass to the limit m ! p? inL2.!/ to get (6.63). As

Z

˝

p?.x/@ O'[email protected]; y/ dxdy D

Z

!

p?.x/Z h.x/

0

@ O'[email protected]; y/ dy dx

DZ

!

p?.x/ O'3.x; h.x//dx DZ

!

p?.x/

2X

iD1O'i.x; h.x//

@h

@xi

!

dx;

we have the equivalence of (6.63) and (6.64). utThe above lemma tells us that if in the limit variational inequality (6.59) we choosea test function belonging to the set

F OK WD f N' 2 ˘. OK/ and N' satisfies condition (D’)g; (6.65)

then this inequality will reduce to the following one,

2X

iD1

1

2

Z

˝

@u?i@y

@. O'i � u?i /

@ydxdy C Ol

Z

!

u?i .x; h.x//. O'i � u?i /.x; h.x// dx C

OkZ

!

.j N' � sj � ju? � sj/ dx � 0 for all N' 2 F OK ; (6.66)

in which the pressure is not present.


Let

F WD fu? W u? is a weak limit of Ou in the topology of Vy � Vyg:

We prove that F is contained in the closure of ˘. OK/ in the topology of Vy � Vy.This will allow us later to get useful conclusions from variational inequalities (6.59)and (6.66), by taking test functions from F .

Lemma 6.9. Each solution u? of the limit variational inequality satisfies condition.D0/, and u? � OG satisfies the following stronger condition (D)

Z

˝

�

.u?1 � OG1/@

@x1.x/C .u?2 � OG2/

@

@x2.x/

�

dxdy D 0 for all 2 C1.!/:

(6.67)

Proof. From div Ou D 0 in ˝ we have, for 2 C10.!/,

0 DZ

˝

.x/

�@Ou1@x1

C @Ou2@x2

C @Ou3@y

�

dxdy D �Z

˝

�

Ou1 @@x1

C Ou2 @@x2

�

dxdy

(as Ou � n D 0 on �F [ !). As Oui ! u?i weakly in Vy, i D 1; 2, we obtain condition(D’) for u?, that is,

Z

˝

�

u?1@

@x1C u?2

@

@x2

�

dxdy D 0 for all 2 C10.!/:

Taking Ou� OG instead of Ou at the beginning of this proof, as .Ou� OG/ �n D 0 on �F [!and Ou � OG D 0 on �L, we obtain (6.67) in the limit. utFrom Lemmas 6.8 and 6.9 it follows that condition (D’) is the right one to get rid ofthe pressure terms in the limit variational inequality. Note that in condition (D’) wedo not assume that N' 2 .H1.˝//2, that is, (6.62) has sense for N' 2 .L2.˝//2, forexample. The solution u? satisfies (D’) and is not in .H1.˝//2.

We shall characterize the set F of solutions u?.

F Dnu? D .u?1 ; u

?2/ 2 .L2.˝//2 W @u?i

@y2 L2.˝/; i D 1; 2; u? satisfies .D0/;

u? satisfies condition .C�F / coming from Ou � n D 0 on �F

o: (6.68)

Our main problem now is to find relations between F OK given by (6.65) and F givenby (6.68). We do not know what is the condition .C�F / in (6.68) in the general case(note that if �F is given by y D const: we have no condition on u? coming fromOu � n D 0 on �F). Thus we define


F1 D�

u? D .u?1 ; u?2/ 2 .L2.˝//2 @u?i

@y2 L2.˝/; i D 1; 2; u? satisfies .D0/

�

:

(6.69)

Then F F1 and it is sufficient to prove the following lemma.

Lemma 6.10. F1 is contained in the closure of ˘. OK/ in the topology of Vy � Vy.

Proof. Let v 2 F o1 D fv D u? � OG W u? 2 F1g. We consider the map

u D ! � .0; 1/ ! ˝ W .x; y/ 7! .x; h.x/y/ D .X;Y/;

and define

v.X;Y/ D v .x; h.x/y/ D Nv.x; y/ and U.x; y/ D Nv.x; y/h.x/:

Then

• v and U are of the same regularity,• “v satisfies (D’) or (D) in ˝” is equivalent to “U satisfies (D’) or (D) in u.”

Indeed for all 2 C1.!/ we have

Z

˝

v.X;Y/r.X/dXdY DZ

uU.x; y/r.x/dxdy;

• “v D 0 on �L” is equivalent to “U D 0 on �L,”

and also

• If Un.x; y/ ! U.x; y/ in Vy.u/ � Vy.u/, then vn.X;Y/ ! v.X;Y/ in Vy.˝/ �Vy.˝/.

Indeed,

Z

˝

jvn.X;Y/ � v.X;Y/j2 dxdy DZ

uj Nvn.x; y/ � Nv.x; y/j2h.x/ dxdy

DZ

uj Nvn.x; y/h.x/ � Nv.x; y/h.x/j2 1

h.x/dxdy

DZ

ujUn.x; y/ � U.x; y/j2 1

h.x/dxdy

� 1

hmin

Z

ujUn.x; y/ � U.x; y/j2 dxdy;


and

Z

˝

ˇˇˇˇ@vn

@Y.X;Y/ � @v

@Y.X;Y/

ˇˇˇˇ

2

dXdY

DZ

u

ˇˇˇˇ@ Nvn

@y.x; y/

1

h.x/� @ [email protected]; y/

1

h.x/

ˇˇˇˇ

2

h.x/ dxdy

DZ

u

ˇˇˇˇh.x/

@ Nvn

@y.x; y/ � h.x/

@ [email protected]; y/

ˇˇˇˇ

21

h3.x/dxdy

� 1

h3min

Z

u

ˇˇˇˇ@Un

@y.x; y/ � @U

@y.x; y/

ˇˇˇˇ

2

dxdy:

Let U� .x; y/ D U.�x; y/, � > 1,

Z

uU.x; y/r.x/ dxdy D 0 for all 2 C1.!/:

Then as �.z/ D 1�2�

z�

� 2 C1.!/, we have

Z

uU� .x; y/r.x/dxdy D

Z

uU.�x; y/r.x/dxdy D

Z

uU.z; y/r

z

�

�� dzdy

�2D 0:

Thus, U� satisfies condition (D). Since it is not smooth enough in x, we take theusual mollification �" ? U� with respect to x variable only which is in ˘0. OK/ forsmall ", " � ".�/. We check condition (D).

Z

u.�" ? U� /.x; y/r.x/dydx D

Z

!

Z 1

0

�Z

!

U� .t; y/�".t � x/dt

�

r.x/dydx

DZ

!

Z 1

0

U� .t; y/

�Z

!

�".x � t/r.x/dx

�

dydt

DZ

uU� .t; y/r.�" ? /.t/dydt D 0:

As �"? 2 C1.!/ and U� satisfies condition (D), then �"?U� also satisfies condition.D/ in u. Now,

kU � �" ? U�kV2y� kU � U�kV2y

C kU� � �" ? U�kV2y;

and we take � close to 1 first and then small ".Thus there exists �" ? U� 2 ˘0. OK/ such that kU � �" ? U�kV2y .u/ ! 0, which is

equivalent to the existence of �" ? v� 2 ˘0. OK/ with kv � �" ? v�kV2y .˝/! 0. ut


6.6 Strong Convergence of Velocities and the Limit Equation

From the Korn inequality (6.24) written in new variables we conclude that if Ou andO' in OK are divergence free then, in particular

2X

iD1

��@.Oui � O'i/

@y

��

2

L2.˝/

� Oa.Ou � O'; Ou � O'/C C.�F; h/Z

�F

jOu � O'j2 dS:

By (6.20),

Oa.Ou � O'; Ou � O'/ D Oa. O'; O' � Ou/C Oa.Ou; Ou � O'/

� Oa. O'; O' � Ou/C OlZ

�F

Ou. O' � Ou/ dS C OkZ

!

.j O' � sj � jOu � sj/ dx;

and we have

2X

iD1

��@.Oui � O'i/

@y

��

2

L2.˝/


�F

Ou. O' � Ou/ dS

C OkZ

!

.j O' � sj � jOu � sj/ dx

C C.�F; h/Z

�F

jOu � O'j2 dS;

where

Oa. O'; O' � Ou/ D 1

2

2X

i;jD1

Z

˝

�

"@ O'i

@xjC "

@ O'j

@xi

��

"@

@xj. O'i � Oui/C "

@

@xi. O'j � Ouj/

�

dxdy

C 1

2

2X

iD1

Z

˝

�@ O'i

@yC "2

@ O'3@xi

��@

@y. O'i � Oui/C "2

@

@xi. O'3 � Ou3/

�

dxdy

C 2

Z

˝

"@ O'3@y"@

@y. O'3 � Ou3/dxdy:

Thus

2X

iD1

��@.Oui � O'i/

@y

��

2

L2.˝/C Ok

Z

!.jOu � sj � j O' � sj/dx


�F

Ou. O' � Ou/ dS C C.�F; h/Z

�F

jOu � O'j2 dS:

6.6 Strong Convergence of Velocities and the Limit Equation 135

Let Nu D .Ou1; Ou2/, u? D .u?1 ; u?2/ as in (6.52) and let N' D . O'1; O'2/ 2 ˘. OK/. Then

lim sup"!0

(2X

iD1

��@.Oui � O'i/

@y

��

2

L2.˝/

C OkZ

!

.jNu � sj � j N' � sj/dx

)

�Z

˝

@ N'@y

@. N' � u?/

@ydxdy C Ol

Z

�F

u?. N' � u?/ dS C C.�F; h/Z

�F

ju? � N'j2 dS;

and

2X

iD1

��@.Oui � O'i/

@y

��

2

L2.˝/

C OkZ

!

.jNu � sj � j N' � sj/dx

�Z

˝

@ N'@y

@. N' � u?/

@ydxdy C Ol

Z

�F

u?. N' � u?/ dS

C C.�F; h/Z

�F

ju? � N'j2 dS C ı for all ı < ".ı/:

By Lemma 6.10, there exists a sequence of functions N' in ˘. OK/ which has u? as alimit in V2

y , whence

2X

iD1

��@.Oui � O'i/

@y

��

2

L2.˝/

C OkZ

!

.jNu � sj � ju? � sj/dx � ı for all ı < ".ı/:

Since ı > 0 is arbitrary, then by the weak lower semicontinuity of the functionalv 7! R

!jv � sj dx, we get

Z

!

.jNu � sj � ju? � sj/dx ! 0:

and from Lemma 6.5 the strong convergence of Nu" to u? in QV2y \ F1 follows.

Now, we obtain from the limit inequality the classical relations for u? and p?.

Theorem 6.3. The limit functions u?; p? satisfy

p?.x1; x2; y/ D p?.x1; x2/ a:e: in ˝; p? 2 H1.!/; (6.70)

� 1

2

@2u?i@y2

C @p?

@xiD 0 .i D 1; 2/ in L2.˝/: (6.71)


Proof. Using Lemma 6.10 we first take O'1 D u?1 ˙ 1 with 1 2 H10.˝/ and

O'2 D u?2 , and then O'1 D u?1 and O'2 D u?2 ˙ 2 with 2 2 H10.˝/ in (6.59), to get

� 1

2

@2u?i@y2

C @p?

@xiD 0 in H�1.˝/; i D 1; 2: (6.72)

Multiplying (6.72) by i.x; y/ D y.y � h.x//.x/ with 2 H10.!/ and using the

Green formula we have

Z

˝

@u?i@y

@ i

@ydxdy D

Z

˝

@u?i@y

@Œy2 � yh.x/�.x/

@ydxdy

DZ

˝

@u?i@yŒ2y � h.x/�.x/dydx

D �Z

˝

2u?i .x; y/.x/dydx CZ

@˝

u?i Œ2y � h.x/�.x/n3dS

D �Z

!

2h.x/Qu?i .x/.x/dx CZ

!

h.x/u?i .x; 0/.x/dx

CZ

!

h.x/u?i .x; h.x//.x/dx;

where

h.x/Qu?i .x/ DZ h.x/

0

u?i .x; y/dy;

and

Z

˝

p?.x/@ i

@xidxdy D

Z

!

p?.x/

Z h.x/

0

@Œy2 � yh.x/�.x/

@xidy

!

dx

DZ

!

p?.x/@

@xi

.x/Z h.x/

0

Œy2 � yh.x/�dy

!

dx

D �16

Z

!

p?.x/@h3.x/.x/

@xidx:

Thus

� h.x/Qu?i .x/C h.x/

2u?i .x; h.x//C h.x/

2u?i .x; 0/ D h3.x/

6

@p?

@xi.x/ (6.73)

in H�1.!/. As u?i 2 Vy, Qu?i 2 L2.!/ and in consequence the left side of (6.73)belongs to L2.!/. This gives (6.70) and (6.71). ut

6.7 Reynolds Equation and the Limit Boundary Conditions 137

6.7 Reynolds Equation and the Limit Boundary Conditions

Theorem 6.4. The pair .u?; p?/ satisfies the following weak form of the Reynoldsequation

Z

!

�h3

12rp? � h

2s? � h

2s?h

�

r'dx CZ

!

s?hrh'dx D �Z

@!

' Qg � n dl (6.74)

for all ' 2 H1.!/, where

s?.x/ D u?.x; 0/; s?h.x/ D u?.x; h.x//;

and

Qg.x/ DZ h.x/

0

Og.x; y/dy for all x 2 @!:

Moreover, the traces �? D @u?

@y .�; 0/, �?h D @u?

@y .�; h.x//, s?h D u?.�; h.x//, ands? D u?.�; 0/ satisfy the following limit form of the Tresca boundary conditions (6.4)and the Fourier boundary condition (6.6)

j�?j D Ok H) 9� � 0 u? D s C ��?

j�?j < Ok H) u? D s

oa:e: in !; (6.75)

�?h rh � n C Ols?h D 0 a:e: on �F: (6.76)

Proof. Integrating twice (6.71) between 0 and y we obtain (for 1 � i � 2)

u?i .x; y/ D y2

2

@p?.x/

@xiC s?i .x/C y�?i .x/; (6.77)

and for y D h.x/ we get

s?h.x/ D h2

2rp?.x/C s?.x/C h�?.x/ a:e: in !: (6.78)

Now, integrating twice (6.71) between y and h.x/ we obtain (for 1 � i � 2)

u?i .x; y/ D�

h2

2� hy C y2

2

�@p?.x/

@xiC s?h;i.x/C .y � h/�?h;i.x/;


and for y D 0 we get

s?.x/ D h2

2rp?.x/C s?h.x/ � h�?h .x/ a:e: in !: (6.79)

Taking now the average

Qv.x/ D 1

h.x/

Z h.x/

0

v.x; y/ dy

of (6.77) we obtain

hQu?i .x/ D h3

6

@p?.x/

@xiC hs?i .x/C h2

2�?i .x/:

Moreover, for all ' 2 H1.!/, we have

0 DZ

˝

' div Ou dydx DZ

!

'.x/Z h

0

2X

iD1

@Oui

@xiC @Ou3@y

!

dydx

DZ

!

'.x/

2X

iD1

@.hQOui/

@xiC Ou3.x; h/ � Ou3.x; 0/

!

dx:

Then as Ou � n D Ou3.x; 0/ D 0 on !, and �F is defined by y D h.x/, we obtain

Ou3.x; h.x// D Ou1.x; h.x// @h

@x1C Ou2.x; h.x// @h

@x2;

and

Z

!

'.x/

2X

iD1

@.hQOui/

@xiC Oui.x; h.x//

@h

@xi

!

dx D 0:

Using the Green formula we have

�2X

iD1

Z

!

hQOui@'

@xidx C

Z

!

'.x/

2X

iD1Oui.x; h.x//

@h

@xi

!

dx C2X

iD1

Z

@!

hQOuini' dl D 0:

As Oui ! u?i weakly in Vy, QOui ! Qu?i weakly in L2.!/, and as @! @˝, we obtain

Z

!

hQOur'dx �Z

!

'.x/ .Ou.x; h.x//rh/ dx DZ

@!

Qg.x/ � n' dl for all ' 2 H1.!/:

6.7 Reynolds Equation and the Limit Boundary Conditions 139

From (6.79) we haveZ

!

�h3

6rp? C hs? C h2

2�?�

r' dx �Z

!

's?h � rh dx DZ

@!

' Qg � n dl (6.80)

for all ' 2 H1.!/. Then using (6.78) and (6.80), we obtain the weak formula-tion (6.74) of the Reynolds equation.

To prove (6.75) we transform (6.59). Using the Green formula, (6.71), conditions'i D Ogi on �L, ni D 0 on ! for i D 1; 2, and ' � n D 0 on ! [ �F, as well as thedensity of D.˝/ in ˘. OK/, and of D.˝/ in L2.˝/, we obtainZ

!

Ok.j'.x; 0/ � sj � js? � sj/ dx

CZ

!

Ols?h.'.x; h.x// � s?h/ dx �Z

!

�?.'.x; 0/ � s?/ dx

CZ

!

�?h .'.x; h.x// � s?h/rh � n dx � 0 for all ' 2 .L2.˝//2: (6.81)

We take first

'.x; y/ D�

s?h.x/ if y D h.x/;si.x/ if y D 0;

then

'.x; y/ D�

s?h.x/ if y D h.x/;2s?i � si if y D 0;

in (6.81), to obtainZ

!

Okjs? � sj � �?.s? � s/�

dx D 0; (6.82)

and from (6.81), with

'.x; y/ D�

s?h.x/ if y D h.x/;�.x/C s if y D 0;

and � 2 .L2.!//2, we obtain

Z

!

Okj�j � �?��

dx �Z

!

Okjs? � sj � �?.s? � s/�

dx: (6.83)

From (6.82) and (6.83) we deduce

Z

!

Okj�j � �?��

dx � 0 for all � 2 .L2.!//2: (6.84)


We will show that j��j � Ok a.e. on !. Suppose that there exists a set of positivemeasure !1 ! such that j��j > Ok on this set. Taking � D 0 on ! n!1 and � D ��on ! in (6.84) we get

Z

!1

j��j.Ok � j��j/ dx � 0:

As the integrand must be a strictly negative function, we arrive at a contradiction.We have thus proved

j�?j � Ok a:e: on !:

Then

Okjs? � sj � j�?jjs? � sj � �? � .s? � s/ a:e: on !

and

Okjs? � sj � �? � .s? � s/ � 0 a:e: on !:

From (6.82) we deduce that

Okjs? � sj � �? � .s? � s/ D 0 a:e: on !: (6.85)

If j�?j D Ok, then from (6.85) we have j�?jjs? � sj D �?.s? � s/ a.e. on !, whichimplies the existence of � � 0 such that s? � s D ��?.

If, in turn, j�?j < Ok, then from (6.85) we have

0 D Okjs? � sj � �?.s? � s/ � .Ok � j�?j/js? � sj a:e: on ! \ fj��j < kg;

whence s? � s D 0 a.e. on ! \ fj��j < kg, and (6.75) follows.We shall prove (6.76). First we take

'.x; y/ D��.x/ if y D h.x/s?.x/ if y D 0

with � 2 .L2.!//2 in (6.81), to get

Z

!

Ols?h.� � s?h/C �?h .� � s?h/rh � n�

dx � 0 for all � 2 .L2.!//2:

Then, taking � D C s?h and � D � C s?h we obtain

Z

!

Ols?h C �?h rh � n� dx D 0 for all 2 .L2.!//2;

which implies (6.76). ut

6.8 Uniqueness 141

Remark 6.5. From formula (6.74) we obtain the following form of the Reynoldsequation (for � D 1),

div

�h3

12rp? � h

2s? � h

2s?h

�

C s?hrh D 0 in H�1.!/:

In the simplest classical one-dimensional case, with ! D .a; b/, s?h D 0, s? Dconst, we obtain the well-known equation describing the pressure distribution in theinterval .a; b/,

� d

dx

�h3

6

dp?

dx

�

C s?dh

dxD 0: (6.86)

Observe how the Reynolds equation depends on the boundary conditions for thevelocity field. When � 6D 1, the second term on the left-hand side of (6.86) shouldbe multiplied by �.

6.8 Uniqueness

Theorem 6.5. There exists a unique solution .u?; p?/ in QVy � .L20.!/ \ H1.!// ofinequality (6.59).

Proof. Let .U1; p1/ and .U2; p2/ be two solutions of (6.59). Taking ' D U2 and' D U1, respectively, as test functions in (6.59) we reduce it to (6.66) and obtain

1

2

Z

˝

ˇˇˇˇ@.U1 � U2/

@y

ˇˇˇˇ

2

dxdy C OlZ

!

jU1.x; h.x// � U2.x; h.x//j2dx � 0:

Using Lemma 6.5 we get

kU2 � U1kV2yD 0 and jU1.x; h.x// � U2.x; h.x//j D 0 a:e: in !: (6.87)

Moreover, from (6.74), we have

Z

!

h3

6r.p2 � p1/r'.x/dx

DZ

!

h.x/ŒU1.x; 0/ � U2.x; 0/C U1.x; h.x// � U2.x; h.x//�r'.x/dx

CZ

!

h.x/.U2.x; h.x// � U1.x; h.x///rh � 'dx for all ' 2 H1.!/:


Now, taking ' D p2 � p1, and by (6.87) we get

kr.p2 � p1/kL2.!/ D 0;

and from the Poincaré inequality, as pi 2 L20.!/we conclude that kp2�p1kL2.!/ D 0.The uniqueness of u? follows then from (6.59). ut


(1) Remarks on boundary conditions. The right choice of boundary conditionsfor the velocity field in hydrodynamical problems is fundamental for applications.In lubrication problems the widely assumed non-slip condition when the fluid hasthe same velocity as surrounding solid boundary is not respected any more when theshear rate becomes too high 107s�1. This phenomenon has been studied in a lot ofmechanical papers, e.g., [120, 121, 214, 223]. Continuous experimental studies areconducted (e.g., [193]) but are still difficult due to thickness of the gap between thesolid surfaces which can be as small as 50 nm.

In such operating conditions, non-slip condition is induced by chemical boundsbetween the lubricant and the surrounding surfaces and by the action of the normalstresses, which are linked to the pressure inside the flow. On the contrary, tangentialstresses are so high that they tend to destroy the chemical bounds and induce slipphenomenon.

This is nothing else than a transposition of the well-known Coulomb law betweentwo solids [90] to the fluid–solid interface.

Although being implicitly used in numerical procedures for lubrication problem,a Reynolds thin film equation taking account of such slip phenomenon have beenhardly studied from the mathematical point of view. In [15, 21], respectively, thesimplified Tresca and Coulomb interface conditions, posed on only one of thesurrounding surfaces of a Newtonian fluid, were considered. This chapter is basedon the results from [23]. Corresponding results for non-Newtonian fluids have beenobtained in [20].

(2) Comments on technical problems. Two technical issues were crucial in themathematical analysis of the problem. First, we could not use the usual Korninequality as we did not assume that the velocity was equal to zero at one of theboundaries (on the top or the bottom) as is usually assumed in lubrication problems.We thus derived an analogue of the Korn inequality suitable for our boundaryconditions and such that the constants could be controlled appropriately as the gapbetween the surfaces approached zero. This led us to the main uniform estimate ofthe velocity fields and to the limit variational inequality, in consequence. Second, tobe able to make use of the latter, we had to characterize precisely the limit solutionspace and the set of admissible test functions. As the limit variational inequalitywas written in terms of the first two components of the velocity field, we had tocharacterize—in this very limit case—projections of the convex sets appearing inthe weak form of the Stokes flow. This allowed us, in particular, to obtain a strongerconvergence of the velocity fields than it is usually expected.

7Autonomous Two-Dimensional Navier–StokesEquations

We put our faith in the tendency for dynamical systems with alarge number of degrees of freedom, and with coupling betweenthose degrees of freedom, to approach a statistical state which isindependent (partially, if not wholly) of the initial conditions.

– George Keith Batchelor

This chapter contains some basic facts about solutions of nonstationaryNavier–Stokes equations

@u

@tC .u � r/u D �1

�rp C ��u C f ;

div u D 0;

in two-dimensional spatial domains. We assume that the system is autonomous, thatis, the external forces do not depend on time and that the domain of the flow isbounded. The boundary conditions are either periodic or homogeneous of Dirichlettype. In what follows, the viscosity � is a positive constant and we set � D 1.

First, we prove the existence of a unique global in time weak solution. Thenwe analyze the time asymptotics of solutions proving the existence of the globalattractor and showing its trivial structure in some cases of special external forces.Finally, we consider the problem of behavior of the kinetic energy of the fluid.To this end we introduce the notion of different scales of the flow, considered inthe theory of turbulence, and corresponding in a way to different spatial dimensionsof vortices of a given turbulent flow.

7.1 Navier–Stokes Equations with Periodic BoundaryConditions

Let Q D Œ0;L�2 and define spaces

H D fu D .u1; u2/ 2 PL2p.Q/ W div u D 0g;


143

144 7 Autonomous Two-Dimensional Navier–Stokes Equations

and

V D fu D .u1; u2/ 2 PH1p.Q/ W div u D 0g;

(cf. Sect. 3.4) with the scalar products

.u; v/ DZ

Qu.x/ � v.x/ dx in H; ..u; v// D

Z

Qru.x/ W rv.x/ dx in V;

and the norms

kukH D juj DsZ

Qju.x/j2dx; juj2 D .u; u/;

and

kukV D kuk DsZ

Qjru.x/j2dx; kuk2 D ..u; u// D .ru;ru/;

respectively. The duality between V and its dual V 0 is denoted by h�; �i.The weak formulation of the initial boundary value problem for the Navier–

Stokes equations is given in Theorem 7.1 below. We adopt the notation fromChap. 4. In particular, the Stokes operator A and the trilinear form b as well asthe associated nonlinear operator B are defined in this chapter the same as inChap. 4 (with the obvious difference that in Chap. 4 the stationary problem is three-dimensional and in present chapter we consider the two-dimensional problem).

Our aim is to prove the following existence and uniqueness result.

Theorem 7.1. Let u0; f 2 PH. Then there exists a unique function u W Œ0;1/ ! Hsuch that

u 2 C.Œ0;T�I H/ \ L2.0;TI V/; u.0/ D u0;

for all T > 0, satisfying the following identity

d

dt.u.t/; v/C �..u.t/; v//C b.u.t/; u.t/; v/ D .f ; v/ for all v 2 V; (7.1)

in the sense of scalar distributions on .0;1/. Moreover, the function u W Œ0;1/ !H is bounded and the map u0 ! u.t/ is continuous as a map in H.

Proof. Let f .x/ D P1jD1 Ofj!j.x/, Ofj D .f ; !j/, where !j, j D 1; 2; : : :, is an

orthonormal sequence in H, a sequence of eigenvectors of the Stokes operator. Weseek our solution u as a suitable limit of the sequence of approximate solutions un,n D 1; 2; : : :, which are finite dimensional solutions of the following problems.

7.1 Navier–Stokes Equations with Periodic Boundary Conditions 145

Problem 7.1. Let n 2 N, fn.x/ D PnjD1 Ofj!j.x/, un.0/ D u0n D Pn

jD1.u0; !j/!j.x/.On the interval .0;1/ find un.t/ D un.�; t/ of the form

un.x; t/ DnX

jD1unj.t/!j.x/

such that

d

dt.un.t/; !j/ C �..un.t/; !j//C b.un.t/; un.t/; !j/ D .fn; !j/ for j D 1; : : : ; n;

un.0/ D u0n: (7.2)

Observe that (7.2) is a system of n ordinary differential equations for the vectorfunctions

t ! .un1.t/; : : : ; unn.t//

with initial condition ..u0; !1/; : : : ; .u0; !n//. From the classical theorem aboutexistence and uniqueness of solutions of the system

Px D F.x/; x.0/ D x0;

with polynomial (hence locally Lipschitz) right-hand side it follows existence of aunique local in time solution un.t/ on some interval Œ0; tn/, tn > 0, for every positiveinteger n. We shall show that the solutions un are defined in fact for all positive times(are global in time). This will follow from the first energy inequality. We multiplythe jth equation in (7.2) by unj.t/ and add the resulting n equations to get

1

2

d

dtjun.t/j2 C �kun.t/k2 C b.un.t/; un.t/; un.t// D .fn; un.t//:

As b.un.t/; un.t/; un.t// D 0, using the Cauchy inequality to the right-hand side aswell as the Poincaré inequality, cf. (3.16),

juj � Ckuk;

we obtain the first energy inequality

d

dtjun.t/j2 C �kun.t/k2 � C.�/jfnj2; (7.3)

and

d

dtjun.t/j2 C k1.�/jun.t/j2 � C.�/jfnj2: (7.4)


Integrating (7.3) in an interval .0; t/ we obtain

jun.t/j2 C �

Z t

0

kun.t/k2ds � C.�/tjfnj2 C jun.0/j2 � C.�/tjf j2 C ju.0/j2:

Hence, for each t > 0, jun.t/j < 1 and thus the local solution can be extended tothe whole positive semiaxis. From this inequality we can see that for each T > 0

our approximate solutions are uniformly bounded in the space

C.Œ0;T�I H/ \ L2.0;TI V/:

Moreover, the approximate solutions are infinitely differentiable functions in xand in t.

Exercise 7.1. Prove that if f D 0 then any approximate solution, that is, solutionstarting from an arbitrary initial data u0 2 H, converges at least exponentially fastto zero in H.

To get the boundedness in H of our approximate solutions we shall use the Gronwallinequality (cf. Lemma 3.20). In our case, from the Gronwall inequality appliedto (7.4), we have

jun.t/j2 � jun.0/j2e�k1t CZ t

0

ek1.s�t/C.�/jfnj2dt

D jun.0/j2e�k1t C C.�/

k1jfnj2.1 � e�k1t/:

Therefore, every approximate solution is bounded on Œ0;1/ as a function withvalues in H.

From the uniform boundedness of the sequence of approximate solutions inC.Œ0;T�I H/ \ L2.0;TI V/ it follows that there exists a subsequence (denoted againby .un/) and some u in L1.0;TI H/ \ L2.0;TI V/ such that

un ! u weakly � � in L1.0;TI H/; (7.5)

which means that

limt!1

Z T

0

Z

Qun.x; t/'.x; t/dxdt D

Z T

0

Z

Qu.x; t/'.x; t/dxdt

for all ' from L1.0;TI H/, and

un ! u weakly in L2.0;TI V/; (7.6)


which means that

limt!1

Z T

0

Z

Qun.x; t/'.x; t/dxdt D

Z T

0

Z

Qu.x; t/'.x; t/dxdt

for all ' from L2.0;TI V/. We shall prove that this limit u is a solution ofProblem 7.1. From (7.2) we have

�Z T

0

.un.t/; v/�0.t/dt C �

Z T

0

..un.t/; v//�.t/dt

CZ T

0

b.un.t/; un.t/; v/�.t/dt DZ T

0

.fn; v/�.t/dt

for all v in Vn D spanf!1; : : : ; !ng and � in C10 ..0;T//. We shall show that this

identity is satisfied for the limit function u and for all v in V .As for the linear terms, we have, for any v 2 Vm, where m is any given integer,

limn!1

Z T

0

.un.t/; v/�0.t/dt D lim

n!1

Z T

0

Z

Qun.x; t/v.x/�

0.t/dxdt

DZ T

0

Z

Qu.x; t/v.x/�0.t/dxdt D

Z T

0

.u.t/; v/�0.t/dt;

which follows from the definition of the weak-star convergence in L1.0;TI H/, asv.x/�0.t/ 2 L1.0;TI H/. Similarly,

limn!1

Z T

0

..un.t/; v//�.t/dt D limn!1

Z T

0

Z

Qrun.x; t/rv.x/�.t/dxdt

DZ T

0

Z

Qru.x; t/rv.x/�.t/dxdt D

Z T

0

..u.t/; v//�.t/dt;

as un ! u weakly in L2.0;TI V/ and v.x/�.t/ 2 L2.0;TI V/. Moreover,

limn!1

Z T

0

.fn; v/�.t/dt DZ T

0

.f ; v/�.t/dt;

as fn ! f strongly in L2.Q/.The only problem which remains is that with the convergence of the nonlinear

term. Weak convergences of un to u as in (7.5) and (7.6) are not sufficient to pass tothe limit in the nonlinear term. This is a typical problem when dealing with nonlinearterms.

Exercise 7.2. Prove that, in general, if un ! u and vn ! v weakly in L2.Q/ then itdoes not follow that unvn ! uv in the sense of distributions on Q.


However, to prove that

limn!1

Z T

0

b.un.t/; un.t/; v/�.t/dt DZ T

0

b.u.t/; u.t/; v/�.t/dt (7.7)

it suffices to know that

un ! u strongly in L2.Q/ (7.8)

for the above subsequence of the sequence of approximate solutions (or for some ofits subsequences).

Lemma 7.1. Let a.�; �/ be a continuous bilinear form in the Banach space X withnorm k � k, un ! u strongly and vn ! v weakly in X. Then limn!1 a.un; vn/ Da.u; v/.

Proof. Since the sequence un is strongly convergent and the sequence vn is boundedas weakly convergent, it is a.un; vn/ � a.u; v/ D a.un � u; vn/ C a.u; vn � v/ andja.un �u; vn/j � Ckun � ukkvnk ! 0. As for the second term, there exists u� 2 V 0,the dual of V such that a.u; vn � v/ D hu�; vn � viV0�V ! 0 since the sequence vn

is weakly convergent in H. utExercise 7.3. Prove (7.7) using (7.6), (7.8), and Lemma 7.1.

Exercise 7.4. Prove (7.8) by proving first that

the sequence f@tung is bounded in L2.0;TI V 0/

and then using Theorem 3.18 and Lemma 7.1.

We have thus

�Z T

0

.u.t/; v/�0.t/dt C �

Z T

0

..u.t/; v//�.t/dt (7.9)

CZ T

0

b.u.t/; u.t/; v/�.t/dt DZ T

0

.f ; v/�.t/dt

for all v in Vm D spanf!1; : : : ; !mg, where m is any positive integer and � is inC10 .0;T/.

Exercise 7.5. Prove that (7.9) holds for all v 2 V using the fact that the union ofall the spaces Vm is dense in V .

To prove that u 2 C.Œ0;T�I H/ for all T > 0 we apply Lemma 3.16.

Exercise 7.6. Prove, using the du Bois-Reymond lemma, that (7.9) implies (7.1).

Exercise 7.7. Prove that u.0/ D u0.


Exercise 7.8. Prove that we have the following equality in V 0,

du.t/

dtC �Au.t/C B.u.t// D f ;

for a.e. t 2 .0;T/. Use Proposition 3.2 and property (3.26) of the evolution tripleV H V 0.

To prove the uniqueness of a solution let us assume, on the contrary, that there existtwo different solutions, u and v, to our problem

du.t/

dtC �Au.t/C B.u.t// D f ; u.0/ D u0

on the interval Œ0;1/. Setting w.t/ D u.t/ � v.t/ we have

dw.t/

dtC �Aw.t/C B.u.t// � B.v.t// D 0 in V 0:


dw.t/

dtC �Aw.t/C B.u.t// � B.v.t//;w.t/

D 0;

that is, see Theorem 3.16,

1

2

d

dtjw.t/j2 C �kw.t/k2 C b.u.t/; u.t/;w.t// � b.v.t/; v.t/;w.t// D 0:

Further, using a property of the trilinear form b, we obtain

1

2

d

dtjw.t/j2 C �kw.t/k2 D �b.w.t/; v.t/;w.t//:

We shall estimate the right-hand side applying to it the Ladyzhenskaya inequality(cf. Exercise 4.8),

kukL4.Q/2 � Cjuj1=2kuk1=2 for u 2 V:

We have

jb.u; v;w/j � kukL4.Q/2kvkkwkL4.Q/2

� c1juj1=2kuk1=2kvkjwj1=2kwk1=2:

Therefore,

b.w.t/; v.t/;w.t// � c1jw.t/kjv.t/kkw.t/k� �

2kw.t/k2 C c2

�jw.t/j2kv.t/k2:


We have then

d

dtjw.t/j2 C �kw.t/k2 � c2

�jw.t/j2kv.t/k2:

Now, in view of the Gronwall inequality we have, for t � 0,

jw.t/j2 � jw.0/j2 exp

�c2�

Z t

0

kv.s/k2ds

�

:

As w.0/ D 0, u.t/ D v.t/ for all t > 0 which proves the uniqueness. From theuniqueness result together with our construction of a weak solution on an arbitraryinterval .0;T/ follows the existence of a unique global solution defined on theinterval .0;1/.

Observe that from the above inequality it follows also the continuous dependenceof solutions on the initial data: the map u0 ! u.t/ is continuous as a map in H fort > 0. This completes the proof of Theorem 7.1. utRemark 7.1. One can prove that under the assumptions of Theorem 7.1 the solutionu belongs to the space L2.�;TI D.A// for all T > 0 and 0 < � < T , where D.A/ isthe domain of the Stokes operator, cf. Sect. 4.1.

Exercise 7.9. Compute the constants C.�/, k1.�/ in inequality (7.4).

Exercise 7.10. Prove an existence and uniqueness theorem analogous to Theo-rem 7.1 for the case of homogeneous Dirichlet boundary conditions u D 0 at@˝ � .0;T/, where ˝ is a bounded open set in R

2, and the spaces H and V inTheorem 7.1 are accordingly modified.

Exercise 7.11. Consider the nonhomogeneous Dirichlet boundary condition u D aat @˝ � .0;T/, where ˝ is a bounded open set in R

2 with sufficiently smoothboundary, and where the function a is defined in ˝, a 2 H1.˝/, div a D 0, so thatu � a 2 V . Formulate and prove an existence and uniqueness theorem analogous toTheorem 7.1 for this case.

Exercise 7.12. Non-autonomous problem. Using the argument of the proof ofTheorem 7.1 prove the following result.

Theorem 7.2. Let V H V 0 be either given by the homogeneous Dirichletboundary conditions on an open and bounded set ˝ R

2 with Lipschitz boundaryor the periodic boundary conditions on a two-dimensional box. Let u0 2 H andf 2 L2loc.R

CI V 0/. Then there exists a unique function u W RC ! H such that

u 2 L2.0;TI V/ \ C.Œ0;T�I H/; u.0/ D u0;

for all T � 0, satisfying the identity

d

dt.u.t/; v/C �..u.t/; v//C b.u.t/; u.t/; v/ D hf .t/; vi for all v 2 V

7.2 Existence of the Global Attractor: Case of Periodic Boundary Conditions 151

in the sense of scalar distributions on .0;1/. Moreover, the map u0 ! u.t/ iscontinuous as a map in H.

Remark 7.2. Consider either the two-dimensional periodic problem or the homo-geneous Dirichlet problem on the bounded domain ˝ R

2 of class C2. Letf 2 L2loc.R

CI H/ and u0 2 H. Then the solution u given in Theorem 7.2 belongsto L2.�;TI D.A// for all 0 < � < T < 1 (see, for example, Theorem 7.4 in [99] orTheorem III.3.10 in [219]).

Remark 7.3. For the case of homogeneous Dirichlet boundary conditions, theexistence results of Theorems 7.1 and 7.2 can be extended to the case of unboundeddomains, where the embedding V H is not compact. For the treatment of this casesee, e.g., [201, 219].

7.2 Existence of the Global Attractor: Case of PeriodicBoundary Conditions

We shall present first some notions from the theory of dynamical systems.

Definition 7.1. Let X be a complete metric space. A family of maps S.t/ W X ! X,t 2 Œ0;1/, denoted as fS.t/gt�0 is called a semigroup in X if S.0/ is the identitymap and for all s; t > 0 it is S.t C s/ D S.t/S.s/. A semigroup is continuous in X ifall the maps are continuous in X.

Definition 7.2. Let fS.t/gt�0 be a continuous semigroup in a complete metricspace X. Then a subset A of X is a global attractor for this semigroup in X ifthe following conditions hold:

.i/ invariance: S.t/A D A for all t > 0,.ii/ compactness: A is a nonempty and compact subset of X,.iii/ attraction: for every bounded subset B of X and every " > 0 there exists

t0 D t0.B; "/ such that for all t � t0, S.t/B O".A /, where O".A / is the"-neighborhood of A ,

O".A / D[

x2AB.x; "/;

and B.x; "/ is the open ball in X with radius " and centered at x.

Definition 7.3. The set A X shall be called invariant with respect to thesemigroup fS.t/gt�0 if S.t/A D A for all t � 0.

Definition 7.4. The Hausdorff semi-distance in the metric space X between twononempty sets A;B X is defined as

distX .A;B/ D supx2A

infy2B�.x; y/:


Exercise 7.13. Prove that the last property in the definition of the global attractor(attraction) is equivalent to say that for every bounded set B X we have

limt!1 distX .S.t/B;A / D 0:

Definition 7.5. A subset B in X is absorbing if for every bounded subset G in Xthere exists a time t0 D t0.G/ such that for all t � t0, S.t/G B.

Schematic illustration of a global attractor and an absorbing set is depicted inFig. 7.1.

Definition 7.6. A semigroup fS.t/gt�0 in X is uniformly compact if for everybounded subset G in X there exists a time t0 D t0.G/ such that the set

St�t0

S.t/Gis precompact in X.

Exercise 7.14. By !.B/ D Ts�0

St�s S.t/B we define the !-limit set !.B/ of a

subset B of X (cf. Fig. 7.2 for an illustration of an !-limit set). Prove that � 2 !.B/if and only if there exists a sequence �n 2 B and a sequence tn ! 1 such thatS.tn/�n ! � as n ! 1.

We shall prove our main theorem about existence of a global attractor.

Theorem 7.3. Let fS.t/gt�0 be a continuous semigroup in a complete metricspace X. If there exists a bounded absorbing set B X and the semigroup isuniformly compact in X, then the !-limit set of B, A D !.B/, is a global attractorfor this semigroup in X.

Proof. Let B be an absorbing subset in X.

Step 1. !.B/ is nonempty and compact in X. As !.B/ D Ts�0

St�s S.t/B, where

the setsS

t�s S.t/B are nonempty and compact for s � t0.B/ (the semigroupis uniformly compact and B is bounded) and constitute a nonincreasing family

Fig. 7.1 The absorbing set Band the global attractor A forthe semigroup fS.t/gt�0. Alltrajectories after some time(depending on the initialcondition u0) are in the set B.They become closer andcloser to the attractor A asthe time goes to infinity


Fig. 7.2 Illustration of the !-limit set of the point u0. The set !.fu0g/ consists of all cluster pointsof all sequences fS.tn/u0g for tn ! 1

of sets as s grows, then !.B/ is not empty and compact in view of the Cantortheorem.Step 2. !.B/ is invariant: S.t/!.B/ D !.B/ for all t > 0.()) S.t/!.B/ !.B/. Let S.t/ D �, 2 !.B/. We shall show that � 2 !.B/.For there exist sequences: n in B and tn ! 1 such that S.t/ n ! . Hence,S.t/S.tn/ n D S.t C tn/ n ! S.t/ D � (as the operator S.t/ is continuous).Moreover, � 2 !.B/ by definition of the !-limit set, as the sequence S.t C tn/ n

converges to �, where t C tn ! 1.(() !.B/ S.t/!.B/. Let � 2 !.B/. We shall show that there exists 2!.B/ such that S.t/ D �. Let S.tn/�n ! �, tn ! 1, �n 2 B. As the setsS

s�tn�t S.s/B are precompact for tn�t � t0 thus the sequence S.tn�t/�n containsa convergent subsequence S.t � t/� ! for some 2 X. But 2 !.B/ as� 2 B and t � t ! 1. Moreover, S.t/S.t � t/� D S.t/� ! � D S.t/ ,from the continuity of S.t/.Step 3. A D !.B/ attracts bounded subsets G of X.Assume that this is not true. Then distX.S.t/G;A / � ı > 0 for t ! 1, thatis, there exist sequences tn ! 1 and �n 2 G such that distX .S.tn/�n;A / �ı > 0 for all n. On the other hand, the sequence S.tn/�n contains a convergentsubsequence, S.t/� ! , 2 A D !.B/ as S.t/� D S.t � t/S.t/� andfor t such that S.t/G B, S.t/� 2 B.

utExercise 7.15. Prove that A is a maximal, in the sense of the inclusion relation,bounded invariant set.

Solution. Assume that A A1, A1 different from A , bounded and invariant. As Babsorbs A1, S.t/A1 B for large t, then A1 B as S.t/A1 D A1 from ourassumption. Hence, !.A1/ D A1 !.B/ D A , and we have a contradiction.


Exercise 7.16. Prove that A is a connected set provided X is connected.

Solution. Assume that this is not true. Then there exist open sets U1, U2 such thatUi \ A 6D ;, i D 1; 2, A U1 [ U2, U1 \ U2 D ;. Let B be the closedconvex hull of A . Then B is compact (and so bounded) and connected. From thecontinuity of S.t/, the sets S.t/B are connected. The set A is a subset of eachS.t/B, as A B ) A D S.t/A S.t/B. As A attracts bounded sets,distX .S.t/B;A / ! 0 as t ! 1. This means that S.t/B U1 [ U2—an openneighborhood of A . But then we have ; 6D Ui \ A Ui \ S.t/B, i D 1; 2,S.t/B U1 [ U2, U1 \ U2 D ;, which means that S.t/B is not connected. We havecome to a contradiction.

Exercise 7.17. Prove that A is a minimal closed set which attracts bounded setsof X.

Solution. Assume, to the contrary, that there exists a bounded set F A , F 6D A ,attracting bounded sets. As A is compact then F is also compact. Let y 2 A n F.For small ", O".y/ \ O".F/ D ;. Let B be an open absorbing set. As F attracts B,S.t/B O".F/ for large t � t."/. But y 2 !.B/ D A whence y D limk!1 S.tk/xk,xk 2 B, tk ! 1, and we have S.tk/xk O".y/ for large k. But we have alsoS.tk/xk 2 O".F/ for large k. A contradiction as O".y/ \ O".F/ D ;.

Remark 7.4. The assumption of uniform compactness in Theorem 7.3 can bereplaced by a weaker assumption of asymptotic compactness. However, we shall usethe stronger version as in the case of the Navier–Stokes problem that we considerin this chapter the uniform compactness property holds and is easy to check. Inthe case of the Navier–Stokes equations in unbounded spatial domains the uniformcompactness property does not hold but we can use the asymptotic compactnessproperty [201].

Definition 7.7. We call a semigroup fS.t/gt�0 asymptotically compact if it has thefollowing property: for any bounded sequence fung, and any sequence ftng, n 2N such that tn ! 1 as n ! 1, the sequence fS.tn/ung contains a convergentsubsequence.

Exercise 7.18. Prove that uniform compactness implies asymptotic compactness.Consider the proof of Theorem 7.3 and show that the condition of uniformcompactness in the hypothesis of the theorem can be replaced by the condition ofasymptotic compactness.

Exercise 7.19. Prove that existence of a global attractor for a given semigroupimplies existence of a bounded absorbing set and asymptotic compactness of thesemigroup.

The global attractor for the semigroup governed by the solution map of the ODEPx D jxj.1 � x/ is presented in Fig. 7.3.Now we can come back to our main considerations. From Theorem 7.1 it followsthat the family of maps S.t/ W H ! H defined for all t � 0 by


Fig. 7.3 Example of asemigroup S.t/ W R ! R

given by the solution map forthe equation Px D jxj.1� x/.The global attractor is givenby A D Œ0; 1�. It containstwo stationary points f0; 1gand all points belonging to acomplete trajectory thatconnects them

S.t/u0 D u.t/; (7.10)

where u.�/ is the unique weak solution of the considered problem with u.0/ D u0,is a continuous semigroup in X. Our aim is to prove the following

Theorem 7.4 (Existence of a Global Attractor in H). Under the assumptionsof Theorem 7.1 there exists a global attractor A for the semigroup fS.t/gt�0in H associated with the two-dimensional Navier–Stokes equations with periodicboundary conditions.

Proof. Step 1. Existence of a bounded absorbing set in H. From the first energyinequality, cf. inequality (7.3),

d

dtju.t/j2 C �ku.t/k2 � C.�/jf j2; (7.11)

we have as an immediate consequence, cf. inequality (7.4),

d

dtju.t/j2 C k1.�/ju.t/j2 � C.�/jf j2:

Applying the Gronwall inequality to the last one we have,

ju.t/j2 � ju0j2e�k1t CZ t

0

ek1.s�t/C.�/jf j2dt D ju0j2e�k1t C C.�/

k1jf j2.1 � e�k1t/;

(7.12)

from which it follows that for �2 > �20 D C.�/k1

jf j2, balls B.0; �/ in the phase spaceH absorb bounded sets in H.

Let u0 2 B1 be a bounded subset in H. Then, ju.t/j2 � �2 for t � t0.B1/.

Exercise 7.20. Let u0 2 B.0; r/. Compute t0 D t0.B.0; r// such that ju.t/j2 � �2

for t � t0.

Step 2. Existence of a bounded absorbing set in V and uniform compactness of thesemigroup. To prove these properties we need the second energy inequality. Taking


the scalar product of the Navier–Stokes equation with Au (for u 2 D.A/;Au D ��u,cf. Remark 7.1) we obtain

d

dtkuk2 C 2�jAuj2 D 2.f ;Au/; (7.13)

as b.u; u;Au/ D 0. The relation (7.13) is called the enstrophy equation.

Exercise 7.21. Prove that for the two-dimensional Navier–Stokes flow with peri-odic boundary conditions we have not only b.u; u; u/ D 0 but also b.u; u;Au/ D 0,cf. [197, 220]. The latter property makes the analysis of the problem much easier.

From (7.13) we obtain

d

dtkuk2 C �jAuj2 � Cjf j2;

and, as kuk � C1jAuj for u 2 D.A/, we have also

d

dtkuk2 C k1kuk2 � Cjf j2: (7.14)

Exercise 7.22. Prove that kuk � C1jAuj for u 2 D.A/ and compute the constantsC, C1, and k1 above.

Now, to estimate the norm ku.t/k we cannot use the Gronwall inequality since ourinitial condition u0 is only in H and not in V . Instead, we shall use the uniformGronwall inequality (cf. Lemma 3.21). To apply the uniform Gronwall lemma, weproceed as follows. First, from inequality (7.11), after integration in the interval.t; t C r/, where t � t0.u0/, we obtain in particular,

�

Z tCr

tku.t/k2ds � C.�/rjf j2 C �2 D a3� (7.15)

for all t � t0.B1/. Then, from inequality (7.14) and the uniform Gronwall lemmawe get directly

ku.t/k2 � a3

rC a2

�for all t � t0 C r; (7.16)

where

a2 DZ tCr

tCjf j2dt D rCjf j2; a3 D 1

�.rCjf j2 C �2/;

[cf. (7.15)]. Estimate (7.16) means that our bounded set B1 is moved by S.t/, fort � t0 C r, into a ball in V ,

kS.t/xk2 � a3

rC a2

�for all t � t0 C r; x 2 B1;

7.3 Convergence to the Stationary Solution: The Simplest Case 157

which is a precompact set in H; due to the Rellich lemma, V is compactly embeddedin H. Thus, the whole trajectory of the set B1 in H, starting from t0 C r,

[

t�t0Cr

S.t/B1 is precompact in H:

This is the condition of the uniform compactness of the semigroup S.t/ in H.From the above considerations and Theorem 7.3 it follows the existence of the

global attractor A H for the two-dimensional Navier–Stokes equations withperiodic boundary conditions. utExercise 7.23. Prove the existence of a global attractor in the case of homogeneousDirichlet boundary conditions. To deal with the trilinear form b.u; u;Au/ apply theAgmon inequality

kuk1 � c2juj1=2jAuj1=2 for u 2 D.A/:

Exercise 7.24. Prove that from the uniform compactness of the semigroup S.t/ inH it follows the existence of a compact absorbing set in H.

Exercise 7.25. Prove that if f 2 L1.RCI H/, then the bounded absorbing set in Vexists for the corresponding non-autonomous problem.

7.3 Convergence to the Stationary Solution:The Simplest Case

Let us recall the stationary problem:

��u C .u � r/u C rp D f in Q;

div u D 0 in Q;


1. Q is a square Œ0;L�2 in R2 and we assume periodic boundary conditions, or


homogeneous boundary condition u D 0 on @Q.

Let H and V be the corresponding function spaces, cf. Sect. 4.1. Then the weakformulation of the above problem is as follows.

Problem 7.2. For f 2 H (or f 2 V 0) find u 2 V such that



or, equivalently,

�Au C Bu D f in H or V 0:

Assuming that the viscosity � is large enough with respect to the volume force f weshall prove convergence of solutions u.t/ of the nonstationary problem to the uniquestationary solution u1 as time goes to infinity.

Theorem 7.5. There exists a constant c1.Q/ such that if �2 > c1.Q/kf kV0 then thestationary solution u1 is unique. Moreover, for each u0 2 H the solution u of theproblem:

u 2 C.Œ0;1/I H/ \ L2loc.0;1I V/; u.0/ D u0;

d

dt.u.t/; v/C �..u.t/; v//C b.u.t/; u.t/; v/ D hf ; vi for all v 2 V;

converges to u1 in H as time goes to infinity,

ju.t/ � u1j ! 0 as t ! 1:

Proof. Let w.t/ D u.t/ � u1. Since �Au1 C B.u1/ D f , we have

dw.t/

dtC �Aw.t/C B.u.t// � B.u1/ D 0 in V 0:


dw.t/

dtC �Aw.t/C B.u.t// � B.u1/;w.t/

D 0;

that is,

1

2

d

dtjw.t/j2 C �kw.t/k2 C b.u.t/; u.t/;w.t// � b.u1; u1;w.t// D 0:

Further, using a property of the trilinear form b, we obtain

1

2

d

dtjw.t/j2 C �kw.t/k2 D �b.w.t/; u1;w.t//: (7.17)

We shall estimate the right-hand side. Using the Ladyzhenskaya inequality weobtain

b.u; v;w/ � kukL4.Q/2kvkkwkL4.Q/2 � c1juj1=2kuk1=2kvkjwj1=2kwk1=2:

7.4 Convergence to the Stationary Solution for Large Forces 159

Thus,

b.w.t/; u1;w.t// � c1jw.t/jku1kkw.t/k� �

2kw.t/k2 C c2

�jw.t/j2ku1k2:

We have then

d

dtjw.t/j2 C �kw.t/k2 � c2

�jw.t/j2ku1k2;

and further, as jw.t/j � Ckw.t/k, we have

d

dtjw.t/j2 C

�C � c2

�ku1k2

�jw.t/j2 � 0: (7.18)

Now, we know that ku1k � 1�kf kV0 , whence

�C � c2�

ku1k2 � �C � c2�3

kf k2V0

> 0

for large � [satisfying �2 > c1.Q/kf kV0 for some constant c1.Q/]. Thus, frominequality (7.18) it follows (for these �) the uniqueness of the stationary solutionu1 and also the exponential decay in H of u.t/ to u1 as t ! 1, and this ends theproof. utExercise 7.26. Prove the above theorem without using the Ladyzhenskaya inequal-ity, by estimating in a different way the right-hand side of Eq. (7.17). Compare theconstants c1.Q/ in both cases for the problem with periodic boundary conditions.

7.4 Convergence to the Stationary Solution for Large Forces

We consider two-dimensional flows on Q D Œ0;L�2 as in the preceding sectionsand prove that for any force of the form f D �

0; ˛ sin�2�L x1

��, where ˛ > 0

can be arbitrarily large, all nonstationary solutions of the Navier–Stokes problemconverge to the unique solution Nu D 1

��1f of the stationary Navier–Stokes problem,

where �1 D 4�2

L2is the first eigenvalue of the Stokes operator.

Remark 7.5. Observe that for our force f the Grashof number

G D jf j�2�1

D Lj˛jp2�2�1

;


which is a dimensionless quantity which measures the strength of the forcingrelative to viscosity, is large for large ˛.

It is easy to see that f has the following properties,

div f D 0; Af D �1f ; B.f ; f / D 0: (7.19)

Let us write the energy equation

1

2

d

dtjuj2 C �kuk2 D .f ; u/;

and the enstrophy equation

1

2

d

dtkuk2 C �jAuj2 D .f ;Au/ D .Af ; u/ D �1.f ; u/:

We have used the property, cf. Exercise 7.21,

b.u; u;Au/ D 0:

Multiplying the energy equation by �1 and subtracting it from the enstrophyequation we have

1

2

d

dt.kuk2 � �1juj2/C �.jAuj2 � �1kuk2/ D 0:

Observe that kuk2 � �1juj2 � 0 in view of the Poincaré inequality. More precisely,recalling that

u D1X

kD1Ouk!k

we have,

kuk2 � �1juj2 D1X

kD5.�k � �1/jOukj2;

and also

jAuj2 � �1kuk2 D1X

kD5�k.�k � �1/jOukj2

� �5

1X

kD5.�k � �1/jOukj2 D �5.kuk2 � �1juj2/:

7.4 Convergence to the Stationary Solution for Large Forces 161

Thus, we have

1

2

d

dt.kuk2 � �1juj2/C ��5.kuk2 � �1juj2/ � 0;

and

limt!1.ku.t/k2 � �1ju.t/j2/ D 0: (7.20)

Now, let u D P4u C Q4u. Then

jQ4uj2 D1X

kD5jOukj2 � 1

�5 � �11X

kD5.�k � �1/jOukj2

D 1

�5 � �1 .kuk2 � �1juj2/: (7.21)

Therefore, from (7.20) and (7.21) we conclude that

limt!1 jQ4u.t/j2 D 0:

We have shown that all higher harmonics go to zero as time goes to infinity. LetNu D 1

��1f be the stationary solution. We define v.t/ D u.t/ � Nu, the difference

between some nonstationary solution and the stationary solution Nu. As P4 Nu D Nuthen Q4v.t/ D Q4u.t/ ! 0 as t ! 1. We shall show that also P4v.t/ ! 0 ast ! 1, so that v.t/ ! 0 and hence u.t/ ! Nu as t ! 1. We have

dv

dtC �Av C B.v; v/C B.v; Nu/C B.Nu; v/ D 0:

We multiply this equation by P4v in H and use kP4vk2 D �1jP4vj2 to get

1

2

d

dtjP4vj2 C ��1jP4vj2 C b.v; v;P4v/C b.v; Nu;P4v/C b.Nu; v;P4v/ D 0:

Let us write v D P4v C Q4v and use jQ4v.t/j ! 0 as t ! 1. We can write theabove equation in the form

1

2

d

dtjP4vj2 C ��1jP4vj2 C b.P4v;P4v;P4v/C b.P4v; Nu;P4v/C b.Nu;P4v;P4v/ D ˇ.t/;

where ˇ.t/ ! 0 as t ! 1 since all terms in ˇ.t/ are of the form b.x; y; z/ where atleast one variable is equal to Q4v.t/. Moreover b.P4v;P4v;P4v/ D 0 and

b.P4v; Nu;P4v/C b.Nu;P4v;P4v/ D �b.P4v;P4v; Nu/ D 0:


Exercise 7.27. Write P4v as a sum of the first four eigenvectors to check directlythat b.P4v;P4v; Nu/ D 0.

In this way we have obtained

1

2

d

dtjP4vj2 C ��1jP4vj2 � ˇ.t/; ˇ.t/ ! 0; t ! 1;

whence

limt!1 jP4u.t/j2 D 0;

and

limt!1 ju.t/ � Nuj D 0:

This relation proves also the uniqueness of the stationary solution.

Exercise 7.28. Write ˇ.t/ explicitly and prove that ˇ.t/ ! 0 as t ! 1.

Exercise 7.29. Assume that the external force f satisfies: B.f ; f / D 0 and Af D �mffor �m large enough. Prove that the stationary solution Nu D 1

��mf is unique and

globally stable, that is, all nonstationary solutions u.t/ converge to Nu in H as t ! 1.

Exercise 7.30. Assume that the external force f D fm C fn satisfies: B.fm; fm/ D 0,B.fn; fn/ D 0, B.fn; fm/ D 0 and Afm D �mfm, Afn D �nfn for �m; �n large enough.Find the unique stationary solution Nu and prove that it is globally stable, that is, allnonstationary solutions u.t/ converge to Nu in H as t ! 1. Generalize the result toany finite combination f D fm1 C : : :C fmk .

Solution to Exercise 7.29. Multiplying the equation

du

dtC �Au C B.u; u/ D f

by �m and using �mf D Af we obtain

�mdu

dtC A.�m�u � f /C �mB.u; u/ D 0: (7.22)

Let v.t/ D �m�u.t/ � f (then u D 1�m�.v C f /). We shall show that v.t/ ! 0 as

t ! 1. Observe that

�mdu

dtD 1

�

dv

dt;

hence, multiplying Eq. (7.22) by � we have the following equation for v.t/,

dv

dtC �Av C B.u; u/ D 0; where u D 1

�m�.v C f /:

7.5 Average Transfer of Energy 163

Writing the nonlinear term in terms of v, and using B.f ; f / D 0 we obtain

dv

dtC �Av C 1

�m�.B.v; v/C B.v; f /C B.f ; v// D 0:

Now, multiplying both sides by v in H we get

1

2

d

dtjvj2 C �kvk2 C 1

�m�b.v; f ; v/ D 0:

Since Af D �mf then kf k2 D �mjf j2, and

ˇˇˇˇ1

�m�b.v; f ; v/

ˇˇˇˇ � c1

�m�kvk2kf k D c1

�m�kvk2

p�mjf j D c1p

�m�kvk2jf j;

whence,

1

2

d

dtjvj2 C

�

� � c1p�m�

jf j�

kvk2 � 0: (7.23)

For any fixed f satisfying (7.19) there exists a natural number m0 such that for allm � m0, �m D � � c1p

�m�jf j > 0 and then all solutions v.t/ converge to zero in

H at least exponentially fast as t ! 1. In fact, using jvj2 � 1�1

kvk2 we obtainfrom (7.23),

jv.t/j2 � jv.0/j2 exp.�2�1�mt/ for t � 0:

This means that u.t/ ! Nu D 1��m

f in H as t ! 1 and proves also that Nu is theunique solution of the stationary problem.

Remark 7.6. Observe that though the norm of our f in H may be large, its norm inH�1 is equal to 1p

�mjf j and for m large is small.

7.5 Average Transfer of Energy

We shall consider the Navier–Stokes system as in Sect. 7.1.Let e.u/ D 1

2juj2 be the kinetic energy of the flow. From the Navier–Stokes

equation we obtain

d

dte.u.t// D ��ku.t/k2 C .f ; u.t//

and we can see that, thanks to the identity b.u; u; u/ D 0, the global balance ofthe kinetic energy of the flow does not depend on the nonlinear term .u � r/u. It is


the same as in the case of the linear (Stokes) flow. In this section we shall studymore precisely the transport of energy in the fluid. We know (cf. Sect. 7.4) that thevelocity u.x; t/ can be represented as a Fourier series

u.x; t/ D1X

kD1Ouk.t/!k.x/;

where !k, k D 1; 2; 3; : : : are eigenfunctions of the Stokes operator constituting anorthonormal family in the phase space H. We have thus,

e.u/ D 1

2

1X

kD1jOukj2:

Let u D y C z, where

y DmX

kD1Ouk!k; z D

1X

kDmC1Ouk!k:

We have,

e.u/ D e.y/C e.z/;

where

e.y/ D 1

2

mX

kD1jOukj2; e.z/ D 1

2

1X

kDmC1jOukj2

are kinetic energies of the large scale component y of the flow and the small scalecomponent z. Now, in the balance of the kinetic energies of components y and z thenonlinear term will not reduce to zero. We shall show how the kinetic energy ofparticular components of the flow is distributed in time among the Fourier modes.Let us assume that the external force is given by

f Dm2X

kDm1

Ofk!k; 1 � m1 � m2 < 1:

Let Pm W u ! y, Qm W u ! z be orthogonal projections, with Qm D I �Pm. Applyingthese projections to the Navier–Stokes system

du

dtC �Au C B.u; u/ D f ;

7.5 Average Transfer of Energy 165

and assuming that

m � m2;

we obtain

dy

dtC �Ay C PmB.y C z; y C z/ D Pmf

and

dz

dtC �Az C QmB.y C z; y C z/ D 0;

as Qmz D 0 (m � m2), PmA D APm, QmA D AQm. The corresponding energyequations are

1

2

d

dtjyj2 C �kyk2 C b.y C z; y C z; y/ D .f ; y/;

and

1

2

d

dtjzj2 C �kzk2 C b.y C z; y C z; z/ D 0:

Define E.u/ D kuk2—the enstrophy. Using the property b.u; v;w/ D �b.u;w; v/we can write

d

dte.y/C �E.y/ D ˚z.y/C .f ; y/; (7.24)

and

d

dte.z/C �E.z/ D ˚y.z/; (7.25)

where

˚z.y/ D �b.z; z; y/C b.y; y; z/; ˚y.z/ D �b.y; y; z/C b.z; z; y/:

Observe that ˚z.y/C˚y.z/ D 0. In (7.24) ˚z.y/ is the net amount of kinetic energyper unit time transferred from small length scales z to large length scales y (similarly,˚y.z/ is the net amount of kinetic energy per unit time transferred from large lengthscales y to small length scales z), while the term �E.y/ represents the rate of energydissipation by viscous effects per unit time for the large length scale and .f ; y/ is theenergy flow injected into the large length scales by the forcing term.


We shall show that on average, over time and space, ˚y.z/ � 0, that is,

if the external forces act in the range of large length scales y then, on average, the net transferof kinetic energy occurs only into the small scales.

This transfer of energy is called direct energy transfer.To prove the above statement, we integrate Eq. (7.25) in t to get

e.z.T//C �

Z T

0

E.z.s//ds DZ T

0

˚y.z.s//ds C e.z.0//:

As the function t ! e.z.t// is bounded for t � 0, dividing both sides by T andpassing with T to infinity we obtain, in particular,

0 � lim infT!1

1

T�

Z T

0

E.z.s//ds D lim infT!1

1

T

Z T

0

˚y.z.s//ds:

Assuming that the limit

limT!1

1

T

Z T

0

˚y.z.s//ds

exists, we have

1

j˝j limT!1

1

T

Z T

0

˚y.z.s//ds � 0:

Exercise 7.31. Let us assume that the external force is given by

f Dm2X

kDm1

Ofk!k;

where now 1 � m � m1, m1 > 1. Thus, the force acts in the range of smalllength scales. Prove that the transfer of energy is then from higher modes (smalllength scales) to lower modes (large length scales), which is called an inverse energytransfer. The inverse energy transfer in the large scales may be responsible for theobserved persistent low-wave number coherent structures in turbulent flows.


In this chapter we introduced and applied some basic tools and techniques of dealingwith the nonstationary Navier–Stokes equations. As in the case of the stationaryNavier–Stokes equations, the problem of existence of solutions was resolved byintroducing the appropriate notion of a weak solution. We proved the uniqueness of a


weak solution and considered, in the simplest context, a problem of regularity of theweak solution. We saw the utility of the Galerkin method of approximate solutions,the importance of compactness theorems, here, the Aubin–Lions compactnesstheorem, in particular. The usefulness of basic functional inequalities: the Poincaréinequality, the Ladyzhenskaya inequality, and the Agmon inequality, as well as therole of the Gronwall and the uniform Gronwall inequalities were demonstrated.

Further considerations and details concerning the existence of solutions andglobal attractors for autonomous two-dimensional Navier–Stokes flows in abounded domain can be found, e.g., in [88, 197, 220]. For the study of three-dimensional case see, e.g., [76]. For the existence and regularity of solutions forproblems on unbounded domains which do not satisfy the Poincaré inequality see[237, 238] and for the problems on cylindrical domains see [192, 196, 233]. InSects. 7.4 and 7.5 we followed closely the argument from [99].

In Chap. 8 we shall show the connections of the global attractor with statisticalsolutions of the Navier–Stokes equations.

In Chap. 9 we shall apply the presented techniques to a problem of a shear flowtaken from the theory of lubrication and shall prove that the global attractor has afinite fractal dimension.

Non-autonomous system of the Navier–Stokes equations and flows in unboundeddomains will be considered in Chaps. 11 and 13, respectively.

8Invariant Measures and Statistical Solutions

In this chapter we prove the existence of invariant measures associated withtwo-dimensional autonomous Navier–Stokes equations. Then we introduce thenotion of a stationary statistical solution and prove that every invariant measureis also such a solution.

8.1 Existence of Invariant Measures

We shall consider flows with homogeneous Dirichlet boundary conditions, in asmooth bounded domain ˝.

Let f 2 H. Then there exists a semigroup fS.t/gt�0 in H such that the solution ofthe problem

ut C �Au C B.u; u/ D f ; u.0/ D u0

is given by u.t/ D S.t/u0 for t � 0. The map .t; u/ ! S.t/u is continuous fromŒ0;1/ � H to H.

Our first aim is to prove the following theorem.

Theorem 8.1. Let LIMT!1 be a fixed Banach generalized limit. Then for everyu0 2 H there exists a probability measure on H such that

LIMT!11

T

Z T

0

�.S.t/u0/dt DZ

H�.u/d.u/ (8.1)

for every continuous function � W H ! R (� 2 C.H/). Moreover, the measure is regular, invariant with respect to the semigroup fS.t/gt�0, that is, for every-measurable set E H it holds .S.t/�1E/ D .E/, and its support is containedin the global attractor A .


169

170 8 Invariant Measures and Statistical Solutions

We shall explain the notion of a Banach generalized limit LIMT!1.

Definition 8.1. LIMT!1—a Banach generalized limit is any linear functional onthe set of all bounded functions W Œ0;1/ ! R ( 2 B.Œ0;1//), such that

• LIMT!1 .T/ � 0 for � 0;

• LIMT!1 .T/ D limt!1 .T/ if the limit on the right-hand side exists.

Remark 8.1. Such functionals exist, there may be many of them, hence in Eq. (8.1)the measure depends not only on the initial condition u0 but also on the choice ofLIMT!1 [note that in (8.1) the right-hand side is defined by the left-hand side].

To prove the existence of LIMT!1 we shall use the Hahn–Banach theorem(cf. Theorem 8.2). Let

G0 D f 2 B.Œ0;1// W limT!1 .T/ existsg:The map �0. / D limT!1 .T/ is a linear functional defined on the subspace G0

of B.Œ0;1//. We can extend it to a linear functional � defined on the whole spaceB.Œ0;1//. In fact, let p. / D lim supt!1 .t/. Then

�0. / � p. / on G0;

and

p. C �/ � p. /C p.�/; p.˛ / D ˛p. /; ˛ � 0

for ; � 2 B.Œ0;1//. In view of the Hahn–Banach theorem there exists an extension� of �0 to the whole space, satisfying �. / � p. / D lim supt!1 .t/ forall 2 B.Œ0;1//. It then follows (cf. also Exercise 8.1) that �. / � 0 for allnonnegative 2 B.Œ0;1//. We denote �. / D LIMT!1 .T/.

Exercise 8.1. Prove that for all 2 B.Œ0;1//,

lim infT!1 .T/ � LIMT!1 .T/ � lim sup

T!1 .T/; 1

whence

jLIMT!1 .T/j � lim supT!1

.T/ � supT�0

j .T/j:

We are going now to prove Theorem 8.1. At first, we assume that there exists ameasure as in (8.1) for some fixed u0 2 H and LIMT!1, and derive its basicproperties which follow from this formula.

1. is a probability measure, i.e., .H/ D 1. Take � � 1 on H to get

1 D LIMT!11

T

Z T

0

1dt DZ

H1d.u/ D .H/:

8.1 Existence of Invariant Measures 171

2. is an invariant measure for the semigroup fS.t/gt�0. We shall prove that

Z

H�.S.t/u/d.u/ D

Z

H�.u/d.u/ (8.2)

for all � 2 C.H/. To this end observe that by the uniform compactness of thesemigroup (cf. Sect. 7.2) we have

Z

H�.S.�/u/d.u/ D LIMT!1

1

T

Z T

0

�.S.t C �/u0/dt

D LIMT!11

T

Z TC�

�

�.S.t/u0/dt

D LIMT!1n 1

T

Z T

0

�.S.t/u0/dt C 1

T

Z TC�

T�.S.t/u0/dt

� 1T

Z �

0

�.S.t/u0/dto

D LIMT!11

T

Z T

0

�.S.t/u0/dt DZ

H�.u/d.u/:

This identity can be extended to all functions � which are-integrable on H. Setting� D �E—the characteristic function of E H, we obtain

Z

H�E.S.t/u/d.u/ D

Z

H�E.u/d.u/;

which can be written as .S.t/�1E/ D .E/. Thus, is invariant.

Exercise 8.2. Let identity (8.2) hold for all � 2 C.H/, be a regular measure[in the sense of Theorem 8.3 (c)] and C.H/ be a dense subset of L1.H; d/.Prove that then (8.2) holds for all functions � which are -integrable on H. Provethat the latter property is equivalent to the property of invariance of given by.S.t/�1E/ D .E/.

Remark 8.2. To prove invariance of , the uniform compactness property is notneeded. In the argument above we need only the boundedness of the function t !�.S.t/u0/, which follows directly from the existence of the global attractor. In fact, if�.S.t/u0/ is not bounded on the positive semiaxis t � 0, then there exists a sequencesn ! 1 such that

j�.S.sn/u0/j � n for each n � 1: (8.3)

But then, as distH.S.t/u0;A / ! 0 as t ! 1, we can find a subsequence s0n and

a sequence vn, vn 2 A , such that distH.S.s0n/u0; vn/ � 1=n for each n � 1. Then,

as A is compact, there is a subsequence of the vn (which we relabel) such that


vn ! v 2 A . But then also S.s0n/u0 ! v, and therefore �.S.s0

n/u0/ ! �.v/ asm ! 1, which contradicts (8.3).

3. The support of is contained in the global attractor A . We shall make use ofthe following lemma.

Lemma 8.1. The support of the measure contains only the nonwandering points,that is, points u 2 H such that for every open neighborhood A of u in H there existsa sequence of times tn ! 1 for which S.tn/A \ A 6D ; for every n.

Remark 8.3. The set M1 of nonwandering points is closed and invariant. If the phasespace is compact, then M1 is a nonempty and compact set. Then, every trajectoryconverges to M1 [186].

Proof (of Lemma 8.1). Let F be the support of the measure , F D supp—thesmallest closed set such that .F/ D 1.

Step 1. First we shall prove that for u 2 F and a set A containing u which isrelatively open in F,.A/ > 0. Assuming that.A/ D 0we obtain.FnA/ D 1,F n A is closed, compact in F, and different from F, hence F is not the support of which gives a contradiction.

Fig. 8.1 Illustration of theset �C.A / n S.T/�C.A /

used in the proof ofLemma 8.1

Step 2. To prove that u 2 F is nonwandering, let us assume to the contrary, thatfor some T and all t � T , S.t/A \ A D ;. Consider the positive trajectory�C.A/ D S

t�0 S.t/A of the set A. By assumption, A \ S.T/�C.A/ D ;.Moreover, S.T/�C.A/ �C.A/, A �C.A/ n S.T/�C.A/, (see Fig. 8.1) andusing step 1 we get

0 < .A/ < .�C.A/ n S.T/�C.A// D .�C.A// � .S.T/�C.A//:

On the other hand,

.S.T/�C.A// D .S.T/�1S.T/�C.A// � .�C.A//;

as always S.T/�1S.T/B B. Thus, we have come to a contradiction. The proof ofLemma 8.1 is complete. ut


Corollary 8.1. The support of the measure is contained in the global attractorA for the semigroup fS.t/gt�0.

Proof. Every point outside of the global attractor is wandering, that is, it is not anonwandering point. utWe have also the following property of the measure .

Corollary 8.2. .S.t/E/ D .E/ for all -measurable E A and t 2 R.

Proof. The semigroup fS.t/gt�0 reduced to A is a complete dynamical system(we have a result about the backward uniqueness on the attractor for the two-dimensional Navier–Stokes equations, see [197], Theorem 12.8), hence, in view ofthe invariance of ,

.E/ D .S.t/�1S.t/E/ D .S.t/E/;

which ends the proof. utProof of the existence of the measure � in (8.1)

Step 1. We shall prove that for every u0 2 H and for every continuous function� W H ! R there exists a limit

L.�/ D LIMT!11

T

Z T

0

�.S.t/u0/dt:

Assume we have chosen the functional LIMT!1. It suffices to show that thefunction

f .T/ D 1

T

Z T

0

�.S.t/u0/dt

is bounded on the interval Œ0;1/. We know that there exists a compact absorbingset X for the semigroup fS.t/gt�0 (cf. Sect. 7.2). Hence, the function t !�.S.t/u0/ is continuous and bounded on the compact set �C.u0/. Therefore, thefunction T ! �.T/ is bounded (and also continuous) on Œ0;1/.Step 2. L.�/ depends only on values of � on any compact absorbing set X. Itmeans that if �1 is another function in C.H/ which is equal to � on X thenL.�/ D L.�1/.To prove this property observe that there exists T0 such that

�.S.t/u0/ � �1.S.t/u0/ D 0

for t � T0, as S.t/u0 2 X for large times. Thus,

L.� � �1/ D LIMT!11

T

Z T

0

.�.S.t/u0/ � �1.S.t/u0//dt

D LIMT!11

T

Z T0

0

.�.S.t/u0/ � �1.S.t/u0//dt D 0:


Step 3. Consider a compact absorbing set X as above. From the Riesz–Kakutanirepresentation theorem (Theorem 8.3) we know that for every positive linearcontinuous functional G on the space of continuous functions on X (G 2 C.X/0),there exists a Borel measure on X such that

G. / DZ

X .u/d.u/ for all 2 C.X/0:

We can extend this measure to the whole phase space setting Q.E/ D .E \ X/for E H. We shall denote this extended measure simply by in what follows.Then, in particular, .H n X/ D 0. Further, from the Tietze extension theorem(Theorem 8.4) we know that every function 2 C.X/ can be extended to acontinuous function on the whole space without enlarging the norm. Thus,

LIMT!11

T

Z T

0

�.S.t/u0/dt D L.�/

D G.�jX/ .L.�/ depends only on �jX/D G. / .�jX D /

DZ

X .u/d.u/ .Riesz–Kakutani theorem/

DZ

H�.u/d.u/ .support of is contained in X/:

Since is a time averaged measure, it is invariant. We have shown that every suchmeasure is supported on the global attractor. Thus, .A / D 1 and

LIMT!11

T

Z T

0

�.S.t/u0/dt D 1

.A /

Z

A�.u/d.u/ for all � 2 C.H/:

The proof of Theorem 8.1 is complete. utRemark 8.4. The existence of a compact absorbing set is not needed to prove theexistence of . We shall provide a simple proof that works in uniformly convexBanach spaces.

Let K be the closed convex hull of the global attractor. As A is compact, itis known that K is also a compact subset of H (see [3], Theorem 5.35). As K iscompact and H is uniformly convex, for each u 2 H there exists a unique ku 2 Ksuch that ju � kuj D infk2K ju � kj and the projection operator P W u 2 H ! ku 2 Kis continuous.

Given u0 2 H consider t ! P.S.t/u0/, the projection onto K of the trajectoryt ! S.t/u0. Since K is compact, the function Œ0;1/ 3 t ! �.P.S.t/u0// 2 R iscontinuous and bounded for � 2 C.H/.


Moreover

j�.S.t/u0/ � �.P.S.t/u0//j ! 0 as s ! 1: (8.4)

We then conclude that

LIMT!11

T

Z T

0

�.S.t/u0/dt D LIMT!11

T

Z T

0

�.P.S.t/u0//dt: (8.5)

The right-hand side defines a linear positive functional on C.K/. The rest of theproof is similar to that in step 3 above, where now K plays the role of the compactabsorbing set X.

Exercise 8.3. Prove that the projection operator P W u 2 H ! ku 2 K is continuous.

Exercise 8.4. Prove (8.4) and (8.5).

Below we present the basic theorems which have been used above.

Theorem 8.2 (Hahn–Banach). Let X be a real linear space, p W X ! R a functionsuch that

p.x C y/ � p.x/C p.y/; p.˛x/ D ˛p.x/

for ˛ � 0, x; y 2 X. Let f be a real-valued linear functional on a subspace Y of Xsuch that f .x/ � p.x/ for x 2 Y. Then there exists a real linear functional F on Xsuch that

F.x/ D f .x/; x 2 Y; F.x/ � p.x/; x 2 X:

Theorem 8.3 (Riesz–Kakutani). Let X be a locally compact Hausdorff space. Let� be a positive linear functional on Cc.X/, the space of real-valued continuousfunctions of compact support on X. Then there exist a � -algebra M on X containingall Borel sets in X and a unique positive measure on M such that:

(a) �f D RX f .u/d.u/ for every f 2 Cc.X/,

(b) .K/ < 1 for every compact set K X,(c) is regular in this sense that for every E M ,

.E/ D inff.�/ W E �; � is openg;

and, for every E 2 M with .E/ < 1,

.E/ D supf.K/ W K E; K is compactg:

(d) is complete, i.e., if E 2 M , A E and .A/ D 0, then A 2 M .


Theorem 8.4 (Tietze). Let X be a normal topological space and A be a closedsubset of X. If f is a continuous and bounded real-valued function defined on A,then there exists a continuous real-valued function F defined on X such that:

(i) F.x/ D f .x/ for x 2 A,(ii) supx2X jF.x/j D supx2X jf .x/j.

8.2 Stationary Statistical Solutions

Let u.t/ D S.t/u, t 2 RC, be a family of maps in .X; 0/. If t.E/ D 0.S.t/�1E/

for all 0-measurable sets E, thenZ

X˚.u/dt.u/ D

Z

X˚.S.t/u/d0.u/ for all ˚ 2 L1.X; t/: (8.6)

If

du.t/

dtD F.u.t//; (8.7)

then from (8.6) by formal differentiation with respect to t and from (8.7) we obtain

d

dt

Z

X˚.u/dt.u/ D d

dt

Z

X˚.S.t/u/d0.u/

DZ

X˚ 0.S.t/u/

d

dt.S.t/u/d0.u/ D

Z

X˚ 0.S.t/u/F.S.t/u/d0.u/

DZ

X˚ 0.u/F.u/dt.u/:

Hence,

d

dt

Z

X˚.u/dt.u/ D

Z

X˚ 0.u/F.u/dt.u/: (8.8)

The above equation we shall call the Liouville equation.If measure 0 is invariant with respect to the semigroup of transformations

fS.t/gt�0, then

0.T�1E/ D 0.E/ D t.E/;

that is, all measures t, t � 0, coincide with 0. In this case the Liouvilleequation (8.8) reduces to the stationary Liouville equation.

Z

X˚ 0.u/F.u/d.u/ D 0; (8.9)

where we denoted by the measure t D 0.

8.2 Stationary Statistical Solutions 177

Let us consider the two-dimensional Navier–Stokes equations

du.t/

dtD F.u.t//; F.u/ D f � �Au � B.u/;

u.0/ D u0:

We proved (in Sect. 8.1) existence of probability measures invariant with respectto the semigroup fS.t/gt�0 associated with the above Navier–Stokes system. Thus,there exists a Borel measure on the phase space H for which .S.t/�1E/ D .E/for Borel sets E in H. Moreover, .H/ D 1. For such measure formula (8.9) holdstrue. We shall give the precise meaning to this formula in the case of the two-dimensional Navier–Stokes dynamical system. Let us define the class of the testfunctions ˚ .

Definition 8.2. To the class T of test functions belong real-valued functionals˚ D ˚.u/ defined on H, which are bounded on bounded subsets of H and suchthat

(i) For any u 2 V , the Fréchet derivative ˚ 0.u/ taken in H along V exists. Moreprecisely, for each u 2 V , there exists an element in H denoted ˚ 0.u/ such that

j˚.u C v/ � ˚.u/ � .˚ 0.u/; v/jjvj ! 0 as jvj ! 0; v 2 V;

(ii) ˚ 0.u/ 2 V for all u 2 V , and u ! ˚ 0.u/ is continuous and bounded as afunction from V to V .

For example, let ˚.u/ D ..u; g1/; : : : ; .u; gn//, where 2 C10.R

n/, gj 2 V . Then˚ 0.u/ D Pn

jD1 @j ..u; g1/; : : : ; .u; gn//gj 2 V .Let denote a probability measure in H such that

Z

Hkuk2d.u/ < 1; (8.10)

where kuk D 1 for u 2 H n V . We thus have .H n V/ D 0, that is, the support of is included in V , and the map u ! .�Au C B.u/� f ; ˚ 0.u// is continuous in V forevery test function ˚ 2 T . Moreover,

j.�Au C B.u/ � f ; ˚ 0.u//j D j�..u; ˚ 0.u///C b.u; u; ˚ 0.u// � .f ; ˚ 0.u//j� �kukk˚ 0.u/k C ckuk2k˚ 0.u/k C jf jj˚ 0.u/j� .�kuk C ckuk2 C �

1=21 jf j/ sup

u2Vk˚ 0.u/k: (8.11)


As.HnV/ D 0, from (8.10) and (8.11) it follows [cf. condition (ii) of the definitionof the class T ] that the integral

Z

H.�Au C B.u/ � f ; ˚ 0.u//d.u/

has sense and is finite.Now we can define the stationary statistical solution of the two-dimensional

Navier–Stokes system.

Definition 8.3. Stationary statistical solution of the Navier–Stokes system is aprobability measure on H such that

(i)R

H kuk2d.u/ < 1,(ii)

RH.F.u/; ˚

0.u//d.u/ D 0 for all ˚ 2 T , where F.u/ D �Au C B.u/ � f ,f 2 H,

(iii)R

E1�juj2<E2f�kuk2 � .f ; u/gd.u/ � 0 for all 0 � E1 < E2 � 1.

Inequality in (iii) is a weak form of the energy inequality. It gives an estimate of theintegral in (i) and an estimate of the support of the measure in H.

Let E D fE1 � juj2 < E2g. Then, from (i),

Z

E.f ; u/d.u/ � jf j

Z

Ejujd.u/ � jf j

�Z

Ejuj2d.u/

� 12

.E/12

� jf j�12

1

�Z

Ekuk2d.u/

� 12

< 1:

Then from (iii),

�

Z

Ekuk2d.u/ �

Z

E.f ; u/d.u/ � jf j

�12

1

�Z

Ekuk2d.u/

� 12

< 1;

and we have,Z

Ekuk2d.u/ � jf j2

�2�1;

and

Z

Ejuj2d.u/ � jf j2

�2�21:

The last inequality we can write in the form

Z

E

�

juj2 � jf j2�2�21

�

d.u/ � 0:

8.2 Stationary Statistical Solutions 179

For E1 D jf j2�2�21

and E2 D 1 we have

juj2 � jf j2�2�21

� 0

for all u 2 E, whence .E/ D 0. Thus

��

u 2 H W juj � jf j��1

��

D 1;

and

supp �

u 2 H W juj � jf j��1

�

\ V:

Now, we shall prove existence of a stationary statistical solution for the two-dimensional Navier–Stokes system.

Theorem 8.5. Let be a probability measure on H invariant with respect to thesemigroup fS.t/gt�0 associated with the two-dimensional Navier–Stokes system.Then is a stationary statistical solution of the system.

Proof. Step 1. Since is invariant, it is concentrated on the global attractor. Thus

Z

Hkuk2d.u/ D

Z

Akuk2d.u/ < 1;

and we have the first property of the stationary statistical solution.Step 2. We shall show that

Z

H.F.u/; ˚ 0.u//d.u/ D 0; (8.12)

where F.u/ D �Au C B.u/ � f , F.u/ D ut, ˚ 2 T . From the invariance of ,for every ˚ 2 T ,

Z

H.F.u/; ˚ 0.u//d.u/ D

Z

H.F.S.t/u/; ˚ 0.S.t/u//d.u/;

and the right-hand side is independent of t � 0. Therefore, from the Fubinitheorem,

Z

H.F.S.t/u/; ˚ 0.S.t/u//d.u/ D 1

T

Z T

0

Z

H.F.S.t/u/; ˚ 0.S.t/u//d.u/dt

DZ

H

1

T

Z T

0

.F.S.t/u/; ˚ 0.S.t/u//dtd.u/:


Thus,

Z

H.F.u/; ˚ 0.u//d.u/ D

Z

H

1

Tf˚.S.t/u/ � ˚.u/gd.u/; (8.13)

as

d

dt˚.u/ D .F.u/; ˚ 0.u//:

The right-hand side of (8.13) tends to 0 as T ! 1, as the function t ! ˚.S.t/u/is bounded. This proves (8.12).Step 3. Now we shall prove that

Z

E1�juj2<E2

f�kuk2 � .f ; u/gd.u/ D 0 (8.14)

for all 0 � E1 < E2 � 1. In particular the left-hand side is not positive whichgives the third property of the stationary statistical solution. From the energyequation

1

2jS.t/uj2 C �

Z t

0

kS.s/uk2ds D 1

2juj2 C

Z t

0

.f ; S.s/u/ds;

we obtain, by regrouping the terms,

Z t

0

f�kS.s/uk2 � .f ; S.s/u/gds D 1

2fjuj2 � jS.t/uj2g (8.15)

for t � 0. From the invariance of , the Fubini theorem, and (8.15) we get

Z

E1�juj2<E2

f�kuk2 � .f ; u/gd.u/ DZ

E1�juj2<E2

f�kS.s/uk2 � .f ; S.s/u/gd.u/

D 1

T

Z T

0

Z

E1�juj2<E2


DZ

E1�juj2<E2

1

T

Z T

0


D 1

2

Z

E1�juj2<E2

1

Tfjuj2 � jS.t/uj2gd.u/

for every T � 0. Passing with T to 1 we obtain zero on the right-hand side. Thisproves (8.14).

ut


Remark 8.5. In fact, the converse is also true. If is a stationary statistical solution,then is a probability measure invariant with respect to the semigroup fS.t/gt�0(cf. Chap. 4 in [99]).


Our presentation follows that in [99], where this subject is treated in greaterdetail. The alternative proof of Theorem 8.1 and its generalization to noncompactsemigroups presented in Remarks 8.2 and 8.4 come from [165]. Theorem 8.5follows also from the results of Chap. 12, cf. [160].

For statistical solutions for the Navier–Stokes equations, see [96, 97, 99, 101, 116,195, 203, 228]. Some recent results for three-dimensional problems can be found in[29, 104].

9Global Attractors and a Lubrication Problem

We start this chapter from necessary background on the theory of fractal dimension.Next, we formulate and study a problem which models the two-dimensionalboundary driven shear flow in lubrication theory. After the derivation of the energydissipation rate estimate and a version of Lieb–Thirring inequality we provide anestimate from above on the global attractor fractal dimension.

9.1 Fractal Dimension

We start this section from the definition and some properties of a fractal dimensionof a relatively compact set K X, where X is a Banach space. By NX

" .K/ we denotethe minimal number of closed balls in X of radius " needed to cover the set K. Fromthe relative compactness of K it follows that this number is finite. Obviously, weexpect the quantity NX

" .K/ to increase as " ! 0. The fractal dimension, given bythe following definition, measures how fast is this increase.

Definition 9.1. Let K X be a relatively compact set. The fractal dimension of K,denoted by dX

f .K/ is a number defined by

dXf .K/ D lim sup

"!0

log NX" .K/

log 1"

:

The value dXf .K/ can be equal to 1 even if K is a compact set. If the set K is

not relatively compact, we set dKf .K/ D 1. In the next result we recall several

properties of dXf .


183

184 9 Global Attractors and a Lubrication Problem

Theorem 9.1 (Cf. [197, Proposition 13.2]).

(i) if Kk X for k D 1; : : : ;N are relatively compact sets, then for every k D1; : : : ;N

dXf .Kk/ � dX

f

N[

kD1Kk

!

� maxkD1;:::;N dX

f .Kk/:

(ii) If f W X ! X is Hölder continuous with the exponent 2 .0; 1�, i.e.,

kf .x/ � f .y/kX � Lkx � ykX for all x; y 2 X;

then

dXf .f .K// � dX

f .K/

for every relatively compact K X:

(iii) We have dXf .K/ D dX

f .K/ for every relatively compact K X, where K is theclosure of K in X.

Note that property (i) does not hold for countable unions. Indeed, let X D R andK D Œ0; 1�\Q. From (iii) it follows that dX

f .K/ D dXf .Œ0; 1�/, and it is easy to verify

that dXf .Œ0; 1�/ D 1, while for every point x 2 K we have dX

f .fxg/ D 0.If we know that an infinite dimensional semiflow fS.t/gt�0 is defined by the

solution maps of an initial and boundary value problem and it has a global attractorA , then we can restrict the dynamical system to the attractor, an invariant set, i.e.,treat the family fS.t/gt�0 as the mappings S.t/ W A ! A , thus interpreting theglobal attractor as the image of the flow after a long time of evolution (keepingin mind that all trajectories are arbitrarily close to the attractor after a long time,but each trajectory does not necessarily have one corresponding trajectory on theattractor, to which it becomes attracted in the norm, such trajectories exist, but onlyon finite time intervals, the length of which goes to infinity as time goes to infinity,see [197] Proposition 10.14 and Corollary 10.15).

An important question that arises is the following: can the dynamics on theattractor be described by means of the time evolution of a finite number of variableswithout loss of information? From the fact that the global attractor exists, we onlyknow that it is compact, and hence it must have an empty interior in an infinitedimensional phase space H, but such compact sets can be still “large.” Indeed, ifa Banach space V embeds compactly in the phase space H, then any set, that isbounded in V , is relatively compact in H, so, potentially, a ball in V is an admissiblecandidate for the attractor. In certain cases, however, it is possible to obtain muchmore knowledge on the attractor than only its compactness: namely it is possibleto show that it has finite fractal dimension. If we could prove that the attractor is afinite dimensional manifold, then it would be natural to use its parameterization, andreduce the dynamics to the finite dimensional one. Unfortunately, the set of finite

9.2 Abstract Theorem on Finite Dimensionality and an Algorithm 185

fractal dimension can be very complicated and it does not have to be a manifold.Still, due to the Hölder–Mañé Theorem, if the fractal dimension of the attractor isfinite we can reduce the dynamics on the attractor to the finite dimensional one.

Theorem 9.2 (Hölder–Mañé, cf. [117]). Let H be a Banach space and let A H be a compact set such that dH

f .A / � m2

. Then, almost every bounded linearfunction � W H ! R

m is a bijection as a map � W A ! �.A / and its inverse��1j�.A / is Hölder continuous with a certain exponent ˛ 2 .0; 1/ depending onlyon m and dH

f .A /.

The expression “almost every” in the last theorem is understood in the senseof prevalence, a generalization on the notion “almost everywhere in the sense ofLebesgue measure” to the infinite dimensional Banach spaces. Here we considerthe prevalence in the space of linear and bounded operators L .HIRm/.

Definition 9.2. The Borel set E X, where X is a Banach space, is said to beprevalent if there exists a compactly supported probability measure such that.E C x/ D 1 for all x 2 X. A non-Borel set that contains a prevalent set is alsoprevalent.

Using a linear map � that satisfies the assumptions of Theorem 9.2 it is possibleto construct the finite dimensional semiflow S�.t/ W �.A / ! �.A / for t � 0

conjugate with the original one according to the formula S�.t/x D �.S.t/��1x/.Unfortunately, this new semiflow is not necessarily Lipschitz continuous, even ifS.t/ are Lipschitz, since ��1 is only Hölder. Therefore it can have more thanone solution. Still, this result plays an important role in the theory of finitedimensional reduction of infinite dimensional dynamical systems, and, clearly, thefinite dimensionality of the global attractor is a strong evidence that the semiflow isin fact finite dimensional. More information about this theory can be found in themonograph [198], and, in context of two-dimensional Navier–Stokes equations, inthe review article [199].

9.2 Abstract Theorem on Finite Dimensionalityand an Algorithm

In this section we present two classes of methods that can be used to prove the finitedimensionality of the global attractor.

The first of this two approaches was introduced by Constantin and Foias in thearticle [74] (also see [57, 58, 220]) and it relies on the linearization of the underlyingPDEs and the fact that if the linearized system contracts m-dimensional volumesuniformly on the global attractor, then its fractal dimension must not exceed m.Below we present an algorithm, based on this idea, that can be used to obtainthe estimate of the fractal dimension of the global attractor for the Navier–Stokesequations. The exposition is based on [57, 220].


Let fS.t/gt�0 be a semiflow on a Hilbert space X, that has a global attractor A ,being a compact, invariant, and attracting set in X.

Definition 9.3. We say that the semiflow fS.t/gt�0 is uniformly quasidifferentiableon A if for every t � 0 and u 2 A there exists a linear and continuous operator�.t; u/ 2 L .XI X/ such that for all t � 0

supu;v2A

sup0<ku�vkX��

kS.t/v � S.t/u ��.t; u/.v � u/kX

kv � ukX! 0 as � ! 0: (9.1)

Note (see Sect. V.2.2 in [220]) that the mapping �.t; u/ does not have to be defineduniquely.

We assume that the semiflow is given by a solution map of the following abstractevolution problem.

du.t/

dtD F.u.t// in X for t > 0;

u.0/ D u0 in X;

where F W W ! X, W being a certain Banach space continuously embedded in X.Then we have u.t/ D S.t/u0. We construct the following linearized problem

dU.t/

dtD F0.S.t/u0/U.t/ in X for t > 0; (9.2)

U.0/ D � in X; (9.3)

where F0 is the Fréchet derivative of F, and we make the following assumption.

Assumption A. The semiflow fS.t/gt�0 is uniformly quasidifferentiable on A , with

supt2Œ0;1�

supu2A

k�.t; u/kL .XIX/ < 1: (9.4)

Moreover, for u0 2 A the mapping X � Œ0;1/ 3 .�; t/ ! �.t; u0/� D U.t/ 2 X isa solution map of the linearized system (9.2)–(9.3).

In practice, the verification of this, technical, assumption depends on the structureof the studied problem and relies on the construction of the linearized problem (9.2)–(9.3) and proving that its solution map is a quasidifferential, i.e., it satisfies (9.1), andthat (9.4) holds (see, for instance, examples in Sects. VI.2.1 and VI.3.1 in [197]).

In addition to Assumption A, for the fractal dimension of a global attractor tobe finite (and equal to d), we need the linearized semiflow to contract uniformlythe d-dimensional volumes. To understand this notion properly we must first recallseveral useful ideas.


If �1; : : : ; �m 2 X, then the tensor product

�1 ˝ �2 ˝ : : :˝ �m

is an m-linear form over X defined as

.�1 ˝ �2 ˝ : : :˝ �m/. 1; : : : ; m/ D .�1; 1/X.�2; 2/X : : : .�m; m/X;

and the space spanned by all tensor products �1˝�2˝ : : :˝�m of m elements fromX is denoted by

Nm X. We will be interested in the subspace ofNm X denoted byVm X D X ^ X ^ : : :^ X (X appears m times on the right-hand side). This subspace,

containing the m-linear antisymmetric forms, is spanned by the forms

�1 ^ �2 ^ : : : ^ �m

being the exterior products of the elements from X, defined by

�1 ^ �2 ^ : : : ^ �m DX

�

sgn.�/��.1/ ˝ ��.2/ ˝ : : :˝ ��.m/;

where the sum is taken over all permutations � of the set f1; : : : ;mg, and sgn.�/is the sign of the permutation � . We define on

Vm X the inner product given, forexterior products of elements from X, by the formula

.�1 ^ �2 ^ : : : ^ �m; 1 ^ 2 ^ : : : ^ m/Vk X D detf.�i; j/Xgi;jD1;:::;m;

and we extend it to the wholeVk X by linearity. The associated norm is given, for

exterior products of elements from X, by

k�1 ^ �2 ^ : : : ^ �mk2Vk XD detf.�i; �j/Xgi;jD1;:::;m:

The importance of the norm of the exterior product k�1 ^ �2 ^ : : : ^ �mkVm X liesin the fact that this quantity is the volume of the m-dimensional parallelepiped withthe edges given by �1; : : : ; �m. We will start the evolution from the volume spannedby the vectors which are the initial conditions of the linearized system (9.2)–(9.3)and we will derive the condition which guarantees that this volume is uniformlycontracted on the attractor by the linearized semiflow. To this end, for a linear andbounded operator L 2 L .XI X/ let us introduce the operator Lm W Xm ! Vm X bythe formula

Lm.�1; : : : ; �m/ D L�1 ^ �2 ^ : : : ^ �m C �1 ^ L�2 ^ : : : ^ �m

C : : :C �1 ^ �2 ^ : : : ^ L�m:


We have the following lemma.

Lemma 9.1 (Cf. [220, Lemma V.1.2]). For any �1; : : : ; �m 2 X we have

.Lm.�1; : : : ; �m/; �1 ^ : : : �m/Vm X D k�1 ^ �2 ^ : : : ^ �mk2Vm XTr.L ı Q/;

where Q is the orthogonal projection on the space spanned by f�igmiD1 and Tr is the

trace of the operator of the finite rank, given by

Tr.L ı Q/ DmX

iD1

�

Li

kikX;i

kikX

�

X

;

where figm1D1 is the orthogonal basis of the space spanned by f�igm

iD1, obtained, forexample, by the Gram–Schmidt procedure.

We fix u0 2 A and we take the linearly independent vectors �1; : : : ; �m 2 H. Next,we solve (9.2)–(9.3) for U1.t/; : : : ;Um.t/ and calculate how the volume spanned byU1.t/; : : : ;Um.t/ is related to the volume spanned by �1; : : : ; �m (see Fig. 9.1). Wehave

Fig. 9.1 Transformation by a linearized semiflow �.t; u0/ of a two-dimensional volume spannedby �1; �2

1

2

d

dtkU1.t/ ^ : : : ^ Um.t/k2Vm X

D�

d

dt.U1.t/ ^ : : : ^ Um.t//;U1.t/ ^ : : : ^ Um.t/

�

Vm X

D .U01.t/ ^ : : : ^ Um.t/;U1.t/ ^ : : : ^ Um.t//Vm X C : : :

C .U1.t/ ^ : : : ^ Um�1.t/ ^ U0m.t/;U1.t/ ^ : : : ^ Um.t//Vm X

D .F0.S.t/u0/U1.t/ ^ : : : ^ Um.t/;U1.t/ ^ : : : ^ Um.t//Vm X C : : :


C .U1.t/ ^ : : : ^ Um�1.t/ ^ F0.S.t/u0/Um.t/;U1.t/ ^ : : : ^ Um.t//Vm X

D ..F0.S.t/u0//m.U1.t/; : : : ;Um.t//;U1.t/ ^ : : : ^ Um.t//Vm X

D kU1.t/ ^ : : : ^ Um.t/k2Vm XTr.F0.S.t/u0/ ı Q.t//;

where Q.t/ D Q.t; u0I �0; : : : ; �m/ is the orthogonal projection onto the spacespanned by U1.t/; : : : ;Um.t/. It follows that

kU1.t/ ^ : : : ^ Um.t/kVm X D k�1 ^ : : : ^ �mkVm Xe.R t0 Tr.F0.S.t/u0/ıQ.t// dt/:

We introduce the quantity

!m.S.t/u0/ D sup�1;:::;�m2X;k�ikX�1

kU1.t/ ^ : : : ^ Um.t/kVm X;

being the largest volume obtained from the m-dimensional parallelepiped withsides no greater than one, by the linearized semiflow. We also introduce auxiliaryquantities

!m.t/ D supu02A

!m.S.t/u0/;

qm.t/ D supu02A

sup�1;:::;�m2X;k�ikX�1

1

t

Z t

0

Tr.F0.S.t/u0/ ı Q.t// dt;

whence we have 1t log!m.t/ � qm.t/. Finally, we introduce the numbers

qm D lim supm!1

qm.t/:

It is easy to deduce the following result

Proposition 9.1. If, for certain m 2 NC,

qm < 0;

then, for some t0 > 0 and all t � t0 we have qm.t/ � �ı < 0, and the volumeelement kU1.t/ ^ : : : ^ Um.t/kVm X decays exponentially as t ! 1 uniformly foru0 2 A and �1; : : : ; �m 2 X, i.e.,

kU1.t/ ^ : : : ^ Um.t/kVm X � ce�ıt for t � t0:

We conclude the presentation of the method with a result that links the values qm

with the fractal dimension of the global attractor.

Theorem 9.3 (Cf. [57, Corollary 2.2]). Let Assumption A hold and let f WŒ0;1/ ! R be a concave function such that for every positive natural numberm we have qm � f .m/ and let d� > 1 be such that f .d�/ D 0. Then


dXf .A / � d�:

The above result gives rise to an algorithm. We need to estimate from above thequantities

Tr.F0.S.t/u0/ ı Q.t//;

and by taking the supremum over �1; : : : ; �m and over the points u0 in the attractor,after the integration over time, we obtain the bound on qm by the concave functionof m. Then, it is enough to find a zero of this function to get the bound onthe global attractor fractal dimension. We conclude with two remarks: firstly, thealgorithm works only if the phase space is a Hilbert space, and secondly, thesemiflow must be smooth, i.e., it must be possible to linearize the underlying initialand boundary value problem and the solution map of the linearized problem mustbe a quasidifferential of the original semiflow in accordance with Definition 9.3.Further in this chapter we will apply the described algorithm to estimate the attractordimension for a shear flow in the lubrication theory. Before we move to this problem,however, we give an account of other techniques useful for the fractal dimensionestimation.

The second class of methods useful for proving the attractor finite dimensionalityrelies on obtaining the estimates on the difference of two trajectories. We presentseveral results from this class of methods.

Theorem 9.4 (Cf. [168, 236]). Let E be a bounded subset of a Banach space H. Letmoreover V be another Banach space compactly embedded in H and let L W H ! Hbe a mapping such that E L.E/, and we have

kLu � LvkV � cku � vkH for all u; v 2 E with c > 0: (9.5)

Then, the fractal dimension of E is finite and

dHf .E/ �

log NH14c.BV.0; 1//

log 2;

where BV.0; 1/ is the closed unit ball in V.

Proof. We choose R > 0 and u 2 E such that E BH.u;R/ (we use the notationBH.x;R/ D fu 2 H W ku � xkH � Rg and BV.x;R/ D fu 2 V W ku � xkV � Rg).We have

E L.E/ BV.Lu; cR/ N[

iD1BH

�

vi;R

4

�

;


where vi 2 H and N is the number of balls in H having the radius R4

needed to coverthe ball in V centered at zero having the radius cR. Note that N D NH

14c.BV.0; 1//.

We can assume that all balls from the last sum have a nonempty intersection with E,and we denote the arbitrary points from these intersections by zi, whence

E N[

iD1BH

�

zi;R

2

�

;

with zi 2 E. It follows that

E DN[

iD1

�

BH

�

zi;R

2

�

\ E

�

:

Now

BH

�

zi;R

2

�

\ E BV

�

Lzi; cR

2

�

:

The ball in V having the radius c R2

can be covered by N balls in H having the radiusR8

, whence

BH

�

zi;R

2

�

\ E N[

jD1BH

�

vij;R

8

�

;

and, if we want the centers to be the points of E, we have

BH

�

zi;R

2

�

\ E N[

jD1BH

�

zij;R

4

�

;

with zij 2 E. Repeating this procedure inductively, we can construct a cover of E byNk balls of radius R

2k . For " > 0 we can always find k 2 N such that R2kC1 < " � R

2k ,whence N".E/ � N R

2kC1.E/ � NkC1, and

log N".E/

log 1"

� .k C 1/ log N

k log 2 � log R:

The assertion follows. utTo get the finite dimensionality of the global attractor by the above theorem, itwould seem easiest to apply it for L D S.t/ for certain t > 0 and E D A . It isthen enough to derive the estimate (9.5), which is known as the smoothing estimateas it implies that in fact we have A V . While this estimate may be difficult to


obtain for L D S.t/, in some cases it is easier to get it for L being the translationoperator defined on the trajectories starting from the attractor (and staying in it,by invariance), and V;H being the appropriately chosen spaces of time-dependentfunctions. This method, known as the method of l-trajectories, was introducedin [168] and will be used for the two-dimensional Navier–Stokes equations withnonmonotone friction in Sect. 10.

The next result, very useful for proving finite dimensionality of global attractorsof infinite dimensional dynamical systems, proved originally by Ladyzhenskaya(see, for example, [147], where, however, the notion of the Hausdorff dimension isused), uses the so-called squeezing property. This property involves the estimateswhich are not, as in Theorem 9.4, in the space compactly embedded in thestate space, but which involve the finite dimensional projectors. The variant ofLadyzhenskaya result, that we present here, comes from [70].

Theorem 9.5 (Cf. [70, Theorem 8.1]). Let H be the Hilbert space and let E Hbe a compact set. Let moreover L W H ! H be a continuous mapping such thatE L.E/ and there exists an orthogonal projector PN W H ! HN onto the subspaceHN H of dimension N, such that

kP.Lu � Lv/kH � lku � vkH for all u; v 2 E;

and

k.I � P/.Lu � Lv/kH � ıku � vkH for all u; v 2 E;

where ı < 1. Then the fractal dimension dHf .E/ is finite and

dHf .E/ � N log

9l

1 � ı�

log2

1 � ı��1

:

The above theorem has also the non-autonomous version (see Theorem 13.5) whichwill be used later in Chap. 13 to prove the finite dimensionality of the pullbackattractor for two-dimensional boundary driven Navier–Stokes flow.

Comments We refer the reader to [198] for more information about dimensions.Fractal dimension is also known as upper box counting dimension (see [198] Def-inition 3.1). Sometimes another notion of dimension, namely, the above mentionedHausdorff dimension, is used to study the global attractors for infinite dimensionaldynamical systems. To define it, one needs to use the s-dimensional Hausdorffmeasure valid for s � 0 and a subset K of a Banach space X, given by

H Xs .K/ D lim

ı!0inf

( 1X

iD1jUijs W K

1[

iD1Ui with jUij � ı

)

;

9.3 An Application to a Shear Flow in Lubrication Theory 193

where jAj D supx;y2A kx � ykX . We define the Hausdorff dimension as

dXH.K/ D inffs � 0 W H K

s .K/ D 0g:

For the properties of dXH see Chap. 2 in [198]. Here we only note that, in contrast to

the fractal dimension, the Hausdorff dimension is stable under the countable unions(see Proposition 2.8 in [198]), and for a set K X, we have

dXH.K/ � dX

f .K/;

whence the ability to estimate from above the fractal dimension gives us also thecorresponding estimate on the Hausdorff dimension, while the opposite implicationdoes not hold (in particular it is possible to construct sets with zero Hausdorffdimension and infinite fractal dimension, see [198], Exercise 13.3). We also notethat both the notions of fractal and Hausdorff dimensions remain valid in metricspaces.

9.3 An Application to a Shear Flow in Lubrication Theory

In this section we consider a two-dimensional Navier–Stokes shear flow for whichthere exists a unique global in time solution as well as the global attractor for theassociated semigroup. Our aim is to estimate from above the fractal dimension ofthe attractor in terms of given data and geometry of the domain of the flow.

The problem we consider is motivated by a typical problem from lubricationtheory where the domain of the flow is usually very thin and the Reynolds numberof the flow is large.

9.3.1 Formulation of the Problem

We consider two-dimensional Navier–Stokes equations,

ut � ��u C .u � r/u C rp D 0; (9.6)

div u D 0; (9.7)

in the channel

˝1 D fx D .x1; x2/ W �1 < x1 < 1; 0 < x2 < h.x1/g;

where h is a function, positive, smooth, and L-periodic in x1.


Let

˝ D fx D .x1; x2/ W 0 < x1 < L; 0 < x2 < h.x1/g;

and @˝ D �C [ �L [ �D, where �C and �D are the bottom and the top, and �L isthe lateral part of the boundary of ˝.

We are interested in solutions of (9.6)–(9.7) in ˝ which are L-periodic withrespect to x1. Moreover, we assume

u D 0 on �D; (9.8)

u D U0e1 D .U0; 0/ on �C; (9.9)

where U0 is a positive constant, and the initial condition

u.x; 0/ D u0.x/ for x 2 ˝: (9.10)

The problem is motivated by a flow in an infinite (rectified) journal bearing˝ � .�1;C1/, where �D � .�1;C1/ represents the outer cylinder, and �C �.�1;C1/ represents the inner, rotating cylinder. In the lubrication problems thegap h between cylinders is never constant. We can assume that the rectification doesnot change the equations as the gap between cylinders is very small with respect totheir radii.

Remark 9.1. When h D const, problem (9.6)–(9.10) was intensively studied inseveral contexts, cf. Sect. 9.4.

In our considerations we use the background flow method and transform the bound-ary condition (9.9) to the homogeneous ones by defining a smooth background flow,a simple version of the Hopf construction, which we shall be described in detail.Let

u.x1; x2; t/ D U.x2/e1 C v.x1; x2; t/; (9.11)

with

U.0/ D U0; U.h.x1// D 0; x1 2 .0;L/: (9.12)

Then v is L-periodic in x1 and satisfies

vt � ��v C .v � r/v C Uv;x1 C.v/2U0e1 C rp D �U00e1; (9.13)

div v D 0; (9.14)

v D 0 on �D [ �C; (9.15)


and the initial condition

v.x; 0/ D v0.x/ D u0.x/ � U.x2/e1: (9.16)

By .v/2 we denoted the second component of v.Now, we define a weak form of the transformed problem above. To this end we

need some notations. Let

QV D fv 2 C1.˝/2 W v is L-periodic in x1; div v D 0; v D 0 on �D [ �Cg;

V D closure of QV in H1.˝/2;

H D closure of QV in L2.˝/2:

We define the scalar products

.u; v/ DZ

˝

u.x/ � v.x/dx;

and

..u; v// D .ru;rv/in H and V , respectively, and the corresponding norms

jvj D .v; v/12 and kvk D ..v; v//

12 :

Let

b.u; v;w/ D ..u � r/v;w/;

and

a.u; v/ D �.ru;rv/:

Then the natural weak formulation of the transformed problem (9.13)–(9.16) is asfollows.

Problem 9.1. Find

v 2 C.Œ0;T�I H/ \ L2.0;TI V/

for each T > 0, such that

d

dt.v.t/;�/C a.v.t/;�/C b.v.t/; v.t/;�/ D F.v.t/;�/ (9.17)


for all � 2 V , and

v.x; 0/ D v0.x/;

where

F.v;�/ D �a.�;�/ � b.�; v;�/ � b.v; �;�/; (9.18)

and � D Ue1 is a suitable background flow.

We have the following existence theorem.

Theorem 9.6. There exists a unique weak solution of Problem 9.1 such that forall �, T, 0 < � < T, v 2 L2.�;T I H2.˝//, and for each t > 0 the map v0 7!v.t/ is continuous as a map in H. Moreover, there exists a global attractor for theassociated semigroup fS.t/gt�0 in the phase space H.

Proof. The standard existence part of the proof is based on an energy inequality,the Galerkin approximations, and the compactness method [220]. We shall notreproduce the standard argument, however, we stop for a moment to mention theproperties of the Stokes operator and the regularity problem. As usual, we definethe Stokes operator as the self-adjoint operator from V to its dual V 0 and also as anunbounded strictly positive operator in H with compact inverse A�1, by

hAu; vi D a.u; v/ for all u; v 2 V:

Then, there exists a complete orthonormal family wj, j 2 N in H, with wj 2 D.A/ Dfu 2 V W Au 2 Hg H2.˝/, such that Awj D �jwj, 0 < �1 � �2 � : : :, �j ! 1as j ! 1. The inclusion D.A/ H2.˝/ can be proved in the usual way by usingthe regularity theory. First, we extend u; p, and f by periodicity to some smooth set˝3 such that ˝ ˝3 ˝1, and then use the classical Cattabriga–Solonnikovestimate [219] to the solution .'u; 'p/ of the related generalized Stokes problem inthe domain ˝3, with ' being a suitable cut-off function that equals one on ˝. Thisargument gives us also the equivalence of the norms kukH2.˝/ and jAuj on D.A/.

Now, defining the Galerkin approximations by vm.x; t/ D PmjD1 Qv.t/wj.x/ we

can see that they belong to H2.˝/ for t > � . Further estimates performed on theGalerkin approximations lead, in a standard way, to v 2 L2.�;TI H2.˝//. utExercise 9.1. Provide the details of the proof of Theorem 9.6.

9.3.2 Energy Dissipation Rate Estimate

Our aim now is to estimate the time averaged energy dissipation rate per unit mass� of the flow u—the weak solution of problem (9.6)–(9.10). We define


� D �

j˝j < kuk2 > D lim supT!C1

�

j˝j1

T

Z T

0

ku.t/k2dt: (9.19)

We estimate first the averaged energy dissipation rate of the flow v, and then use therelation, cf. (9.11),

ku.t/k2 D kv.t/k2 C 2

Z

˝

U0.v/1;x2 dx CZ

˝

jU0j2dx: (9.20)

To estimate the right-hand side of (9.20) we use Eq. (9.17). Taking� D v in (9.17),we obtain

1

2

d

dtjvj2 C a.v; v/C b.v; v; v/ D F.v; v/:

Since v D 0 on �D [�C, is L-periodic in x1, and div v D 0, we have b.v; v; v/ D 0,and

1

2

d

dtjvj2 C �kvk2 D F.v; v/: (9.21)

Integrating the above equation in t on the interval .0;T/, dividing by T , and takinglim sup of both sides we estimate the averaged energy dissipation rate of v. Before,however, we have to estimate carefully the term F.v; v/ on the right-hand sideof (9.21). By (9.18),

F.v; v/ D �a.�; v/ � b.v; �; v/: (9.22)

We have

ja.�; v/j � �k�kkvk � �k�k2 C �

4kvk2: (9.23)

To estimate the last term in (9.22) we use the following lemma.

Lemma 9.2. For any � > 0 there exists a smooth extension

� D �.x2/ D U.x2/e1 D .U.x2/; 0/

of the boundary condition for u, such that

jb.v; �; v/j � �kvk2 for all v 2 V:

Proof. Let � W Œ0;1/ ! Œ0; 1� be a smooth function such that

�.0/ D 1; supp � Œ0; 1=2�; max j�0.s/j � p8;


and let, for 0 < " � 1,

U.x2/ D U0�.x2=.h0"//; h0 D min0�x1�L

h.x1/:

Then the function U matches the boundary conditions (9.12), is L-periodic in x1,and div Ue1 D 0. Moreover, for Q" D .0;L/ � .0; h0"=2/,

jb.v;Ue1; v/j D jb.v; v;Ue1/j ��v

x2

��

L2.Q"/

krvkL2.Q"/kx2U.x2/kL1.Q"/:

Now,

kx2U.x2/kL1.Q"/ � h0"

2U0;

and

��v

x2

��

2

L2.Q"/

�Z L

0

Z h.x1/

0

ˇˇˇˇv.x1; x2/

x2

ˇˇˇˇ

2

dx2dx1 � 4

Z L

0

Z h.x1/

0

ˇˇˇˇ@v.x1; x2/

@x2

ˇˇˇˇ

2

dx2dx1

by the Hardy inequality (cf. Theorem 3.12 and Exercise 3.12). Thus, we obtain

jb.v;Ue1; v/j � �kvk2 with � D 2h0U0": ut

In particular,

jb.v; �; v/j � �

4kvk2 for " D minf�=.8h0U0/; 1g: (9.24)

In view of estimates (9.23) and (9.24),

jF.v; v/j � �k�k2 C �

2kvk2;

and, by (9.21),

1

2

d

dtjvj2 C �

2kvk2 � �k�k2: (9.25)

To estimate the right-hand side in terms of the data, we use

Lemma 9.3. Let U be as in Lemma 9.2. ThenZ

˝

jU.x2/j2dx1dx2 � 1

2Lh0U

20"; (9.26)


and

Z

˝

jU0.x2/j2dx1dx2 � 4LU20

h0

1

": (9.27)

Proof. We have jU.x2/j � U0 and supp U Q", which gives (9.26). Moreover,

Z

˝

jU0.x2/j2dx D U20

h20"2

Z

˝

ˇˇˇˇ�

0�

x2h0"

�ˇˇˇˇ

2

dx

D U20

h20"2

LZ h0"

2

0

ˇˇˇˇ�

0�

x2h0"

�ˇˇˇˇ

2

dx2 � 4LU20

h0

1

";

as j�0j � p8, which gives the second inequality of the lemma. ut

Applying Lemma 9.3 to inequality (9.25) we obtain

1

2

d

dtjvj2 C �

2kvk2 � �

4LU20

h0

1

": (9.28)

Let hM D max0�x1�L h.x1/ and hM=h0 ' 1. Then we can define the Reynoldsnumber of the flow u by Re D .h0U0/=�. Observe that for a turbulent flow u (of somelarge Reynolds number) " in (9.24) is equal to �=.8h0U0/. Therefore, from (9.28)we obtain, as indicated above,

Lemma 9.4. If Re � 1 then the time averaged energy dissipation rate per unitmass �.v/ for the flow v can be estimated as follows:

�.v/ � 64L

j˝j U30 � 64

h0U30:

In the end, we have the following theorem.

Theorem 9.7. For the Navier–Stokes turbulent flow u with Re � 1 the timeaveraged energy dissipation rate per unit mass � defined in (9.19) can be estimatedas follows:

� � CU30

h0; (9.29)

where C is a numeric constant.

Proof. By (9.20) we have

< kuk2 > � 2 < kvk2 > C2Z

˝

jU0j2dx;

and then we use (9.27) to estimate the second term on the right-hand side. ut


Estimate (9.29) has the same form as the usual in turbulence theory estimate of theaveraged energy dissipation rate for the Navier–Stokes boundary driven flow in arectangular domain [87, 88, 99].

In the sequel we shall use the estimates of � to find an upper bound of thedimension of the global attractor.

9.3.3 A Version of the Lieb–Thirring Inequality

Let

QH1 D fv 2 C1.˝/2 W v is L -periodic in x1; v D 0 on �D [ �Cgand

H1 D closure of QH1 in H1.˝/2:

Lemma 9.5. Let 'j 2 H1, j D 1; : : : ;m be an orthonormal family in L2.˝/ and lethM D max0�x1�L h.x1/. Then the following inequality holds:

Z

˝

0

@mX

jD1'2j

1

A

2

dx � �

1C�

hM

L

�2!

mX

jD1

Z

˝

jr'jj2dx; (9.30)

where � is an absolute constant.

Proof. Let˝1 D .0;L/� .0; h0/, and let j 2 H1.˝1/, j D 1; : : : ;m, be a family offunctions that are orthonormal in L2.˝1/, with j.0; x2/ D j.L; x2/ and j D 0 forx2 D 0, x2 D h0. We know [89] that for this family there exist absolute constants C0

0

and C01, such that

Z

˝1

0

@mX

jD1 2

j

1

A

2

dx � C00

0

@mX

jD1

Z

˝1

ˇˇˇˇ@ j

@x1

ˇˇˇˇ

2

dx C C01m

L2

1

A

120

@mX

jD1

Z

˝1

ˇˇˇˇ@ j

@x2

ˇˇˇˇ

2

dx

1

A

12

:

We have also the Poincaré inequality

Z

˝1

2j dx � h2M

Z

˝1

ˇˇˇˇ@ j

@x2

ˇˇˇˇ

2

dx:

From the above two inequalities and the observation that

m DmX

jD1

Z

˝1

2j dx;

(9.30) follows. ut


9.3.4 Dimension Estimate of the Global Attractor

We rewrite Eq. (9.17) for the transformed flow v in the short form

dv

dtD L.v/; v.0/ D v0;

where, for all � 2 V ,

hL.v.t//;�i D �a.v.t/;�/ � b.v.t/; v.t/;�/C F.v.t/;�/;

and F is defined in (9.18). Now, to estimate from above the dimension of the globalattractor we follow the procedure described in Sect. 9.2. First, for an integer m > 1

and vectors �j 2 H, j D 1; : : : ;m, we consider the corresponding linearized aroundthe orbit v.t/ problems

d

dtUj D L0.v/Uj and Uj.0/ D �j for j D 1; : : : ;m; (9.31)

where L0.v/ is the Fréchet derivative of L at v D v.t/, with

.L0.v/Uj; �/ D �a.Uj; �/ � b.v;Uj; �/ � b.Uj; v;�/ � b.�;Uj; �/ � b.Uj; �; �/

D �a.Uj; �/ � b.u;Uj; �/ � b.Uj; u; �/: (9.32)

Let, for a particular time � , �j D �j.�/, j 2 N be an orthonormal basis of H with�1.�/; : : : ; �m.�/ spanning

Qm.�/H D Qm.�; v0; �1; : : : ; �m/H D spanfU1.�/; : : : ;Um.�/g;

Qm.�/ being the orthogonal projector of H onto the space spanned by Uj.�/ forj D 1; : : : ;m, solutions of (9.31). We then have �j.�/ 2 V , j D 1; : : : ;m, for a.e.� 2 RC.

The trace of L0.v.�// ı Qm.�/ is

Tr�L0.v.�// ı Qm.�/

� DmX

jD1

�L0.v.�//�j.�/;�j.�/

�: (9.33)

Let A be the global attractor for the transformed flow v.�/ D S.�/v0, and let

qm D lim supT!1

qm.T/;


where

qm.T/ D supv02A

sup�j2H;

j�jj�1;jD1;:::;m

�1

T

Z T

0

Tr�L0.v.�// ı Qm.�/

�d�

�

:

Now, our aim is to obtain estimate (9.37) and then inequality (9.38).

Lemma 9.6. The following estimate holds:

Tr�L0.v/ ı Qm

� � ��mX

jD1k�jk2 C j�jL2.˝/kuk; (9.34)

where

�.x/ DmX

jD1j�j.x/j2:

Proof. From (9.32) and (9.33) we obtain

mX

jD1

�L0.v/�j; �j

� � ��mX

jD1k�jk2 �

mX

jD1b.�j; u; �j/

� ��mX

jD1k�jk2 C

Z

˝

jruj0

@mX

jD1j�j.x/j2

1

A dx;

whence (9.34) follows. utNow, in our estimates of the trace we use Lemma 9.5. By this lemma, taking intoaccount that hM L for a thin domain ˝, and from

Z

˝

�.x/dx D m

we have

m2

j˝j � j�j2L2.˝/ � �1

mX

jD1k�jk2; (9.35)

where �1 D 2� . Moreover,

j�jL2.˝/kuk � �

2�1j�j2L2.˝/ C �1

2�kuk2 � �

2

mX

jD1k�jk2 C �1

2�kuk2 (9.36)


by the second inequality in (9.35). Thus, by (9.34), (9.36)

Tr�L0.v/ ı Qm

� � ��2

mX

jD1k�jk2 C �1

2�kuk2;

and by (9.35) again, we obtain in the end

Tr�L0.v.�// ı Qm.�/

� � � �

2�1j˝jm2 C �1

2�kuk2: (9.37)

From (9.37), recalling � defined in (9.19), we have

qm � � �

2�1j˝jm2 C �1j˝j2�2

�: (9.38)

Define

a D �

2�1j˝j ; c D �1j˝j2�2

�:

We can write

qm � �am2 C c:

By Theorem 9.3 we have

dHf .A / � p;

where p > 0 is such that �ap2 C c D 0. Obviously, we have

p Dr

c

a:

Further, taking into account definitions of a and c, as well as estimate (9.29) of �,we obtain

p � �1j˝jh20.Re/3=2: (9.39)

From the above estimates we conclude the following theorem about stronglyturbulent flows in thin (or elongated) domains.


Theorem 9.8. Consider the Navier–Stokes flow u as described in this section.Assume that the domain ˝ is thin and that the flow is strongly turbulent, namely

hM

L 1 and Re � 1: (9.40)

Then the fractal dimension of the global attractor A for this flow can be estimatedas follows:

dHf .A / �

j˝jh20.Re/3=2; (9.41)

where is some absolute constant. For a rectangular domain ˝ D .0;L/ � .0; h0/we obtain, in particular,

dHf .A / �

L

h0.Re/3=2:

Proof. By (9.39), (9.40), and by the definition of m0, (9.41) follows. ut


The standard procedure of estimating the global attractor dimension based on thedynamical system theory [88, 99, 220], we used in Sect. 9.3, involves two importantingredients: estimate of the time averaged energy dissipation rate � and a Lieb–Thirring-like inequality. The precision and physical soundness of an estimate of thenumber of degrees of freedom of a given flow (expressed by an estimate of its globalattractor) depends directly on the quality of the estimate of � and a good choice ofthe Lieb–Thirring-like inequality which depends, in particular, on the geometry ofthe domain and on the boundary conditions of the flow.

There is a quickly growing literature devoted to better and better estimates ofaveraged parameters and attractor dimension of a variety of flows. We mention onlya few positions which are related to the problem considered in Sect. 9.3 and someother to give a larger context. Estimate (9.41) is a direct generalization of that in[89], where the domain of the flow is an elongated rectangle ˝ D .0;L/ � .0; h/,L � h. In this case the attractor dimension can be estimated from above by c L

h Re3=2,where c is a universal constant, and Re D Uh

�is the Reynolds number. Here, the

dependence on the geometry of the domain is explicit. (To obtain such result theauthors prove a suitable anisotropic form of the Lieb–Thirring inequality. For avariety of forms of this inequality cf. [220]. Moreover, a discussion on agreementof their estimates with Kolmogorov and Kraichnan scaling in turbulence theory isprovided in [89].)


In [241] optimal bounds of the attractor dimension are given for a flow ina rectangle .0; 2�L/ � .0; 2�L=˛/, with periodic boundary conditions and givenexternal forcing. The estimates are of the form c0=˛ � dH

f .A / � c1=˛, see also[180]. A free boundary conditions are considered in [242], see also [222], andan upper bound on the attractor dimension established with the use of a suitableanisotropic version of the Lieb–Thirring inequality.

Estimates of the energy dissipation rate for boundary driven flows, essential inturbulence theory, are investigated, e.g., in [230] for ˝ D .0;Lx/ � .0;Ly/ � .0; h/,and in [229] for a smooth and bounded three-dimensional domains. The bestestimates come from variational methods, cf. [88], and, in particular [134]. Othercontexts and problems can be found, e.g., in monographs [62, 88, 99, 197, 220], andthe literature quoted there.

Boundary driven flows in smooth and bounded two-dimensional domains fora non-autonomous Navier–Stokes system are considered in [178], by using anapproach of Chepyzhov and Vishik, cf. [62]. In Chap. 13 we shall consider a shearflow with time-dependent boundary driving.

10Exponential Attractors in Contact Problems

In this chapter we consider two examples of contact problems. First, we study theproblem of time asymptotics for a class of two-dimensional turbulent boundarydriven flows subject to the Tresca friction law which naturally appears in lubricationtheory. Then we analyze the problem with the generalized Tresca law, where thefriction coefficient can depend on the tangential slip rate.

While the first problem is governed by a variational inequality, since theunderlying potential is convex, the problem with the generalized Tresca law due tothe lack of potential convexity leads to a differential inclusion where the multivaluedterm has the form of the Clarke subdifferential.

We prove that in both cases the global attractors exist and they have finite fractaldimensions. Moreover, there exists an object, called exponential attractor. It containsthe global attractor, has finite fractal dimension and attracts the trajectories exponen-tially fast in time. This property allows, among other things, to locate the exponentialand thus the global attractor in the phase space by using numerical analysis.

We start the chapter with the section in which we remind a general method ofproving the existence of exponential attractors.

10.1 Exponential Attractors and Fractal Dimension

Let us consider an abstract autonomous evolutionary problem

dv.t/

dtD F.v.t// in Z for a.e. t 2 R

C; (10.1)

v.0/ D v0;


207

208 10 Exponential Attractors in Contact Problems

where Z is a Banach space, F is a nonlinear operator, and v0 is in a Banach space Xthat embeds in Z. We assume that the above problem has a global in time uniquesolution R

C 3 t ! v.t/ 2 X for every v0 2 X. In this case one can associate withthe problem a semigroup fS.t/gt�0 of (nonlinear) operators S.t/ W X ! X settingS.t/v0 D v.t/, where v.t/, t > 0, is the unique solution of (10.1).

Below we recall the notion of the global attractor and its properties (cf. Sect. 7.2).We know that from the properties of the semigroup of operators fS.t/gt�0 we can

derive the information on the behavior of solutions of problem (10.1), in particular,on their time asymptotics. One of the objects, the existence of which characterizesthe asymptotic behavior of solutions, is the global attractor. It is a compact andinvariant with respect to operators S.t/ subset of the phase space X (in general, ametric space) that uniformly attracts all bounded subsets of X.

Definition 10.1. A global attractor for a semigroup fS.t/gt�0 in a Banach space Xis a subset A of X such that:

• A is compact in X,• A is invariant, i.e., S.t/A D A for every t � 0,• for every " > 0 and every bounded set B in X there exists t0 D t0.B; "/ such

thatS

t�t0S.t/B is a subset of the "-neighborhood of the attractor A (uniform

attraction property).

The global attractor defined above is uniquely determined by the semigroup.Moreover, it is connected and also has the following properties: it is the maximalcompact invariant set and the minimal closed set that attracts all bounded sets. Theglobal attractor may have a very complex structure. However, as a compact set (in aninfinite dimensional Banach space) its interior is empty.

In Sect. 7.2 we proved the following theorem that guarantees the global attractorexistence (see Theorem 7.3 and Remark 7.4).

Theorem 10.1. Let fS.t/gt�0 be a semigroup of operators in a Banach space X suchthat:

• for all t � 0 the operators S.t/ W X ! X are continuous,• fS.t/gt�0 is dissipative, i.e., there exists a bounded set B0 X such that for every

bounded set B X there exists t0 D t0.B/ such thatS

t�t0S.t/B B0,

• fS.t/gt�0 is asymptotically compact, i.e., for every bounded set B X and allsequences tk ! 1 and yk 2 S.tk/B, the sequence fykg is relatively compact in X.

Then fS.t/gt�0 has a global attractor A .

In this chapter, however, we shall make use of another general theorem on theexistence of the global attractor, cf. Theorem 10.2.

As we have seen in Chap. 9, for some dissipative dynamical systems the globalattractor has a finite fractal dimension. This fact has a number of importantconsequences for the behavior of the flow generated by the semigroup [197, 198].

10.1 Exponential Attractors and Fractal Dimension 209

Definition 10.2. The fractal dimension (also known as upper box counting dimen-sion) of a compact set K in a Banach space X is defined as

dXf .K/ D lim sup

"!0

log NX" .K/

log 1"

;

where NX" .K/ is the minimal number of balls of radius " in X needed to cover K.

Another important property that holds for many dynamical systems is the existenceof an exponential attractor.

Definition 10.3. An exponential attractor for a semigroup fS.t/gt�0 in a Banachspace X is a subset M of X such that:

• M is compact in X,• M is positively invariant, i.e., S.t/M M for every t � 0,• fractal dimension of M is finite, i.e., dX

f .M / < 1,• M attracts exponentially all bounded subsets of X, i.e., there exist a universal

constant c1 and a monotone function ˚ such that for every bounded set B in X,its image S.t/B is a subset of the ".t/-neighborhood of M for all t � t0, where".t/ D ˚.kBkX/e�c1t (exponential attraction property).

Since the global attractor A is the minimal compact attracting set it follows that ifboth global and exponential attractors exist, then A M and the fractal dimensionof A must be finite. Moreover, in contrast to the global attractor, an exponentialattractor does not have to be unique.

In the following we will remind the abstract framework of [168] (see also [162])that uses the so-called method of l-trajectories (or short trajectories) to prove theexistence of global and exponential attractors.

Let X, Y , and Z be three Banach spaces such that

Y X with compact imbedding and X Z:

We assume, moreover, that X is reflexive and separable.For � > 0, let

X� D L2.0; � I X/;

and

Y� D�

u 2 L2.0; � I Y/ W du

dt2 L2.0; � I Z/

�

:

By Cw.Œ0; ��I X/ we denote the space of weakly continuous functions from theinterval Œ0; �� to the Banach space X, and we assume that the solutions of (10.1) areat least in Cw.Œ0;T�I X/ for all T > 0. Then by an l-trajectory we mean the restrictionof any solution to the time interval Œ0; l�. If v D v.t/; t � 0, is the solution of (10.1)


then both � D vjŒ0;l� and all shifts Lt.�/ D vjŒt;lCt� for t > 0 are l-trajectories. Notethat the mapping Lt is defined as Lt.�/.�/ D v.� C t/ for t > 0 and � 2 Œ0; l�, wherev is a unique solution such that vjŒ0;l� D �.

We can now formulate a theorem which gives criteria for the existence of a globalattractor A for the semigroup fS.t/gt�0 in X and its finite dimensionality. Thefollowing criteria are stated as assumptions .A1/–.A8/ in [168]:

.A1/ for any v0 2 X and arbitrary T > 0 there exists (not necessarily unique)v 2 Cw.Œ0;T�I X/ \ YT , a solution of the evolutionary problem on Œ0;T� withv.0/ D v0; moreover, for any solution v, the estimates of kvkYT are uniform withrespect to kv0kX ,.A2/ there exists a bounded set B0 X with the following properties: if v is anarbitrary solution with initial condition v0 2 X then

1. there exists t0 D t0.kv0kX/ such that v.t/ 2 B0 for all t � t0,2. if v0 2 B0 then v.t/ 2 B0 for all t � 0,

.A3/ each l-trajectory has a unique continuation among all solutions of theevolution problem which means that from an endpoint of an l-trajectory therestarts at most one solution,.A4/ for all t > 0, Lt W Xl ! Xl is continuous on B0

l —the set of all l-trajectoriesstarting at any point of B0 from .A2/,.A5/ for some � > 0, the closure in Xl of the set L� .B0

l / is contained in B0l ,

.A6/ there exists a space Wl such that Wl Xl with compact embedding, and� > 0 such that L� W Xl ! Wl is Lipschitz continuous on B1

l —the closure ofL� .B0

l / in Xl,.A7/ the map e W Xl ! X, e.�/ D �.l/, is continuous on B1

l ,.A8/ the map e W Xl ! X is Hölder-continuous on B1

l .

Theorem 10.2. Let the assumptions .A1/–.A5/, .A7/ hold. Then there exists aglobal attractor A for the semigroup fS.t/gt�0 in X. Moreover, if the assumptions.A6/, .A8/ are satisfied then the fractal dimension of the attractor is finite.

For the existence of an exponential attractor we need two additional properties,where now X is a Hilbert space (cf. [168]):

.A9/ for all � > 0 the operators Lt W Xl ! Xl are (uniformly with respect tot 2 Œ0; ��) Lipschitz continuous on B1

l ,.A10/ for all � > 0 there exists c > 0 and ˇ 2 .0; 1� such that for all � 2 B1

l andt1; t2 2 Œ0; �� it holds that

kLt1� � Lt2�kXl � cjt1 � t2jˇ: (10.2)

Theorem 10.3. Let X be a separable Hilbert space and let the assumptions .A1/–.A6/ and .A8/–.A10/ hold. Then there exists an exponential attractor M for thesemigroup fS.t/gt�0 in X.

For the proofs of Theorems 10.2 and 10.3 we refer the readers to correspondingtheorems in [168].

10.2 Planar Shear Flows with the Tresca Friction Condition 211

10.2 Planar Shear Flows with the Tresca Friction Condition

10.2.1 Problem Formulation

We start this section from the description of the problem for which we will laterprove the exponential attractor existence. The flow of an incompressible fluid in atwo-dimensional domain ˝ is described by the equation of motion

ut � ��u C .u � r/u C rp D 0 in ˝ � RC; (10.3)

and the incompressibility condition

div u D 0 in ˝ � RC: (10.4)

To define the domain ˝ of the flow, let ˝1 be the channel

˝1 D fx D .x1; x2/ W �1 < x1 < 1; 0 < x2 < h.x1/g;

where h is a positive function, smooth, and L-periodic in x1. Then we set

˝ D fx D .x1; x2/ W 0 < x1 < L; 0 < x2 < h.x1/g;

and @˝ D �C [�L [�D, where �C and �D are the bottom and the top, and �L is thelateral part of the boundary of ˝. By n we will denote the unit outward normal to@˝ and by T D T.u; p/ the stress tensor given by T.u; p/ D .Tij.u; p//2i;jD1, whereTij.u; p/ D �pıij C �

�ui;j C uj;i

�.

We are interested in solutions of (10.3)–(10.4), such that the velocity u and theCauchy stress vector T.u; p/n on �L are L-periodic with respect to x1. Note thatif both velocity and pressure are smooth functions defined on ˝1 and L-periodicwith respect to x1, then the periodicity of the Cauchy stress vector on �L followsautomatically from the constitutive law.

We assume that

u D 0 on �D � RC: (10.5)

Moreover, we assume the no flux condition across �C so that the normal componentof the velocity on �C satisfies

un D u � n D 0 on �C � RC; (10.6)

and that the tangential component of the velocity u� on �C is unknown and satisfiesthe Tresca friction law with a constant and positive maximal friction coefficient k.This means that, cf., e.g., [91, 215],


jT� .u; p/j � k

jT� .u; p/j < k ) u� D U0e1

jT� .u; p/j D k ) 9� � 0 such that u� D U0e1 � �T� .u; p/

9>>>>>=

>>>>>;

on �C � RC;

(10.7)

where U0e1 D .U0; 0/, U0 2 R, is the velocity of the lower surface producing thedriving force of the flow and T� is the tangential component of the stress vector on�C given by

T� .u; p/ D T.u; p/n � ..T.u; p/n/ � n/n; (10.8)

and u� is the tangential velocity on �C given by u� D u�unn. From (10.6) it followsthat u� D u.

Finally, the initial condition for the velocity field is

u.x; 0/ D u0.x/ for x 2 ˝:The problem is motivated by the examination of a certain two-dimensional flowin an infinite (rectified) journal bearing ˝ � .�1;C1/, where �D � .�1;C1/

represents the outer cylinder, and �C � .�1;C1/ represents the inner, rotatingcylinder. In the lubrication problems the gap h between cylinders is never constant.We can assume that the rectification does not change the equations as the gapbetween cylinders is very small with respect to their radii.

Since the widely used no-slip boundary condition when the fluid has the samevelocity as surrounding solid boundary is not respected if the shear rate becomestoo high we can model such situation by a transposition of the well-known frictionlaws between two solids [215] to the fluid–solid interface. The Tresca friction lawis one of them.

To work with the above problem it is convenient to transform the boundarycondition (10.7) to the homogeneous one. To this end, let

u.x1; x2; t/ D U.x2/e1 C v.x1; x2; t/; (10.9)

with

U.0/ D U0; U.h.x1// D 0; x1 2 .0;L/: (10.10)

The new vector field v is L-periodic in x1 and satisfies the equation of motion

vt � ��v C .v � r/v C rp D G.v/; (10.11)

with

G.v/ D �Uv;x1 �.v/2 U;x2 e1 C �U;x2x2 e1;


where by .v/2 we denoted the second component of v. As div.Ue1/ D 0 we get

div v D 0 in ˝ � RC: (10.12)

From (10.9)–(10.10) we obtain

v D 0 on �D � RC (10.13)

and

vn D v � n D 0 on �C � RC: (10.14)

Moreover,

T� .v; p/ D T� .u; p/C

�@U.x2/

@x2

ˇˇˇˇx2D0

; 0

!

on �C � RC:

Since we can define the extension U in such a way that

@U.x2/

@x2

ˇˇˇˇx2D0

D 0;

the Tresca condition (10.7) transforms to

jT� .v; p/j � k

jT� .v; p/j < k ) v� D 0

jT� .v; p/j D k ) 9� � 0 such that v� D ��T� .v; p/

9>>>>>=

>>>>>;

on �C � RC:

(10.15)Finally, the initial condition becomes

v.x; 0/ D v0.x/ D u0.x/ � U.x2/e1 for x 2 ˝: (10.16)

The Tresca condition (10.15) is a particular case of an important in contactmechanics class of subdifferential boundary conditions of the form, cf., e.g., [188],

'.��.x//�'.v� .x; t// � �T� .x; t/.��.x/�v� .x; t// on �C�RC; (10.17)

where �� belongs to a certain set of admissible functions (in our case the tangentialcomponents of the traces on �C of functions from the space V defined below) and 'is a convex functional. For '.v� / D kjv� j the last condition is equivalent to (10.15).


We pass to the variational formulation of the homogeneous problem (10.11)–(10.16). Then, for the convenience of the reader, we describe the relations betweenthe classical and the weak formulations.

We begin with some basic definitions. Let

QV D fv 2 C1.˝/2 W div v D 0 in ˝; v is L-periodic in x1;

v D 0 on �D; v � n D 0 on �Cg;

and

V D closure of QV in H1.˝/2; H D closure of QV in L2.˝/2:

We define the scalar products in H and V , respectively, by

.u ; v/ DZ

˝

u.x/ � v.x/ dx and .ru ; rv/ DZ

˝

ru.x/W rv.x/ dx;

and their associated norms by

jvj D .v; v/12 and kvk D .rv ; rv/ 12 :

Let, for u; v, and w in V ,

a.u ; v/ D .ru ; rv/ and b.u ; v ; w/ D ..u � r/v ; w/:

We denote the trace operator from V to L2.�C/2 by � . Its norm is denoted by

k�k D k�kL .VIL2.�C/2/. Finally, let us define the convex and continuous functional˚ W V ! R by

˚.u/ DZ

�C

kju.x1; 0/jdx1 DZ

�C

'.u� / dS:

The variational formulation of the homogeneous problem (10.11)–(10.16) is asfollows:

Problem 10.1. Given v0 2 H, find v W Œ0;1/ ! H such that:

.i/ for all T > 0,

v 2 C.Œ0;T�I H/ \ L2.0;TI V/ with vt 2 L2.0;TI V 0/;

where V 0 is the dual space to V ,


.ii/ for all � in V , and for almost all t in the interval .0;1/, the followingvariational inequality holds

hvt.t/;� � v.t/i C �a.v.t/;� � v.t// C b.v.t/; v.t/;� � v.t//C ˚.�/ � ˚.v.t// � hL .v.t//;� � v.t/i; (10.18)

.iii/ the following initial condition holds

v.x; 0/ D v0.x/ for a.e. x 2 ˝: (10.19)

In (10.18) the operator L W V ! V 0 is defined by

hL .v/;�i D ��a.�;�/ � b.�; v;�/ � b.v; �;�/;

where � D Ue1 is a suitable smooth background flow.

We have the following relations between the classical and weak formulations.

Proposition 10.1. Every classical solution of Problem (10.11)–(10.16) is also asolution of Problem 10.1. On the other hand, every solution of Problem 10.1 whichis smooth enough is also a classical solution of Problem (10.11)–(10.16).

Proof. As the proof is quite technical it is only sketched, and the details are leftto the reader. Let v be a classical solution of Problem (10.11)–(10.16). As it is(by assumption) sufficiently regular, we have to check only (10.18). Remark firstthat (10.11) can be written as

vt.t/ � div T.v.t/; p.t//C .v.t/ � r/v.t/ D G.v.t//: (10.20)

Let� 2 V . Multiplying (10.20) by��v.t/ and using the Green formula we obtainZ

˝vt.t/ � .� � v.t// dx C

Z

˝Tij.v.t/; p.t//.� � v.t//i;j dx C b.v.t/; v.t/;� � v.t//

DZ

@˝Tij.v.t/; p.t//nj.� � v.t//i dS C

Z

˝G.v.t// � .� � v.t// dx (10.21)

for t 2 .0;T/. As v.t/ and � are in V , after some calculations we obtainZ

˝

Tij.v.t/; p.t//.� � v.t//i;j dx D �a.v.t/ ; � � v.t//: (10.22)

By (10.17) with '.v� / D kjv� j, taking into account the boundary conditions, we getZ

@˝

Tij.v.t/; p.t//nj.� � v.t//i dS DZ

�C

T� .v.t/; p.t// � .�� v� .t// dS

� �Z

�C

k.j�� j � jv� .t/j/ dS: (10.23)


Finally,Z

˝

G.v.t// � .� � v.t//dx D hL .v.t// ; � � v.t/i: (10.24)

From (10.22)–(10.24) we see that (10.21) yields (10.18), and (10.19) is the sameas (10.16).

Conversely, suppose that v is a sufficiently smooth solution to Problem 10.1. Wehave immediately (10.12)–(10.14) and (10.16). Let ' be in the space

.H1div.˝//

2 D f' 2 V W ' D 0 on @˝g:

We take � D v.t/˙ ' in (10.18) to get

hvt.t/ � ��v.t/C .v.t/ � r/v.t/ � G.v.t// ; 'i D 0 for every ' 2 .H1div.˝//

2:

Thus, there exists a distribution p.t/ on ˝ such that

vt.t/ � ��v.t/C .v.t/ � r/v.t/ � G.v.t// D rp.t/ in ˝; (10.25)

so that (10.11) holds. It follows that p is smooth and without loss of generality wemay assume that it is also L-periodic with respect to x1. Now, we shall derive theTresca boundary condition (10.15) from the weak formulation. We have

Z

˝

Tij.v.t/; p.t//.� � v.t//i;j dx D �Z

˝

div T.v.t/; p.t// � .� � v.t// dx

CZ

@˝

T.v.t/; p.t//n � .� � v.t// dS: (10.26)

Applying (10.22) and (10.26) to (10.18) we get

Z

˝

.vt.t/ � div T.v.t/; p.t//C .v.t/ � r/v.t/ � G.v.t/// � .� � v.t// dx

CZ

@˝

T.v.t/; p.t//n � .� � v.t// dS � ˚.v.t// � ˚.�/:

By (10.25) we have (10.20) and so the first integral on the left-hand side vanishes.Thus we obtain condition (10.23). Due to the fact that �n � vn.t/ D 0 on �C, weconcludeZ

@˝

T.v.t/; p.t//n � .� � v.t//i dS

DZ

�C

T� .v.t/; p.t// � .�� v� .t// dS CZ

�L

T.v.t/; p.t//n � .� � v.t// dS;


and then,

Z

�L

T.v.t/; p.t//n � .� � v.t// dS CZ

�C

T� .v.t/; p.t// � .�� v� .t// dS

� �Z

�C

k.j�� j � jv� .t/j/ dS;

where � is any element of V . We assume that the weak solution v is a sufficientlysmooth and L-periodic with respect to x1 function defined on the infinite strip ˝1.Then, as the associated pressure is also L-periodic, it follows that the Cauchy stressvector T.v.t/; p.t//n must be L-periodic with respect to x1 and the integral on �L

vanishes, whereas

Z

�C

T� .v.t/; p.t// � .�� v� .t// dS � �Z

�C

k.j�� j � jv� .t/j/ dS: (10.27)

Taking � D 0 and � D 2v, respectively, we get

Z

�C

T� � v� dS D �Z

�C

kjv� j dS; (10.28)

whence it follows thatZ

�C

T� �� dS � �Z

�C

kj�� j dS for every � 2 V: (10.29)

Note that only the second coordinate of T� and v� is nonzero on �C, so by a slightabuse of notation we can consider these functions as scalars. If�� is smooth on �C,using the methods of Sect. 1.2 in [146] it is possible to construct� 2 V such that itstangential part is equal to �� on �C. From density of smooth functions in L2.�C/ itfollows that the following inequality analogous to (10.29)

Z

�C

T�� dS � �Z

�C

kj� j dS

holds for all � 2 L2.�C/. From the above inequality it follows that the pointwiseinequality jT� j � k must hold a.e. on �C. Indeed, if the opposite inequality holds ona set S of positive measure then we arrive at contradiction by taking � D 0 outsideS and � D � T�jT� j on S. Now, (10.28) implies that the Tresca condition (10.15) holdsand the proof is complete. ut


10.2.2 Existence and Uniqueness of a Global in Time Solution

In this subsection we establish the existence and uniqueness of a global in timesolution for Problem 10.1. First, we present two lemmas.

Lemma 10.1 (cf. [28]). There exists a smooth extension

�.x1; x2/ D U.x2/e1

of U0e1 from �C to ˝ satisfying: (10.10),

@U.x2/

@x2

ˇˇˇˇx2D0

D 0;

and such that

jb.v ; �; v/j � �

4kvk2 for all v 2 V:

Moreover,

j�j2 C jr�j2 DZ

˝

jU.x2/j2dx1dx2 CZ

˝

jU;x2 .x2/j2dx1dx2 � F;

where F is a constant depending on �;˝, and U0.

Exercise 10.1. Provide a proof of Lemma 10.1, cf. Lemma 9.2.

Lemma 10.2. For all v in V we have the Ladyzhenskaya inequality

kvkL4.˝/ � C.˝/jvj 12 kvk 12 : (10.30)

Proof. Let v 2 V and r 2 C1.Œ�L;L�/ such that r D 1 on Œ0 ; L� and r D 0 atx1 D �L. Define ' D rv (keep in mind that v can be periodically extended to whole˝1), and extend ' by 0 to ˝1 D .�L ; L/ � .0 ; h/, where h D max0�x1�L h.x1/.We obtain

'2.x1; x2/ D 2

Z x1

�L'.t1; x2/

@'

@t1.t1; x2/ dt1 � 2

Z L

�Lj'.x1; x2/j

ˇˇˇˇ@'

@x1.x1; x2/

ˇˇˇˇ dx1

and

'2.x1; x2/ D �2Z h

x2

'.x1; t2/@'

@t2.x1; t2/ dt2 � 2

Z h

0

j'.x1; x2/jˇˇˇˇ@'

@x2.x1; x2/

ˇˇˇˇ dx2;


whence

k'k4L4.˝1/ DZ

˝1

'2.x1; x2/'2.x1; x2/ dx1dx2

� Z h

0

sup�L�x1�L

'2.x1; x2/ dx2

! Z L

�Lsup

0�x2�h'2.x1; x2/ dx1

!

� 4

�Z h

0

Z L

�Lj'jˇˇˇˇ@'

@x1

ˇˇˇˇ dx1dx2

�

��Z L

�L

Z h

0

j'jˇˇˇˇ@'

@x2

ˇˇˇˇ dx2dx1

�

:

By the Cauchy–Schwartz inequality,

k'k4L4.˝1/ � 4j'j2L2.˝1/ˇˇˇˇ@'

@x1

ˇˇˇˇL2.˝1/

ˇˇˇˇ@'

@x2

ˇˇˇˇL2.˝1/

� 2j'j2L2.˝1/ ˇˇˇˇ@'

@x1

ˇˇˇˇ

2

L2.˝1/

Cˇˇˇˇ@'

@x2

ˇˇˇˇ

2

L2.˝1/

!

� 2j'j2L2.˝1/jr'j2L2.˝1/:We use the fact that jrj � 1 and the Poincaré inequality to get

kvkL4.˝/ � k'kL4.˝1/; j'jL2.˝1/ � 2jvj; and jr'jL2.˝1/ � Ckvkfor some constant C, whence (10.30) holds. utTheorem 10.4. For any v0 2 H and U0 2 R there exists a solution of Problem 10.1.

Proof. We provide only the main steps of the proof as it is quite standard and, onthe other hand, long. The estimates we obtain will be used further in the section.

Observe that the functional ˚ is convex, lower semicontinuous (in fact it iscontinuous) but nondifferentiable at zero. To overcome this difficulty we use thefollowing approach (see, e.g., [91, 113, 188]). For ı > 0 let ˚ı W V ! R be afunctional defined by

V 3 w 7! ˚ı.w/ D 1

1C ı

Z

�C

kjw� j1Cı dx:

This functional is convex and lower semicontinuous on V . Moreover, for vı ! v

weakly in L2.0;TI V/,

lim infı!0C

Z T

0

˚ı.vı.t// dt �Z T

0

˚.v.t// dt;

and

limı!0C

˚ı.w/ D ˚.w/


for all w 2 V . The functional ˚ı is Gâteaux differentiable in V , with

h˚ 0ı.v/ ; �i D

Z

�C

kjv� jı�1 v� � �� for all � 2 V:

Let us consider the following equation,

dvı.t/

dt; �

C �a.vı.t/;�/C b.vı.t/; vı.t/;�/C h˚ 0ı.vı.t//;�i

D ��a.�;�/ � b.�; vı.t/;�/ � b.vı.t/; �;�/ (10.31)

for any test function � 2 V , with the initial condition

vı.0/ D v0: (10.32)

For ı > 0, we establish a priori estimates of vı . Since h˚ 0ı.vı/; vıi � 0, vı 2 V , and

b.vı; vı; vı/ D b.�; vı; vı/ D 0, taking � D vı.t/ in (10.31) we get

1

2

d

dtjvı.t/j2 C �kvı.t/k2 � ��a.�; vı.t// � b.vı.t/; �; vı.t//:

In view of Lemma 10.1 we obtain

1

2

d

dtjvı.t/j2 C �

2kvı.t/k2 � �k�k2:

We estimate the right-hand side in terms of the data using Lemma 10.1 to get

1

2

d

dtjvı.t/j2 C �

2kvı.t/k2 � F; (10.33)

with F D F.�;˝;U0/. From (10.33) we conclude that

jvı.t/j2 C �

Z t

0

kvı.s/k2ds � jv.0/j2 C 2tF; (10.34)

whence

vı is bounded in L2.0;TI V/ \ L1.0;TI H/ independently of ı: (10.35)

The existence of vı satisfying (10.31)–(10.32) is based on inequality (10.33), theGalerkin approximations, and the compactness method. Moreover, from (10.34) wecan deduce that

dvıdt

is bounded in L2.0;TI V 0/: (10.36)


From (10.35) and (10.36) we conclude that there exists v such that (possibly for asubsequence)

vı ! v weakly in L2.0;TI V/;dvıdt

! dv

dtweakly in L2.0;TI V 0/: (10.37)

In view of (10.37), v 2 C.Œ0;T�I H/, and

vı ! v strongly in L2.0;TI H/:

We can now pass to the limit ı ! 0 in (10.31)–(10.32) as in [91] to obtain thevariational inequality (10.18) for almost every t 2 .0;T/. Thus the existence of asolution of Problem 10.1 is established. utTheorem 10.5. Under the hypotheses of Theorem 10.4, the solution v of Prob-lem 10.1 is unique and the map v.�/ ! v.t/, for t > � � 0, is Lipschitz continuousin H.

Proof. Let v and w be two solutions of Problem 10.1. Set � D w in the variationalinequality for v, � D v in the variational inequality for w, and add the twothus obtained inequalities. The terms with the boundary functionals reduce and foru.t/ D w.t/ � v.t/ we obtain

1

2

d

dtju.t/j2 C �ku.t/k2 � b.u.t/;w.t/; u.t//C b.u.t/; �; u.t//:

By Lemma 10.1 and the Ladyzhenskaya inequality (10.30) we obtain

d

dtju.t/j2 C �

2ku.t/k2 � 2

�C.˝/4kw.t/k2ju.t/j2; (10.38)

and then in view of the Poincaré inequality we conclude

d

dtju.t/j2 C �

2ju.t/j2 � 2

�C.˝/4kw.t/k2ju.t/j2;

where � > 0 is a constant. Using the Gronwall inequality, we obtain

ju.t/j2 � ju.�/j2 exp

�

�Z t

�

��

2� 2

�C.˝/4kw.s/k2

�

ds

�

: (10.39)

From (10.37) it follows that the solution w of Problem 10.1 belongs to L2.�; tI V/.By (10.39) the map v.�/ ! v.t/, t > � � 0, in H is Lipschitz continuous, with

jw.t/ � v.t/j � Cjw.�/ � v.�/j (10.40)

uniformly for t; � in a given interval Œ0;T� and initial conditions w.0/; v.0/ in a givenbounded set B in H.


In particular, as u.0/ D w.0/ � v.0/ D 0, the solution v of Problem 10.1 isunique. This ends the proof of Theorem 10.5. ut

10.2.3 Existence of Finite Dimensional Global Attractor

In this subsection we prove the following theorem:

Theorem 10.6. There exists a global attractor of a finite fractal dimension for thesemigroup fS.t/gt�0 associated with Problem 10.1.

Proof. From the considerations in the previous section it follows that to prove thetheorem it suffices to check assumptions .A1/–.A6/, .A8/. For the convenience ofthe reader we repeat their statements in appropriate places.

Assumption .A1/. For any v0 2 X and arbitrary T > 0 there exists (not necessarilyunique) v 2 Cw.Œ0;T�I X/ \ YT , a solution of the evolutionary problem on Œ0;T�with v.0/ D v0. Moreover, for any solution the estimates of kvkYT are uniform withrespect to kv.0/kX .In our case, set X D H, Y D V , and YT D fu 2 L2.0;TI V/ W u0 2 L2.0;TI V 0/g.From Theorems 10.4 and 10.5 we know that for any v0 2 H and arbitrary T > 0

there exists a unique v 2 C.Œ0;T�I H/ \ YT , solution of Problem 10.1. We shallobtain the needed estimates directly from the variational inequality, cf. (10.18),

hvt.t/;� � v.t/i C �a.v.t/;� � v.t// C b.v.t/; v.t/;� � v.t//C ˚.�/ � ˚.v.t// � hL .v.t//;� � v.t/i: (10.41)

Set � D 0 in (10.41) to get

1

2

d

dtjvj2 C �kvk2 C ˚.v/ � hL .v/; vi

as ˚.0/ D 0. Since, by Lemma 10.1,

hL .v/; vi � �

2kvk2 C �k�k2

we obtain

d

dtjvj2 C �kvk2 C 2˚.v/ � 2�k�k2 D F: (10.42)


Integration in t gives

jv.t/j2 C �

Z t

0

kv.s/k2ds � jv.0/j2 C 2tF; (10.43)

and we deduce that

v is bounded in L2.0;TI V/ \ L1.0;TI H/

uniformly with respect to jv.0/j.To get a uniform with respect to jv.0/j estimate of v0 in L2.0;TI V 0/ set� D v� , 2 V , in (10.41). We then have

hv0; i � hL .v/; i � �a.v; / � b.v; v; /C ˚.v � / � ˚.v/: (10.44)

Using the Poincaré and the trace inequalities we obtain k�.v/kL2.@˝/ � Ckvk, and

˚.v � / � ˚.v/ D kZ

�C

.jv� � � j � jv� j/ dS � kZ

�C

j � j dS � C.�C/k k:(10.45)

Moreover, using the Ladyzhenskaya inequality (10.30) to the nonlinear term, wehave

hL .v/; i � �a.v; / � b.v; v; / � C1.kvk C jvj kvk C 1/k k: (10.46)

From (10.44)–(10.46) we obtain

kv0kV0 � C2.kvk C jvj kvk C 1/;

and

kv0k2L2.0;TIV0/DZ T

0

kv0.t/k2V0

dt

� C2

�Z T

0

kv.t/k2 dtCkvk2L1.0;TIH/Z T

0

kv.t/k2 dtCT

�

�C.jv.0/j/:

Thus, .A1/ holds true.

Assumption .A2/. There exists a bounded set B0 X with the following properties:if v is an arbitrary solution with initial condition v0 2 X then

(i) there exists t0 D t0.kv0kX/ such that v.t/ 2 B0 for all t � t0,(ii) if v0 2 B0 then v.t/ 2 B0 for all t � 0.

From (10.42) and the Poincaré inequality,

d

dtjvj2 C ��1jvj2 � F;


and then by the Gronwall inequality (cf. Lemma 3.20)

jv.t/j2 � jv.0/j2e��1t C F

��1:

Thus, there exists a bounded absorbing set (e.g., the ball BH.0; �/ with �2 D 2 F��1

)

in H. Let B0 be defined as B0 D St�0 S.t/BH.0; �/

H. If v0 2 B0 then there exist

sequences tk and vk ! v0 in H such that vk 2 S.tk/BH.0; �/. From the continuity ofS.t/ it follows that S.t C tk/BH.0; �/ 3 S.t/vk ! S.t/v0 in H and hence S.t/v0 2 B0

for all t � 0. Thus, .A2/ holds true. Note that we need B0 to be closed in H to beable to satisfy assumption .A5/ below.

Assumption .A3/. Each l-trajectory has among all solutions a unique continuation.We recall that by the l-trajectory we mean any solution on the time interval Œ0; l�,and the unique continuation means that from an endpoint of an l-trajectory therestarts at most one solution. In our case .A3/ is satisfied as the solutions are unique(Theorem 10.5).

Assumption .A4/. For all t > 0, Lt W Xl ! Xl is continuous on B0l .

B0l is defined as the set of all l-trajectories starting at any point of B0 from .A2/,

and Xl D L2.0; lI X/. The semigroup fLtgt�0 acts on the set of l-trajectories as theshifts operators: .Lt�/.�/ D v.t C �/ for 0 � � � l, where v is the unique solutionon Œ0; l C �� such that vjŒ0;l� D �.In our case, X D H, hence Xl D Hl D L2.0; lI H/. In view of inequality (10.40) themap S.t/ W B0 ! B0 is Lipschitz continuous for every t > 0, whence for any two�1; �2 in B0

l ,

Z l

0

jLt�1.s/ � Lt�2.s/j2ds � C2.t/Z l

0

j�1.s/ � �2.s/j2ds: (10.47)

Thus, .A4/ holds true.

Assumption .A5/. For some � > 0, the closure in Xl of the set L� .B0l / is included

in B0l .

As L� .B0l / B0

l , it is enough to check that the set B0l is closed in Xl. We have to

prove that if f�kg is a sequence in B0l converging to some � in Xl then � is also a

trajectory and that �.0/ 2 B0.From assumption .A1/ it follows that the sequence f�kg is bounded in Yl and thus

contains a subsequence (relabeled f�kg) such that

�k ! � weakly in L2.0; lI V/ andd�k

dt! d�

dtweakly in L2.0; lI V 0/: (10.48)

Moreover, by the Aubin–Lions compactness theorem,

�k ! � strongly in L2.0; lI H/: (10.49)


We have

h�0k.t/;� � �k.t/i C �a.�k.t/;� � �k.t// C b.�k.t/; �k.t/;� � �k.t//

C ˚.�/ � ˚.�k.t// � hL .�k.t// ; � � �k.t/i:

We multiply both sides by a nonnegative smooth function � D �.t/ with support inthe interval .0; l/ and integrate with respect to t in this interval. We shall prove thattaking lim infk!1 of both sides and using (10.48) and (10.49) we obtain

Z l

0

h�0.t/;� � �.t/i�.t/ dt C �

Z l

0

a.�.t/;� � �.t//�.t/dt

CZ l

0

b.�.t/; �.t/;� � �.t//�.t/ dt CZ l

0

˚.�/�.t/ dt �Z l

0

˚.�.t//�.t/ dt

�Z l

0

hL .�.t//;� � �.t/i�.t/ dt: (10.50)

First we shall show (without using the convexity argument) that

limn!1

Z l

0

˚.�k.t//�.t/dt DZ l

0

˚.�.t//�.t/dt: (10.51)

Let � 2 �0; 1

2

�. From the fact that H1.˝/ embeds in H1��.˝/ compactly, the trace

operator �1�� W H1��.˝/ ! L2.@˝/ is linear and continuous, and from the Ehrlinglemma (cf. Lemma 3.17) for the spaces H1.˝/;H1��.˝/, and L2.˝/ it follows thatfor every " > 0 there exists C" > 0 such that for all v 2 H1.˝/,

k�.v/kL2.@˝/ � "jrvj C C"jvj:

We have thus

k�.�k/ � �.�/k2L2.�C/� "k�k � �k2 C C0

"j�k � �j2;

and

Z l

0

k�.�k/ � �.�/k2L2.�C/dt � "

Z l

0

k�k � �k2 dt C C0"

Z l

0

j�k � �j2 dt:

As, in view of (10.48), there exists M > 0 such that for all n,

Z l

0

k�k.t/ � �.t/k2 dt � M;


we obtain, using (10.49),

lim supk!1

Z l

0

k�.�k/ � �.�/k2L2.�C/dt � "M:

Now, as " is any positive number, we obtain

limk!1

Z l

0

k�.�k/ � �.�/k2L2.�C/dt D 0: (10.52)

From (10.52), (10.51) easily follows.We have also

limk!1

Z l

0

h�0k; �ki�.t/ dt D lim

k!1

Z l

0

1

2

d

dtj�kj2�.t/ dt

D � limk!1

Z l

0

1

2j�kj2�.t/0 dt D �

Z l

0

1

2j�j2�.t/0 dt

DZ l

0

h�0; �i�.t/ dt:

In view of (10.48) and (10.49), there are no problems to get the other terms in (10.50)and finally, (10.50) itself. As inequality (10.50) is equivalent to the inequality

h�.t/0; � � �i C �a.�.t/;� � �.t// C b.�.t/; �.t/;� � �.t//C ˚.�/ � ˚.�.t// � hL .�.t//;� � �.t/i;

satisfied for almost all t 2 .0; l/, � is a solution with �.0/ in H. To end the proof wehave to show that �.0/ belongs to the set B0. We have �k.t/ 2 B0 for all t 2 Œ0; l�and, by (10.49), for a subsequence, �k.t/ ! �.t/ for almost all t 2 .0; l/. As B0 isclosed, �.t/ 2 B0 for almost all t 2 .0; l/. Now, from the continuity of � W Œ0; l� ! Hand the closedness of B0 it follows that �.0/ is in B0. Thus, assumption .A5/ holds.

Assumption .A6/. There exists a space Wl such that Wl Xl with compactembedding, and � > 0 such that L� W Xl ! Wl is Lipschitz continuous on B1

l —theclosure in Xl of L� .B0

l /.Define

Wl D fu 2 L2.0; lI V/ W u0 2 L1.0; lI U0/g;

where U D f 2 V W D 0 on �Cg. We have, Wl Xl, with compact embeddingand we shall prove that Ll W Xl ! Wl is Lipschitz continuous on B1

l .Let w and v be two solutions of Problem 10.1 starting from B0 and let u D w�v.

Then, cf. (10.38),


d

dtju.t/j2 C �

2ku.t/k2 � 2

�C.˝/4kw.t/k2ju.t/j2:

Take s 2 .0; l/ and integrate this inequality over � 2 .s; 2l/ to get

ju.2l/j2 C �

2

Z 2l

sku.�/k2d� � 2

�C.˝/4

Z 2l

sku.�/k2ju.�/j2 d� C ju.s/j2:

(10.53)

From (10.40) we conclude that

ju.�/j2 � C1ju.s/j2

for � 2 .s; 2l/ and from (10.43) we have

Z 2l

sku.�/k2 d� �

Z 2l

0

.kv.�/k2 C kw.�/k2/ d� � 1

�.jv.0/j2 C jw.0/j2 C 8lF/;

whenceZ 2l

sku.�/k2 d� � C2

uniformly for w.s/; v.s/ 2 B0. Therefore, from (10.53) we obtain

Z 2l

lku.�/k2 d� � C3ju.s/j2:

Integrating over s 2 .0; l/ we obtain

Z 2l

lku.�/k2 d� � C3

l

Z l

0

ju.s/j2 ds;

and therefore

kLl�1 � Ll�2kL2.0;lIV/ �r

C3l

k�1 � �2kL2.0;lIH/ (10.54)

for any �1; �2 in B0l .

In order to prove that

k.Ll�1 � Ll�2/0kL1.0;lIU0/ � Ck�1 � �2kL2.0;lIH/

for some C > 0, it is sufficient, in view of (10.54), to prove that

k.�1 � �2/0kL1.0;lIU0/ � C0k�1 � �2kL2.0;lIV/ (10.55)

with some C0 > 0.


We have

hv0; � � vi C �a.v;� � v/C b.v; v;� � v/C ˚.�/ � ˚.v/ � hL .v/;� � vi;

and

hw0; � � wi C �a.w; � � w/C b.w;w; � � w/C ˚.�/ � ˚.w/ � hL .w/;� � wi:

Setting � D v � in the first inequality and � D w C in the second one, where 2 U, k k � 1, and adding the obtained two inequalities we get

hu0;� i � N .u;w; vI /; (10.56)

where

N .u;w; vI / D b.�; u; /C b.u; �; /C �a.u; /C b.w; u; /C b.u; v; /:

Estimating the right-hand side of (10.56) we get

hu0;� i � C00.1C kvk C kwk/kuk k k;

whence

ku0.t/kU0 D supfhu0.t/;� i W k k � 1g � C00.1C kv.t/k C kw.t/k/ku.t/k:

Finally, integration over t 2 .0; l/ gives

Z l

0

ku0.t/kU0 dt � C000�Z l

0

ku.t/k2 dt

�1=2

with

C000 D C0

�Z l

0

.1C kv.t/k2 C kw.t/k2/ dt

�1=2

with a constant C0 > 0;

uniformly for trajectories starting from B0. This proves (10.55) and ends the proofof the Lipschitz continuity of the map Ll W Xl ! Wl. Assumption .A6/ holds true.

Assumption .A8/. The map e W Xl ! X is Hölder-continuous on B1l .

Assumption .A8/ follows directly from the Lipschitz continuity of the mape W Xl ! X, e.�/ D �.l/. To check the latter, let w; v be two solutions as above,starting from B0. From (10.40) we have, in particular,

jw.l/ � v.l/j � Cjw.�/ � v.�/j


for � 2 .0; l/. Integrating this inequality in � on the interval .0; l/ we obtain

jw.l/ � v.l/j � Cplkw � vkL2.0;lIH/:

This ends the proof of Theorem 10.6. ut

10.2.4 Existence of an Exponential Attractor

Now we can prove the main theorem of Sect. 10.2.

Theorem 10.7. There exists an exponential attractor for the semigroup fS.t/gt�0associated with Problem 10.1.

Proof. In view of Theorem 10.3 and the considerations in the previous section itsuffices to check conditions .A9/ and .A10/. The first one follows immediately frominequality (10.47). Thus it remains to prove that

kLt1� � Lt2�kXl � cjt1 � t2jˇ (10.57)

holds for all � > 0, 0 � t2 � t1 � � , � 2 B1l , some c > 0 and some ˇ 2 .0; 1�,

where, in our case, Xl D Hl.To obtain (10.57) it suffices to know that �0, the time derivatives of � 2 B1

l , areuniformly bounded in Lq.0; lI H/ for some 1 < q � 1, cf. [168]. In fact, we havethen

jLt1�.s/ � Lt2�.s/j D ju.t1 C s/ � u.t2 C s/j

D��

Z t1Cs

t2Csu0.�/ d�

��

� jt1 � t2j1�1q ku0kLq.t2Cs;t1CsIH/:

and integration with respect to s in the interval Œ0; l� gives (10.57) with c dependingon � and l.

We shall prove that there exists M > 0 such that

k�0kL1.0;lIH/ � M for all � 2 B1l :

The formal a priori estimates that follow can be performed on the smooth in thetime variable Galerkin approximations vn

ı .t/, n D 1; 2; 3; : : :, of the regularizedproblem (10.31)–(10.32), as in [148]. The obtained estimates are preserved bysolutions vı.t/ of problem (10.31)–(10.32), with bounds independent of ı whenı ! 0, and in the end, by solutions v.t/ of Problem 10.1.


Let us consider the solution v D v.t/ of problem (10.31)–(10.32) (we drop thesubscript ı for short),

dv.t/

dt; �

C �a.v.t/;�/C b.v.t/; v.t/;�/C h˚ 0ı.v.t//;�i

D ��a.�;�/ � b.�; v.t/;�/ � b.v.t/; �;�/; (10.58)

with the initial condition

v.0/ D v0:

Our aim is to derive, following the method used in [148], two a priori estimatesfollowing from (10.58). To get the first one, set � D vt (vt D v0) in (10.58). Weobtain

jvtj2 C d

dt

�

2kvk2 C d

dt˚ı.v/ D hL .v/; vti � b.v; v; vt/: (10.59)

Using the Ladyzhenskaya inequality (10.30) we have

hL .v/; vti � c1.kvtk C kvkjvtj1=2kvtk1=2 C jvj1=2kvk1=2jvtj1=2kvtk1=2/: (10.60)

Now, we differentiate (10.58) with respect to the time variable and set � D vt, toget

1

2

d

dtjvtj2 C �kvtk2 � �b.vt; �; vt/ � b.vt; v; vt/; (10.61)

as b.�; vt; vt/ D 0, b.v; vt; vt/ D 0, and, cf. [91, 188],

d

dt˚ 0ı.v/; vt

� 0:

Using Lemma 10.1 and the Ladyzhenskaya inequality (10.30) to estimate the right-hand side of (10.61) we obtain

d

dtjvtj2 C �kvtk2 � c2kvk2jvtj2: (10.62)

Now, we multiply (10.62) by t2 to get

d

dt.t2jvtj2/C t2�kvtk2 � c2kvk2.t2jvtj2/C 2tjvtj2: (10.63)

10.3 Planar Shear Flows with Generalized Tresca Type Friction Law 231

To get rid of the last term on the right-hand side we add to (10.63) Eq. (10.59)multiplied by 2t. After simple calculations and using (10.60) we obtain

d

dt.t2jvtj2 C t�kvk2 C 2t˚ı.v//C t2�kvtk2

� 2tc1.kvtk C kvkjvtj1=2kvtk1=2 C jvj1=2kvk1=2jvtj1=2kvtk1=2/C c2kvk2.t2jvtj2/C �kvk2 C 2˚ı.v/

C c3jvj1=2kvk3=2.tjvtj/1=2.tkvtk/1=2: (10.64)

Define y D t2jvtj2C t�kvk2C2t˚ı.v/. Using the Young inequality to the right-handside of (10.64) and observing that ˚ı.v/ � C.kvk2 C 1/ for 0 < ı � 1, we obtainthe inequality of the form

d

dty.t/C t2�

2kvtk2 � C1.t/y.t/C C2.t/;

where the coefficients Ci.�/, i D 1; 2, are locally integrable and do not depend on ı.They are also independent of the initial conditions for v in a given bounded sets in H.This proves that the time derivative of solutions of the regularized problems isuniformly bounded with respect to ı in L1.�;TI H/ \ L2.�;TI V/ for all intervalsŒ�;T�, 0 < � < T . As a consequence, this property holds for all � 2 B1

l . In viewof the above considerations this ends the proof of the existence of an exponentialattractor. ut

10.3 Planar Shear Flows with Generalized TrescaType Friction Law

The problem considered in this section is based on the results of Sect. 10.2, cf. also[162]. As we show, the arguments used there can be generalized to a class ofproblems with the friction coefficient dependent on the slip rate.

10.3.1 Classical Formulation of the Problem

We will consider the planar flow of an incompressible viscous fluid governed by theequation of momentum

vt � ��v C .v � r/v C rp D f in ˝ � RC; (10.65)



div v D 0 in ˝ � RC (10.66)

in domain ˝ given by

˝ D fx D .x1; x2/ W 0 < x1 < L; 0 < x2 < h.x1/g;

with boundary @˝ D �D[�C[�L, where �D is the top part of the boundary given by�D D f.x1; h.x1// W x1 2 .0;L/g, �C is the bottom part of the boundary given by�C D .0;L/ � f0g, and �L is the lateral one given by �L D f0;Lg � .0; h.0//.The function h is smooth and L-periodic such that h.x1/ � " > 0 for all x1 2 R

with a constant " > 0. We will use the notation e1 D .1; 0/ and e2 D .0; 1/ forthe canonical basis of R2. Note that on �C the outer normal unit vector is given byn D �e2.

The unknowns are the velocity field v W ˝ � RC ! R

2 and the pressure fieldp W ˝ � R

C ! R, � > 0 is the viscosity coefficient and f W ˝ ! R2 is the mass

force density. The stress tensor T is related to the velocity and pressure through thelinear constitutive law Tij D �pıij C �.vi;j C vj;i/.

We are looking for the solutions of (10.65)–(10.66) which are L-periodic in thesense that v.0; x2; t/ D v.L; x2; t/,

@v2.0;x2;t/@x1

D @v2.L;x2;t/@x1

, and p.0; x2; t/ D p.L; x2; t/

for x2 2 Œ0; h.0/� and t 2 RC. The first condition represents the L-periodicity of

velocities, while the latter two ones, the L-periodicity of the Cauchy stress vectorTn on �L in the space of divergence free functions. Moreover we assume that

v D 0 on �D � RC: (10.67)

On the contact boundary �C we decompose the velocity into the normal componentvn D v � n and the tangential one v� D v � e1. Note that since the domain ˝ istwo-dimensional it is possible to consider the tangential components as scalars, forthe sake of the ease of notation. Likewise, we decompose the stress on the boundary�C into its normal component Tn D Tn � n and the tangential one T� D Tn � e1.

We assume that there is no flux across �C and hence

vn D 0 on �C � RC: (10.68)

The boundary �C is assumed to be moving with the constant velocity U0e1 D.U0; 0/ which, together with the mass force, produces the driving force of the flow.The friction coefficient k is assumed to be related to the slip rate through the relationk D k.jv� � U0j/, where k W RC ! R

C. If there is no slip between the fluid and theboundary then the friction force is bounded by the friction threshold k.0/

v� D U0 ) jT� j � k.0/ on �C � RC; (10.69)


while if there is a slip, then the friction force is given by the expression

v� ¤ U0 ) �T� D k.jv� � U0j/ v� � U0

jv� � U0j on �C � RC: (10.70)

Note that (10.69)–(10.70) generalize the Tresca law considered in Sect. 10.2, wherek was assumed to be a constant. Here k depends on the slip rate, this dependencerepresents the fact that the kinetic friction is less than the static one, which holdsif k is a decreasing function. Similar friction law is used, for example, in the studyof the motion of tectonic plates, see [119, 190, 207] and the references therein. Wemake the following assumptions on the friction coefficient k:

.k1/ k 2 C.RCIRC/,

.k2/ k.s/ � ˛.1C s/ for all s 2 RC with ˛ > 0,

.k3/ k.s/ � k.r/ � �.s � r/ for all s > r � 0 with > 0.

Note that the assumption that k has values in RC has a clear physical interpretation,

namely it means that the friction force is dissipative.Finally, the initial condition for the velocity field is

v.x; 0/ D v0.x/ for x 2 ˝: (10.71)

In the next section we present a weak formulation of the problem in the form ofan evolutionary differential inclusion with a suitable potential corresponding to thegeneralized Tresca condition.

10.3.2 Weak Formulation of the Problem

To be able to define a weak solution of problem (10.65)–(10.71) we introduce somenotation. Let

QV D fv 2 C1.˝/2 W div v D 0 in ˝; v is L-periodic in x1;

v D 0 on �D; vn D 0 on �Cg;

and


We define scalar products in H and V , respectively, by

.u ; v/ DZ

˝

u.x/v.x/ dx and .ru ; rv/ DZ

˝

ru.x/W rv.x/ dx;



jvj D .v; v/12 and kvk D .rv ; rv/ 12 :

We denote the trace operator from V to L2.�C/2 by � and its norm by

k�k D k�kL .VIL2.�C/2/. Moreover, let, for u; v, and w in V ,

a.u ; v/ D .ru ; rv/ and b.u ; v ; w/ D ..u � r/v ; w/:

Finally, we define the functional j W R ! R corresponding to the generalized Trescacondition (10.69)–(10.70) by

j.v/ DZ jv�U0j

0

k.s/ ds: (10.72)

Note that the functional j is not necessarily convex, but it is locally Lipschitz. By @jwe will denote its Clarke subdifferential (see Sect. 3.7). The variational formulationof problem (10.65)–(10.71) can be derived by the calculation analogous to the proofof Proposition 10.1.

Problem 10.2. Given v0 2 H and f 2 H, find v W RC ! H such that:

.i/ v 2 C.RCI H/ \ L2loc.RCI V/, with vt 2 L2loc.R

CI V 0/,.ii/ for all � in V and for almost all t 2 R

C, the following equality holds

hvt.t/;�i C �a.v.t/;�/C b.v.t/; v.t/;�/C .�.t/;��/L2.�C/ D .f ; �/;(10.73)

with �.t/ 2 [email protected]� .t//for a.e. t 2 R

C, where for w 2 L2.�C/ we denote by [email protected]/

the set of all functions � 2 L2.�C/ such that �.x/ 2 @j.w.x// for a.e. x 2 �C,.iii/ the following initial condition holds:

v.0/ D v0: (10.74)

The above definition is justified by the assertion .i/ of the following lemma.

Lemma 10.3. Under assumptions .k1/–.k3/ the functional j defined by (10.72)satisfies the following properties:

.i/ j is locally Lipschitz and conditions (10.69)–(10.70) are equivalent to

� T� 2 @j.v� /; (10.75)

.ii/ j�j � Q .1C jwj/ for all w 2 R and � 2 @j.w/, Q > 0,


.iii/ m.@j/ � �, where m.@j/ is the one sided Lipschitz constant defined by

m.@j/ D infu;v2R;u¤v;

�[email protected]/;�[email protected]/

.� � �/ � .u � v/ju � vj2 ;

.iv/ for all w 2 R and � 2 @j.w/, � � w � �ˇ.1C jwj/, with a constant ˇ > 0.

Proof. .i/ The fact that j is locally Lipschitz follows from the continuity of k andthe direct calculation. To see that (10.69)–(10.70) is equivalent to (10.75) observethat for v ¤ U0 the functional j is continuously differentiable on a neighborhoodof v and its derivative is given as j0.v/ D k.jv � U0j/ v�U0jv�U0j , whence (10.70) isequivalent to (10.75) by the characterization of the Clarke subdifferential givenin Theorem 3.19. The same characterization implies that (10.69) is equivalentto (10.75) if v D U0.

Assertion .ii/ follows in a straightforward way from .k2/.Assertion .iii/ can be obtained by a following computation. Let � 2 @j.u/,

� 2 @j.v/, where u ¤ U0 and v ¤ U0. Then

.� � �/ � .u � v/ D�

k.ju � U0j/ u � U0

ju � U0j � k.jv � U0j/ v � U0

jv � U0j�

� .u � v/

D k.ju � U0j/ju � U0j � k.ju � U0j/ .u � U0/ � .v � U0/

ju � U0j

�k.jv � U0j/ .u � U0/ � .v � U0/

jv � U0j C k.jv � U0j/jv � U0j

� .k.ju � U0j/ � k.jv � U0j// .ju � U0j � jv � U0j/� �.ju � U0j � jv � U0j/2 � �ju � vj2:

The calculation in the case, when either u D U0 or v D U0 is similar.Proof of assertion .iv/ is also straightforward. Indeed, for w D 0 the assertion

obviously holds. Let w ¤ 0 and � 2 @j.w/. We have, by .k1/–.k2/,

� � w D k.jw � U0j/ w � U0

jw � U0j .w � U0 C U0/

� k.jw � U0j/jw � U0j � k.jw � U0j/jU0j � �˛.1C jwj C jU0j/jU0j;

and .iv/ follows. The proof is complete. utRemark 10.1. Observe that assertion .iii/ of Lemma 10.3 is equivalent to the claim

that the functional j is semiconvex, i.e., the functional s ! j.s/ C s2

2is in fact

convex (see Definition 10 in [18]).


10.3.3 Existence and Properties of Solutions

We begin with some estimates that are satisfied by the solutions of Problem 10.2.We define the auxiliary operators A W V ! V 0 and B W V ! V 0 by hAu; vi D a.u; v/and hBu; vi D b.u; u; v/.

Lemma 10.4. Let v be a solution of Problem 10.2. Then, for all t � 0,

maxs2Œ0;t� jv.s/j

2 CZ t

0

kv.s/k2 ds � C.t; jv0j/; (10.76)

Z t

0

kv0.s/k2V0

ds � C.t; jv0j/; (10.77)

where C.t; jv0j/ is a nondecreasing function of t and jv0j.Proof. Take � D v.t/ in (10.73). Since, for v 2 V it is b.v; v; v/ D 0, we get, withan arbitrary " > 0,

1

2

d

dtjv.t/j2 C �kv.t/k2 � jf jjv.t/j C k�.t/kL2.�C/kv� .t/kL2.�C/

� "jv.t/j2 C jf j24"

C 3

2Q kv.t/k2L2.�C/

C Qm1.�C/;

where we used (ii) of Lemma 10.3 and m1.�C/ D L is the one-dimensionalboundary measure of �C.

Denoting Z D V \ H1�ı.˝/2 with some small ı 2 �0; 1

2

�endowed with the

norm H1�ı.˝/2, observe that the embedding V Z is compact and the trace map�C W Z ! L2.�C/

2 is continuous. We denote k�Ck D k�CkL .ZIL2.�C/2/. Fromthe Ehrling lemma (cf. Lemma 3.17) we get, for an arbitrary " > 0 independent onv 2 V and C."/ > 0 that

kvk2Z � "kvk2 C C."/jvj2:

Hence, noting that the Poincaré inequality

�1jvj2 � kvk2

is valid for v 2 V , we get

d

dtjv.t/j2 C 2�kv.t/k2 � 2"

�1kv.t/k2 C jf j

2"

C 3 Q k�Ck2"kv.t/k2 C 3 Q k�Ck2C."/jv.t/j2 C 2 QL:


It is clear that we can choose " small enough such that

d

dtjv.t/k2 C �kv.t/k2 � C1 C C2jv.t/j2;

with C1;C2 > 0. Using the Gronwall inequality (cf. Lemma 3.20) we get (10.76).For the proof of (10.77) observe that for u 2 V , kBukV0 � Cjujkuk. Indeed, by

the Hölder inequality and the Ladyzhenskaya inequality,

jhBu; zij D jb.u; u; z/j D jb.u; z; u/j� kuk2L4.˝/2 jrzj � Ckukjujkzk for u; z 2 V: (10.78)

Hence (10.77) follows by (10.76) and the straightforward computation that uses theassertion .ii/ of Lemma 10.3 to estimate the multivalued term. utWe formulate and prove the theorem on existence of solutions to Problem 10.2.

Theorem 10.8. For any u0 2 H Problem 10.2 has at least one solution.

Proof. Since the proof of solution existence, based on the Galerkin method, isstandard and quite long, we provide only its main steps.

Let % 2 C10 ..�1; 1// be a mollifier such that

R 1�1 %.s/ ds D 1 and %.s/ � 0.

We define %k W R ! R by %k.x/ D k%.kx/ for k 2 N and x 2 R. Then supp %k �� 1k ;

1k

�. We consider jk W R ! R defined by the convolution

jk.r/ DZ

R

%k.s/j.r � s/ ds for r 2 R: (10.79)

Note that jk 2 C1.R/. From the assertion .B/ of Theorem 3.21 it follows that forall n 2 N and r 2 R we have jj0k.r/j � ˛.1C jrj/, where ˛ is different from Q in .ii/of Lemma 10.3 above but independent of k.

Let us furthermore take the sequence Vk of finite dimensional spaces such thatVk D spanfz1; : : : ; zkg, where fzig are orthonormal in H eigenfunctions of the Stokesoperator with the Dirichlet and periodic boundary conditions given in the definitionof the space V . Then fVkg1

kD1 approximate V from inside, i.e.,S1

kD1 Vk D V .We denote by Pk W V 0 ! Vk the orthogonal projection on Vk defined as Pku DPk

iD1hu; ziizi. Using Theorem 3.17 we conclude that Pkv ! v strongly in V 0 forany v 2 V 0.

We formulate the regularized Galerkin problems for k 2 N.Find vk 2 C.RCI Vk/ such that for a.e. t 2 R

C, vk is differentiable and

hv0k.t/;�i C �a.vk.t/;�/C b.vk.t/; vk.t/;�/

C .j0k.vn� .t/.�//;��/L2.�C/ D .f ; �/; (10.80)

vk.0/ D Pkv0 (10.81)

for a.e. t 2 RC and all � 2 Vk.


Note that due to the fact that j0k; a; b are smooth, the solution of (10.80)–(10.81), if itexists, is a continuously differentiable function of time variable, with values in Vk.

We first show that if vk solves (10.80) then estimates analogous to the ones inLemma 10.4 hold. Taking vk.t/ in place of � in (10.80) and using the argumentsimilar to that in the proof of (10.76) in Lemma 10.4 we obtain that, for a givenT > 0,

kvkkL1.0;TIH/ � const; (10.82)

kvkkL2.0;TIV/ � const: (10.83)

Now, denoting by � W V ! L2.�C/ the map �v D .�v/� and by �� the adjoint to �,we observe that (10.80) is equivalent to the following equation in V 0,

v0k.t/C �Avk.t/C PkB.vk.t//C Pk�

�j0k.vk� .t/.�// D Pkf : (10.84)

Since, by Theorem 3.17, for w 2 V 0 we have kPkwkV0 � kwkV0 , then, using theestimate kBwkV0 � Cjwjkwk valid for w 2 V , the growth condition on j0k as wellas (10.82) and (10.83), from Eq. (10.84) we get

kv0kkL2.0;TIV0/ � const: (10.85)

The existence of the solution to the Galerkin problem (10.80) follows by theCarathéodory theorem and estimates (10.82)–(10.83).

Since the bounds (10.82), (10.83), and (10.85) are independent of k then, for asubsequence, we get

vk ! v weakly � � in L1.0;TI H/;

vk ! v weakly in L2.0;TI V/;

v0k ! v0 weakly in L2.0;TI V 0/;

vk� ! v� strongly in L2.0;TI L2.�C//; (10.86)

vk ! v strongly in L2.0;TI H/; (10.87)

where (10.86) is a consequence of the Aubin–Lions compactness theorem usedfor the spaces V Z V 0 and (10.87) is a consequence of the Aubin–Lionscompactness theorem used for the spaces V H V 0.

From (10.83) and the growth condition on j0k it follows that

kj0k.vn� /kL2.0;TIL2.�C// � const:

Hence, perhaps for another subsequence,

j0k.vk� / ! � weakly in L2.0;TI L2.�C//:


In a standard way (see, for example, Sects. 7.4.3 and 9.4 in [197]) we can pass tothe limit in (10.84) and obtain that v and � satisfy (10.73) for a.e. t 2 .0;T/. Theassertion .C/ of Theorem 3.21 implies that �.t/ 2 [email protected]� .t//

for a.e. t 2 .0;T/.We shall show that v.0/ D v0 in H. We have, for t 2 .0;T/,

vk.t/ D Pkv0 CZ t

0

v0k.s/ ds; v.t/ D v.0/C

Z t

0

v0.s/ ds;

where the equalities hold in V 0. Take � 2 V . Then

Z T

0

hvk.t/ � v.t/; �i dt

DZ T

0

hPkv0 � v.0/; �i dt CZ T

0

Z t

0

.v0k.s/ � v0.s// ds; �

dt

D ThPkv0 � v.0/; �i CZ T

0

hv0k.s/ � v0.s/; .T � s/�i ds:

Since vk ! v weakly in L2.0;TI V/ and v0k ! v0 weakly in L2.0;TI V 0/, it follows

that Pkv0 ! v.0/ weakly in V 0. We know that Pkv0 ! v0 strongly in V 0 and hencev.0/ D v0.

Finally, the solution can be extended from Œ0;T� to the whole RC. Indeed, we cantake the value of the solution at the time point T as an initial condition and solve theresulting problem, obtaining another solution on Œ0;T�. Then, we concatenate thetwo solutions, obtaining the solution defined on the interval Œ0; 2T�. Repeating thisprocedure we get the solution on the whole R

C (see, for example, [93], Theorem 2in Sect. 9.2.1). utRemark 10.2. In the proof of Theorem 10.8 we used only assertions .i/ and .ii/ ofLemma 10.3 and therefore the theorem holds for a class of potentials j that satisfythese two conditions and are not necessarily given by (10.72). This is a new resultof independent interest since it weakens the conditions required for the existence ofsolutions provided, for example, in [174]. Indeed, the sign condition (see H(j)(iv)p. 583 in [174]) is not needed here for the existence proof.

Next lemma shows that assertion .iii/ of Lemma 10.3 gives the Lipschitz continuityof the solution map on bounded sets as well as the solution uniqueness.

Lemma 10.5. The solution of Problem 10.2 is unique and if v, w are two solutionsof Problem 10.2 with the initial conditions v0;w0, respectively, then for anyt > s � 0 we have

jv.t/ � w.t/j � D.t; jw0j/jv.s/ � w.s/j; (10.88)

where D.t; jw0j/ > 0 is a nondecreasing function of t; jw0j.


Proof. Let v and w be two solutions of Problem 10.2 with the initial conditions v0and w0. We denote u.t/ D v.t/� w.t/. Subtracting (10.73) for v and w, respectively,and taking u.t/ as a test function we get, for a.e. t 2 R

C,

1

2

d

dtju.t/j2 C �ku.t/k2 C b.u.t/;w.t/; u.t//C .�.t/� �.t/; v.t/� � w.t/� /L2.�C/ D 0;

where �.t/ 2 [email protected]� .t//and �.t/ 2 [email protected]� .t//

.Using estimate (10.78) to estimate the convective term and also assertion .iii/ in

Lemma 10.3 to estimate the multivalued one we get, for a.e. t 2 RC,

1

2

d

dtju.t/j2 C �ku.t/k2 � ku� .t/k2L2.�C/

� Cju.t/jku.t/kkw.t/k: (10.89)

As in the proof of Lemma 10.4 we use the fact that the embedding V Z is compactand the trace �C W Z ! L2.�C/

2 is continuous. We get, for a.e. t 2 RC,

1

2

d

dtju.t/j2 C �ku.t/k2 � "ku.t/k2 C C."/ju.t/j2kw.t/k2

C k�Ck2."ku.t/k2 C C."/ju.t/j2/ (10.90)

with an arbitrary " > 0 and the constant C."/ > 0 independent on u, w, and t. Bychoosing " > 0 small enough we get

d

dtju.t/j2 � ju.t/j2.C1kw.t/k2 C C2/ for a.e. t 2 R

C;

with the constants C1;C2 > 0. The Gronwall inequality (cf. Lemma 3.20) gives

ju.t/j2 � ju.s/j2eC2.t�s/CC1R t

s kw.�/k2 d� � ju.s/j2eC2tCC1R t0 kw.�/k2 d� :

Using (10.76) we get

jv.t/ � w.t/j � jv.s/ � w.s/je 12C2tC 1

2C1C.t;jw0j/; (10.91)

and the proof of (10.88) is complete. Taking s D 0 and v.0/ D w.0/ in (10.88) weobtain the uniqueness. utNow we shall prove that assertion .iv/ of Lemma 10.3 gives additional, dissipative,a priori estimates.

Lemma 10.6. If v is a solution of Problem 10.2 with the initial condition v0, then,

jv.t/j � jv0je�C1t C C2 for all t � 0;

with positive constants C1;C2 independent of t; v0.


Proof. We proceed as in the proof of (10.76) in Lemma 10.3. Taking the testfunction � D v.t/ in (10.73) we get

1

2

d

dtjv.t/j2 C �kv.t/k2 C .�.t/; v� .t//L2.�C/ � jf jjv.t/j

for a.e. t 2 RC. We use assertion .iv/ of Lemma 10.3 and the Poincaré inequality

�1jvj2 � kvk2 to get

1

2

d

dtjv.t/j2 C �kv.t/k2 � ˇL

�

1C 1

4"1

�

� ˇ"1kv� .t/k2L2.�C/

� jf j24"2

C "2

�1kv.t/k2

for a.e. t 2 RC, with an arbitrary "1 > 0 and "2 > 0. Using the trace inequality and

the Poincaré inequality again, we can choose "1 and "2 small enough to get

d

dtjv.t/j2 C ��1jv.t/j2 � C

for a.e. t 2 RC, where C > 0 is a constant. Finally, the assertion follows from the

Gronwall inequality (cf. Lemma 3.20). ut

10.3.4 Existence of Finite Dimensional Global Attractor

In this subsection we prove the following theorem.

Theorem 10.9. The semigroup fS.t/gt�0 associated with Problem 10.2 has a globalattractor A H of finite fractal dimension in the space H.

Proof. In view of Theorem 10.2 we need to verify assumptions .A1/–.A8/. We setY D V , X D H, Z D V 0, YT D fu 2 L2.0;TI V/ W u0 2 L2.0;TI V 0/g for T > 0, andl > 0, Xl D L2.0; lI H/.

Assumption .A1/. From Theorem 10.8 it follows that for any u0 2 H thereexists at least one solution of Problem 10.2. This solution restricted to the interval.0;T/ belongs to YT and C.Œ0;T�I H/. Uniform bounds in YT follow directly fromLemma 10.4.

Assumption .A2/. In view of Lemma 10.6 the closed ball BH.0;C2 C 1/ absorbs

in finite time all bounded sets in H. Let B0 D St�0 S.t/BH.0;C2 C 1/

H. Obviously

this set is closed in H (we will need this fact to verify the assumption .A5/ below).From Lemma 10.6 it follows that B0 is bounded. Let us assume that u 2 B0 andv D S.t/u. Then, for a sequence uk 2 S.tk/BH.0;C2 C 1/ with certain tk we have


uk ! u in H. From Lemma 10.5 it follows that S.t/uk ! S.t/u D v in H. ButS.t/uk 2 S.t C tk/BH.0;C2 C 1/ B0. Hence, from the closedness of B0 it followsthat v 2 B0 and the assertion is proved.

In the sequel of the proof the generic constant depending on B0; l and the problemdata will be denoted by Cl;B0 .

Assumption .A3/. Recall that the l-trajectory is a restriction of any solution ofProblem 10.2 to the time interval Œ0; l�. The fact that assumption .A3/ holds followsimmediately from Lemma 10.5.

Assumption .A4/. Define the shift operator Lt W Xl ! Xl by .Lt�/.s/ D u.s C t/ fors 2 Œ0; l�, where u is the unique solution on Œ0; l C t� such that ujŒ0;l� D �. We willprove that this operator is continuous on B0

l for all t � 0 and l > 0, where B0l is the

set of all l-trajectories with the initial condition in the set B0 from assumption .A2/.Let u; v be two solutions with the initial conditions u0; v0 2 B0. By Lemma 10.5 wehave

Z l

0

jv.s C t/ � u.s C t/j2 ds � D2.t C l; jB0j/Z l

0

jv.s/ � u.s/j2 ds;

where jB0j D supv2B0 jvj. Hence, denoting �1 D ujŒ0;l� and �2 D vjŒ0;l�, we get

kLt�1 � Lt�2k2Xl� D2.t C l; jB0j/k�1 � �2k2Xl

; (10.92)

and the proof of .A4/ is complete.

Assumption .A5/. We need to prove that, for some � > 0, L� .B0l /

Xl B0l .

Let u0 2 B0. From the fact that for all � > 0, S.�/u0 2 B0, it follows thatL� .B0

l / B0l . To obtain the assertion we must show that B0

l is closed in Xl. To thisend assume that uk is a sequence of solutions to Problem 10.2 such that uk.0/ 2 B0

and that

�k ! � strongly in L2.0; lI H/; (10.93)

where �k D ukjŒ0;l�. From .A1/ it follows that

�k ! � weakly in L2.0; lI V/; (10.94)

�0k ! �0 weakly in L2.0; lI V 0/: (10.95)

We shall show that � 2 B0l . Let us first prove that �.0/ 2 B0. Indeed, from (10.93)

it follows that, for a subsequence, �k.t/ ! �.t/ strongly in H for a.e. t 2 .0; l/.Since �k.t/ 2 B0 for all k 2 N and all t 2 Œ0; l�, then �.t/ belongs to the closed setB0 for a.e. t 2 .0; l/. The closedness of B0 and the fact that � 2 C.Œ0; l�I H/ implythat �.t/ 2 B0 for all t 2 Œ0; l� and, in particular, �.0/ 2 B0. We need to prove that �satisfies (10.73) for a.e. t 2 .0; l/. It is enough to show that for all w 2 L2.0; lI V 0/,


Z l

0

h�0.t/;w.t/i dt C �

Z l

0

hA�.t/;w.t/i dt

CZ l

0

hB�.t/;w.t/i dt CZ l

0

.�.t/;w� .t//L2.�C/ dt DZ l

0

.f ;w.t// dt (10.96)

with �.t/ 2 S2@j.�� .t//for a.e. t 2 .0; l/. From the fact that �k satisfies (10.73) for a.e.

t 2 .0; l/ we have

Z l

0

h�0k.t/;w.t/i dt C �

Z l

0

hA�k.t/;w.t/i dt

CZ l

0

hB�k.t/;w.t/i dt CZ l

0

.�k.t/;w� .t//L2.�C/ dt DZ l

0

.f ;w.t// dt;

(10.97)

with �k.t/ 2 S2@j.�k� .t//for a.e. t 2 .0; l/. Then, from the growth condition .ii/ in

Lemma 10.3 it follows that k�kkL2.0;lIL2.�C// � p2lL Q C p

2 Q k�k�kL2.0;lIL2.�C//.Since �k is bounded in L2.0; lI V/, then �k� is bounded in L2.0; lI L2.�C// and hence�k is bounded in the same space. Therefore, for a subsequence,

�k ! � weakly in L2.0; lI L2.�C//: (10.98)

Now (10.93), (10.94), and (10.98) are sufficient to pass to the limit in all the termsin (10.97) and thus to prove (10.96). It remains to show that �.t/ 2 S2@j.�n� .t//

for a.e.

t 2 .0; l/. Let Z D V\H1�ı.˝/2 be equipped with H1�ı topology, where ı 2 �0; 12

�.

Then for the triple V Z V 0 we can use the Aubin–Lions compactness theoremand conclude from (10.93) and (10.94) that �k ! � strongly in L2.0; lI Z/. Now,since the trace operator �C W Z ! L2.�C/

2 is linear and continuous, and so is thecorresponding Nemytskii operator,

�k� ! �� strongly in L2.0; lI L2.�C//: (10.99)

We use assertion .H2/ in Theorem 3.24 to deduce that �.x; t/ 2 @j.�� .x; t// a.e. in�C � .0; l/. This completes the proof of .A5/.

Assumption .A6/. We define W D H10.˝/

2 \ V , where we equip W with the normof V . Then V 0 W 0. By the Aubin–Lions compactness theorem the space

Wl D f� 2 L2.0; lI V/ W �0 2 L1.0;TI W 0/g

is embedded compactly in Xl. We must show that the shift operator L� W Xl ! Wl

is Lipschitz continuous on B1l D L� .B0

l /Xl

for some � > 0. In fact, we shall provethat L� is Lipschitz continuous on the larger set B0

l for � D l. Indeed, let w; v be twosolutions of Problem 10.2 starting from B0. Denote u D v � w. Then, by (10.90),


d

dtju.t/j2 C �ku.t/k2 � Cju.t/j2.1C kw.t/k2/ for a.e. t 2 R

C;

where the constant C depends only on the problem data. We fix s 2 .0; l/ andintegrate the last inequality over interval .s; 2l/ to obtain

ju.2l/j2 C �

Z 2l

sku.t/k2 dt � C

Z 2l

sju.t/j2.1C kw.t/k2/ dt C ju.s/j2:

Using (10.91) we get

�

Z 2l

lku.t/k2 � ju.s/j2

�

2Cl;B0 l C Cl;B0

Z 2l

0

kw.t/k2 dt C 1

�

:

Since, by .A1/,

Z 2l

0

kw.t/k2 dt � Cl;B0 ;

it follows that

Z 2l

lku.t/k2 dt � Cl;B0 ju.s/j2:

Integrating this inequality over interval .0; l/ with respect to the variable s it followsthat

lZ 2l

lku.t/k2 dt � Cl;B0

Z l

0

ju.s/j2 ds;

and therefore

kLlv � LlwkL2.0;lIV/ �r

Cl;B0

lkv � wkXl : (10.100)

It remains to prove that

k.Llv � Llw/0kL2.0;lIW0/ � Cl;B0kv � wkXl :

In view of (10.100) it is sufficient to prove that

k.v � w/0kL2.0;lIW0/ � Cl;B0kv � wkL2.0;lIV/:


To this end, we subtract (10.73) for w and v, respectively, and take � 2 W as a testfunction. We get

hu0.t/;�i C �hAu.t/;�i C hBv.t/ � Bw.t/;�i D 0:

We have, for a constant k > 0 dependent only on the problem data,

hBv.t/ � Bw.t/;�i D b.u.t/;w.t/;�/C b.w.t/; u.t/;�/C b.u.t/; u.t/;�/

� k.ku.t/kkw.t/k C ku.t/k2/k�k� k.2kw.t/k C kv.t/k/ku.t/kk�k;

whence, for a constant C > 0 dependent only on the problem data,

ku0.t/kW0 D sup�2W;k�kD1

j�hAu.t/;�i C hBv.t/ � Bw.t/;�ij

� �kAu.t/kV0 C kBv.t/ � Bw.t/kV0

� C.1C kw.t/k C kv.t/k/ku.t/k:

It follows that

ku0kL1.0;lIW0/ � CkukL2.0;lIV/

sZ l

0

.1C kw.t/k C kv.t/k/2 dt:

The assertion follows from .A1/.

Assumptions .A7/ and .A8/. Since .A8/ implies .A7/ we prove only .A8/. We shallshow that the mapping e W Xl ! X defined as e.�/ D �.l/ is Lipschitz on B0

l .Indeed, by (10.91), for any two solutions v;w of Problem 10.2 such that their initialconditions belong to B0, we get

jw.l/ � v.l/j � Cl;B0 jw.s/ � v.s/j for all s 2 .0; l/:

Integrating the square of this inequality over s 2 .0; l/ we obtain

ljw.l/ � v.l/j2 � C2l;B0kw � vk2Xl

;

whence

jw.l/ � v.l/j � Cl;B0pl

kw � vkXl ;

and .A8/ holds.


This completes the proof Theorem 10.9 about the existence of the global attractorof a finite fractal dimension. utIn the following subsection we shall prove the existence of an exponential attractor.

10.3.5 Existence of an Exponential Attractor

Our goal is to prove the following:

Theorem 10.10. The semigroup fS.t/gt�0 associated with Problem 10.2 has anexponential attractor in the space H.

Proof. Since we have shown, in the proof of Theorem 10.9, that assumptions .A1/–.A8/ hold, in view of Theorem 10.3 it is enough to prove .A9/ and .A10/.

Assumption .A9/. Lipschitz continuity of Lt W Xl ! Xl, uniform with respect tot 2 Œ0; �� for all � > 0, follows immediately from inequality (10.92).

Assumption .A10/. It remains to prove that the inequality

kLt1� � Lt2�kXl � cjt1 � t2jı

holds for all t0 > 0, 0 � t2 � t1 � t0 and � 2 B1l with ı 2 .0; 1� and c > 0.

Since B1l D L� .B0

l /Xl

for certain � > 0, it is enough to prove the assertion for� 2 L� .B0

l /. Let v be the unique solution of Problem 10.2 on Œ0; � C l C t0� withv.0/ 2 B0 such that vjŒ�;�Cl� D �. It is enough to obtain the uniform bound on v0 inthe space L1.�; � C l C t0I H/. Indeed,

kLt1� � Lt2�kXl DsZ l

0

jv.� C s C t2/ � v.� C s C t1/j2 ds

�sZ l

0

�Z �CsCt2

�CsCt1

jv0.r/j dr

�2

ds

� jt1 � t2jp

lkv0kL1.�;�ClCt0IH/:

The a priori estimate will be computed for the smooth solutions of the regularizedGalerkin problem (10.80)–(10.81) formulated in the proof of Theorem 10.8. Weshall show that, for the Galerkin approximations vk, for any T > � , the functionsv0

k are bounded in L1.�;TI H/ independently of k and, in consequence, the desiredestimate will be preserved by their limit v, a solution of Problem 10.2.

The argument follows the lines of the proof of Theorem 10.7 (compare [148]).First we take � D v0

k.t/ in (10.80) which gives us


jv0k.t/j2 C �

1

2

d

dtkvk.t/k2 C b.vk.t/; vk.t/; v

0k.t//

C d

dt

Z

�C

jk.vk� .x; t// dS D .f ; v0k.t//:

Estimating the convective term by using the Ladyzhenskaya inequality (10.30) weget

jv0k.t/j2 C �

d

dtkvk.t/k2 C 2

d

dt

Z

�C

jk.vk� .x; t// dS (10.101)

� jf j2 C Cjvk.t/j 12 kvk.t/k 32 jv0

k.t/j12 kv0

k.t/k12 ;

where C > 0 is a constant.Now, a straightforward and technical calculation that uses the Fatou lemma, the

Lebourg mean value theorem (cf. Theorem 3.20) and the conditions .ii/ and .iii/ inLemma 10.3, shows, that for all r1; r2 2 R,

.j0k.r2/ � j0k.r1//.r2 � r1/ � �jr2 � r2j2;

where the constant is the same as in .iii/ of Lemma 10.3. Hence, since vk is acontinuously differentiable function of time variable,

�d

dtj0k.vk� .x; t//

�

v0k� .x; t/ � �jv0

k� .x; t/j2 for all .x; t/ 2 ˝ � .0;T/:(10.102)

We differentiate both sides of (10.80) with respect to time and take � D v0k.t/,

which gives us

1

2

d

dtjv0

k.t/j2 C �kv0k.t/k2 C

Z

�C

�d

dtj0k.vk� .x; t//

�

v0k� .x; t/ dS

C b.v0k.t/; vk.t/; v

0k.t//C b.vk.t/; v

0k.t/; v

0k.t// D 0:

Using (10.102) and the fact that b.vk.t/; v0k.t/; v

0k.t// D 0 we get

1

2

d

dtjv0

k.t/j2 C �kv0k.t/k2 � kv0

k� .t/k2L2.�C/C b.v0

k.t/; vk.t/; v0k.t// � 0:

Proceeding exactly as in the proof of (10.89) we get, for an arbitrary " > 0 and aconstant C."/ > 0,

1

2

d

dtjv0

k.t/j2 C �kv0k.t/k2 � C."/kvk.t/k2jv0

k.t/j2

C "kv0k.t/k2 C C."/jv0

k.t/j2;


whence

d

dtjv0

k.t/j2 C �kv0k.t/k2 � Cjv0

k.t/j2.kvk.t/k2 C 1/:

Multiplying this inequality by t2 we get

d

dt.t2jv0

k.t/j2/C �t2kv0k.t/k2 � Ct2jv0

k.t/j2.kvk.t/k2 C 1/C 2tjv0k.t/j2:

We add this inequality to inequality (10.101) multiplied by 2t to obtain, after simplecalculations that use the fact that

R�C

jk.vk� .x; t// dS � C.1C kvk.t/k2/,

d

dt

�

t2jv0k.t/j2 C 2tkv0

k.t/k2 C 4tZ

�C

jk.vk� .x; t// dS

�

C �t2kv0k.t/k2

� C1kvk.t/k2.tjv0k.t/j/2 C C2.tjv0

k.t/j/2 C C3 C C4kvk.t/k2

C C5jvk.t/j 12 kvk.t/k 32 .tjv0

k.t/j/12 .tkv0

k.t/k/12 ;

with Ci > 0 for i D 1; : : : ; 5 independent on k and initial data. Denoting

yk.t/ D t2jv0k.t/j2 C 2tkv0

k.t/k2 C 4tZ

�C

jk.vk� .x; t// dS;

using the bound of Lemma 10.6 for jvk.t/j, Young inequality, and the fact that jkassumes nonnegative values, we get

d

dtyk.t/C �

2t2kv0

k.t/k2 � �C6kvk.t/k2 C C2

�yk.t/C C3 C C7kvk.t/k2;

where C6;C7 > 0. Finally, the Gronwall inequality (cf. Lemma 3.20) and the boundin (10.76) of Lemma 10.4 for

R T0

kvk.s/k2 ds yield the uniform boundedness of v0k

in L1.�;TI H/ \ L2.�;TI V/ for all intervals Œ�;T�, 0 < � < T . This completes theproof of .A10/ and thus the existence of an exponential attractor for the semigroupfS.t/gt�0 associated with Problem 10.2. ut


Large time behavior of solutions of problems in contact mechanics is an importantthough somewhat neglected part of the theory. In fact, remarking on future directionsof research in the field of contact mechanics, in their book [215], the authorswrote: “The infinite-dimensional dynamical systems approach to contact problemsis virtually nonexistent. (. . . ) This topic certainly deserves further consideration.”


From the mathematical point of view, a considerable difficulty in analysis ofcontact mechanics problems, and dynamical problems in particular, comes from thepresence of involved boundary constraints which are often modeled by multivaluedboundary conditions of a subdifferential type and lead to a formulation of theconsidered problem in terms of a variational inequality or differential inclusion with,frequently, nondifferentiable boundary functionals.

To establish the existence of the global attractor of a finite fractal dimension weused the method of l-trajectories as presented in [168] (cf. also [167]). This methodappears very useful when one deals with variational inequalities, cf. [208], as itovercomes obstacles coming from the usual methods presented in [62, 69, 111, 197,220]. One does not need to check directly neither compactness of the dynamicswhich results from the second energy inequality nor asymptotic compactness, cf.,e.g., [28, 220], which results from the energy equation. In the case of variationalinequalities it is sometimes not possible to get the second energy inequality and thedifferentiability of the associated semigroup due to the presence of nondifferentiableboundary functionals.

While there are other methods to establish the existence of the global attractor,where the problem of the lack of regularity appears, such as, e.g., the approach basedon the notion of the Kuratowski measure of noncompactness of bounded sets, wherewe do not need even the strong continuity of the semigroup associated with a givendynamical problem, cf., e.g., [240], the problem of a finite dimensionality of theattractor is more involved.

The method of l-trajectories allows to prove the existence of an even moredesirable object, the exponential attractor, for many problems for which there existsa finite dimensional global attractor [179]. It contains the global attractor and thusits existence implies the finite dimensionality of the global attractor itself. Its crucialproperty is an exponential rate of attraction of solution trajectories [94, 179]. Theproof of the existence of an exponential attractor requires the solution to be regularenough to ensure the Hölder continuity of the semigroup in the time variable [168].We established this property by providing additional a priori estimates of solutions.

In the end we would like to mention some related problems. First, there is aquestion of the existence of global and exponential attractors for other contactproblems with multivalued boundary conditions in form of Clarke subdifferentials.As concerns exponential attractors, the main difficulty seems to be produced byassumption .A10/. Usually (10.2) follows from some better regularity of the timederivative of the solution, e.g., it easily follows if we know that u0 2 Lq.�;TI X/ forT > � > 0 and some q > 1. While some regularity results for contact problems areknown [91, 215], such a property is not known to hold for most cases, and hence aquestion of the solution regularity for contact problems still demand further study.In particular we needed the convexity of underlying potential in the Tresca case,and .k3/, the one sided Lipschitz condition, in the generalized Tresca case. Theseproperties guaranteed us the solution uniqueness. It is, to our knowledge, an openproblem, to obtain results on the global attractor fractal dimension for the problemswithout the solution uniqueness, for example, if we do not assume .k3/.


For some visco-plastic flows (e.g., Bingham flows) governed by variationalinequalities (however, only with homogeneous or periodic boundary conditions) theregularity problem in question was solved, e.g., in [148, 211] and used to study thetime asymptotics of solutions.

In this context, there are other important open problems, namely these of the timeasymptotics of solutions of the Navier–Stokes equations, visco-plastic, and otherfluid models governed by evolution variational or even hemivariational inequalities(cf., e.g., [176]) that take into account involved boundary conditions coming from avariety of applications in mechanics.

Note that for the dissipative second order (in time) problems with multivaluedboundary conditions the situation can be totally different from that for the firstorder problems. Namely, even for the case of a very simple convex potential andthe gradient system, it is possible to show that the global attractor can have infinitefractal dimension (see [126]).

11Non-autonomous Navier–Stokes Equationsand Pullback Attractors

Running water has within itself an infinite number of movementswhich are greater or less than its principal course.

– Leonardo da Vinci

In this chapter we study the time asymptotics of solutions to the two-dimensionalNavier–Stokes equations. In the first two sections we prove two properties of theequations in a bounded domain, concerning the existence of determining modes andnodes. Then we study the equations in an unbounded domain, in the framework ofthe theory of infinite dimensional non-autonomous dynamical systems and pullbackattractors.

11.1 Determining Modes

Our aim in this section is to prove that if a number of Fourier modes of two differentsolutions of the Navier–Stokes equations have the same asymptotic behavior ast ! 1 then, under the condition that the corresponding external forces have thesame asymptotic behavior as t ! 1, the remaining infinite number of modes alsohave the same asymptotic behavior. This means that both solutions have the sametime asymptotics.

We shall consider first two-dimensional flows with homogeneous Dirichletboundary conditions, in an open, bounded, and connected domain ˝ of class C2.

To be more precise, let

@u

@tC ��u C .u � r/u C rp D f ; div u D 0;

and

@v

@tC ��v C .v � r/v C rq D g; div v D 0;


251

252 11 Non-autonomous Navier–Stokes Equations and Pullback Attractors

and let

u.x; t/ D1X

kD1Ouk.t/!k.x/; Pmu.x; t/ D

mX

kD1Ouk.t/!k.x/;

and

v.x; t/ D1X

kD1Ovk.t/!k.x/; Pmv.x; t/ D

mX

kD1Ovk.t/!k.x/;

where the functions!k, k D 1; 2; 3; : : :, are the eigenfunctions of the Stokes operatorA, A!k D �k!k, cf. Chap. 4.

We assume that the forcing terms f and g, f ; g 2 L1.0;1I H/, have the sameasymptotic behavior as time goes to infinity,

Z

˝

jf .x; t/ � g.x; t/j2dx ! 0 as t ! 1: (11.1)

Definition 11.1. Let (11.1) hold. Then the first m modes associated with Pm arecalled the determining modes if the condition

Z

˝

jPmu.x; t/ � Pmv.x; t/j2dx ! 0 as t ! 1

implies

Z

˝

ju.x; t/ � v.x; t/j2dx ! 0 as t ! 1: (11.2)

Remark 11.1. Relation (11.2) implies further a similar convergence in the normof V .

Let

lim supt!1

�Z

˝

jf .x; t/j2dx

�1=2

D F:

We shall give the explicit estimate of the number m of determining modes in termsof the Grashof number G, which is a nondimensional parameter, defined here as

G D F

�1�2for the space dimension n D 2:

We shall prove the following

11.1 Determining Modes 253

Theorem 11.1. Suppose that m 2 N is such that m � cG2 where G is the Grashofnumber, G D F

�1�2, and c is a constant depending only on the shape of ˝. Then the

first m modes are determining (in the sense defined above) for the two-dimensionalNavier–Stokes system with non-slip boundary conditions.

Proof. The proof is based on the generalized Gronwall lemma (Lemma 3.23),cf. [99]. We shall apply the lemma to �.t/ D jQmw.t/j2, where w.t/ D u.t/� v.t/ isthe difference of the two solutions and Qm D I�Pm, showing that if m is sufficientlylarge and jPmw.t/j ! 0 together with jf .t/� g.t/j ! 0 as t ! 1 then (3.36) holds.

We have the following equation for the difference of solutions,

dw

dtC �Aw C B.w; u/C B.v;w/ D f .t/ � g.t/:

Taking the inner product with Qmw in H we obtain

1

2

d

dtjQmwj2 C �kQmwk2 C b.w; u;Qmw/ C b.v;w;Qmw/

D .f .t/ � g.t/;Qmw/: (11.3)

We estimate the nonlinear terms as follows. Observe that

b.w; u;Qmw/ D b.Pmw; u;Qmw/C b.Qmw; u;Qmw/: (11.4)

We estimate the first term on the right-hand side using the Ladyzhenskaya inequal-ity (3.8) to get

jb.Pmw; u;Qmw/j D jb.Pmw;Qmw; u/j� c1jPmwj1=2kPmwk1=2juj1=2kuk1=2kQmwk:

The second term on the right-hand side of (11.4) is estimated similarly to the firstterm, obtaining

jb.Qmw; u;Qmw/j � c1jQmwjkQmwkkuk

� c212�

jQmwj2kuk2 C �

2kQmwk2:

Observe that

jb.v;w;Qmw/j D jb.v;Pmw;Qmw/j� c1jvj1=2kvk1=2jPmwj1=2kPmwk1=2kQmwk:


For the right-hand side of (11.3) we have

j.f .t/ � g.t/;Qmw/j � jf .t/ � g.t/jjQmwj:

From the above estimates we conclude that

1

2

d

dtjQmwj2 C �

2kQmwk2 � c21

2�kuk2jQmwj2

� c1jPmwj1=2kPmwk1=2juj1=2kuk1=2kQmwk (11.5)

Cc1jvj1=2kvk1=2jPmwj1=2kPmwk1=2kQmwk C jf .t/ � g.t/jjQmwj:

We use

�mC1jQmwj2 � kQmwk2 (11.6)

in the second term on the left-hand side of (11.5) to obtain

d�.t/

dtC ˛.t/�.t/ � ˇ.t/;

with

˛.t/ D ��mC1 � c212�

ku.t/k2;

and

ˇ D 2 � fthe right-hand side of (11.5)g:

We shall check the conditions of the generalized Gronwall lemma (i.e.,Lemma 3.23). The existence of uniquely determined u and v follows fromTheorem 7.2. Remark 7.2 implies that both u and v belong to L2.�;TI D.A//for all 0 < � < T < 1. Now, as f ; g 2 L1.RCI H/ we use an argument basedon the enstrophy estimate and the uniform Gronwall inequality, similar as in theproof of Theorem 7.4, to deduce that the solutions u.t/ and v.t/ are uniformlybounded in t away from zero in both H and V norms (for the initial conditions inH), see Exercise 7.25. Therefore, if jPmw.t/j ! 0 and jf .t/ � g.t/j ! 0 as t ! 1then from inequality (11.5) it follows that ˇ.t/ ! 0 as t ! 1, which impliescondition (3.35).

Second, from the energy equation

1

2ju.t/j2 C �

Z t

t0

ku.s/k2ds D 1

2ju.t0/j2 C

Z t

t0

.f .s/; u.s//ds;

11.1 Determining Modes 255

the boundedness of ju.t/j, and the Poincaré inequality we have

1

T

Z tCT

tku.s/k2ds � 2

�2�1kf k2L1.t;tCTIH/ for large T: (11.7)

By (11.7), condition (3.34) is satisfied. We have also

lim inft!1

1

T

Z tCT

t˛.�/d� � ��m � c21

�lim sup

t!11

T

Z tCT

tku.�/k2d�

� ��m � 2c21F2

�3�1:

Thus, for large m,

�mC1 >2c21F

2

�4�1; (11.8)

and (3.33) is satisfied. In view of Lemma 3.23, �.t/ ! 0 as t ! 1 for sufficientlylarge m. For m ! 1 we have �m � c0

1�1m where c01 is a nondimensional constant

(cf. [99]), and we can see that (11.8) is satisfied for m � cG2, where c is anondimensional number depending only on the shape of ˝. utExercise 11.1. Prove inequality (11.6).


Determining Modes in the Space-Periodic Case In this case we can prove in asimilar way a stronger result due to better properties of the trilinear form b.

Theorem 11.2. Suppose that m 2 N is such that m � cG where G is the Grashofnumber, G D F

�1�2, and c is a constant depending only on the shape of ˝ (i.e., the

ratio between the periods in the two space directions). Then the first m modes aredetermining in the sense that

Z

˝

jru.x; t/ � rv.x; t/j2dx ! 0 as t ! 1:

for the two-dimensional Navier–Stokes system with periodic boundary conditionsand vanishing space average.

Proof. The proof is more involved than that of Theorem 11.1 and can be foundin [99]. The idea is again to use the generalized Gronwall lemma to the function�.t/ D Qmw.t/. We start with equation

dw

dtC �Aw C B.w; u/C B.v;w/ D f .t/ � g.t/;


take the inner product with AQmw in H (we know that u.t/ 2 D.A/ for t > 0) toobtain

1

2

d

dtkQmwk2 C �kQmAwk2 C b.w; u;AQmw/ C b.v;w;AQmw/

D .f .t/ � g.t/;AQmw/;

and then proceed as above using further properties (as these in Exercises 11.3and 11.4) of solutions of two-dimensional Navier–Stokes equations with space-periodic boundary conditions. utExercise 11.3. Prove that for u; v 2 D.A/

b.v; u;Au/C b.u; v;Au/C b.u; u;Av/ D 0:

Exercise 11.4. Prove that

1

T

Z tCT

tjAu.s/j2ds � 2

�2kf k2L1.t;tCTIH/ for large T:

(cf. Exercise 11.6).

11.2 Determining Nodes

In many practical situations the experimental data is collected from measurementsat a finite number of nodal points in the physical space.

A set of points in the physical space (in the domain filled by the fluid) is calleda set of determining nodes if, whenever the difference between the measurementsat those points of the velocity field of any two flows goes to zero as time goes toinfinity, then the difference between those velocity fields goes to zero uniformly onthe domain (in a certain norm).

We shall prove that there exists a finite number of determining nodes and shallestimate their lowest number.

Let˝ be an open, bounded, and connected two-dimensional domain of class C2.We consider two velocity fields u D u.x; t/ and v D v.x; t/ satisfying in ˝ theNavier–Stokes equations,

@u

@tC ��u C .u � r/u C rp D f ; div u D 0;

and

@v

@tC ��v C .v � r/v C rq D g; div v D 0;

11.2 Determining Nodes 257

with the same homogeneous Dirichlet boundary conditions; p; q and f ; g are thecorresponding pressures and external forces, respectively.

We assume that the forcing terms f and g, f ; g 2 L1.0;1I H/, have the sameasymptotic behavior as time goes to infinity,

Z

˝

jf .x; t/ � g.x; t/j2dx ! 0 as t ! 1:

Let ˝ be covered by N identical squares Q1; : : : ;QN (for example, by a regularmesh) and let each square Qi contain exactly one measurement point xi, i D1; : : : ;N. We thus have a set of N measurement points E D fx1; : : : ; xNg. Let

maxjD1;:::;N ju.xj; t/ � v.xj; t/j ! 0 as t ! 1: (11.9)

The set E is then called a set of determining nodes if (11.9) implies

Z

˝

ju.x; t/ � v.x; t/j2dx ! 0 as t ! 1:

We shall show that if (11.9) holds for some N then also

Z

˝

jru.x; t/ � rv.x; t/j2dx ! 0 as t ! 1:

Denote

�.w/ D max1�j�N

jw.xj/j:

To prove the existence of a finite number of determining nodes we shall use thegeneralized Gronwall lemma (cf. Lemma 3.23) as well as inequality (11.10) of thefollowing lemma.

Lemma 11.1. Let the domain ˝ be covered by N identical squares. Consider theset E D fx1; : : : ; xNg of points in ˝, distributed one in each square. Then for eachvector field w in D.A/ the following inequalities hold,

jwj2 � c

�1�.w/2 C c

�21N2jAwj2;

kwk2 � cN�.w/2 C c

�1NjAwj2;

kwk2L1.˝/ � cN�.w/2 C c

�1NjAwj2; (11.10)

where the constant c depends only on the shape of the domain ˝.


A proof of the lemma can be found in [99].

Theorem 11.3. Let the domain ˝ be covered by N identical squares. Consider theset E D fx1; : : : ; xNg of points in ˝, distributed one in each square. Let f and g betwo forcing terms in L1.0;1I H/ that satisfy

Z

˝

jf .x; t/ � g.x; t/j2dx ! 0 as t ! 1;

and let

F D lim supt!1

jf .t/j D lim supt!1

jg.t/j as t ! 1:

Then there exists a constant C D C.F; �;˝/ such that if

N � C D C.F; �;˝/;

then E is a set of determining nodes in the sense defined above for the two-dimensional Navier–Stokes equations with homogeneous Dirichlet boundaryconditions.

Proof. Let w D u � v. We know that �.w.t// ! 1 as t ! 1. We have to showthat jw.t/j ! 0 as well. Actually, we will show that

kw.t/k ! 0 as t ! 1:

We have, from the Navier–Stokes equations,

dw.t/

dtC �Aw.t/C B.w.t/; u.t//C B.v.t/;w.t// D f .t/ � g.t/:

Taking the inner product with Aw.t/ in H we obtain

1

2

d

dtkw.t/k2 C �jAw.t/j2 C b.w.t/; u.t/;Aw.t//C b.v.t/;w.t/;Aw.t//

D .f .t/ � g.t/;Aw.t//: (11.11)

By C we shall denote a generic constant that might depend on ˝; f ; g; �, but not onthe initial conditions u0; v0. By c we denote constants that might depend only on˝.

We estimate the nonlinear terms in the following way:

jb.w; u;Aw/j � c1jwj1=2kukjAwj3=2 � �

6jAwj2 C c

�3kuk4jwj2;

jb.v;w;Aw/j � c1jvj1=2jAvj1=2kwkjAwj � �

6jAwj2 C c

�jvjjAvjkwk2:

11.2 Determining Nodes 259

Exercise 11.5. Prove the above two inequalities by using the following Agmoninequality:

kuk1 � c2juj1=2jAuj1=2 for u 2 D.A/: (11.12)

The right-hand side of (11.11) is estimated as

j.f � g;Aw/j � �

6jAuj2 C 3

2�jf � gj2:

Thus, (11.11) yields

d

dtkwk2 C �jAwj2 � c

�3kuk4jwj2 C c

�jvjjAvjkwk2 C 3

�jf � gj2: (11.13)

From (11.10) we have

jAwj2 � c�1�1Nkwk2 � �1N2�.w/2:

Using the Poincaré inequality in the first term on the right-hand side of (11.13) weget, in the end,

d

dtkwk2 C

�

c�1�1N� � c

�3�1kuk4 � c

�jvjjAvj

�

kwk2

� 3

�jf � gj2 C �1N

2��.w/2:

This inequality can be written in the form

d�

dtC ˛� � ˇ where � D kw.t/k2: (11.14)

We see that

ˇ.t/ ! 0 as t ! 1;

and hence

limt!1

1

T

Z tCT

tˇC.�/d� D 0: (11.15)

As the functions

t ! ku.t/k and t ! jv.t/j (11.16)


are bounded on Œt0;1/ for t0 > 0, and

1

T

Z tCT

tjAv.�/jd� � C.jgj2L1.t;tCTIH/; �;˝/; (11.17)

we deduce that

lim inft!1

1

T

Z tCT

t˛.�/d� � c�1��1N � C.f ; g; �;˝/:

For N sufficiently large we have

lim inft!1

1

T

Z tCT

t˛.�/d� > 0: (11.18)

Also

lim supt!1

1

T

Z tCT

t˛�.�/d� < 1: (11.19)

In view of (11.14) together with (11.15), (11.18), and (11.19) we deduce that

�.t/ D kw.t/k ! 0 as t ! 1:

Hence, the set E is a set of determining nodes. utExercise 11.6. Prove (11.17) from

1

2

d

dtkv.t/k2 C �jAv.t/j2 C b.v.t/; v.t/;Av.t// D .g.t/;Av.t//;

knowing that the functions in (11.16) are bounded.

11.3 Pullback Attractors for Asymptotically CompactNon-autonomous Dynamical Systems

Let ˝ be a nonempty set and let ftgt2R be a family of mappings t W ˝ ! ˝

satisfying:

1. 0! D ! for all ! 2 ˝,2. t.�!/ D tC�! for all ! 2 ˝ and all t; � 2 R:

The mappings t are called the shift operators.Let moreover X be a metric space with the distance d.�; �/; and let � be a -

cocycle on X, i.e., a mapping � W RC �˝ � X ! X; which satisfies:

11.3 Pullback Attractors for Asymptotically Compact Non-autonomous. . . 261

(a) �.0; !; x/ D x for all .!; x/ 2 ˝ � X,(b) �.tC�; !; x/ D �.t; �!; �.�; !; x// for all t; � 2 RC and .!; x/ 2 ˝�X.

The -cocycle � is said to be continuous if for all .t; !/ 2 RC �˝, the mapping�.t; !; �/ W X ! X is continuous.

The sets parameterized by the index ! 2 ˝ will be called the parameterized setsand denoted bD D fD.!/g!2˝ , while their particular case, the sets parameterized bytime t 2 R will be called the non-autonomous sets and denoted D D fD.t/gt2R: Thisdifferent notation stresses the special case˝ D R and ts D sC t. Let P.X/ denotethe family of all nonempty subsets of X, and S the class of all parameterized setsbD D fD.!/g!2˝ such that D.!/ 2 P.X/ for all ! 2 ˝. We say that the fibersD.!/ of the parameterized set bD are nonempty.

We consider a given nonempty subclass D S : This class will be called anattraction universe.

Definition 11.2. The -cocycle � is said to be pullback D-asymptotically compact.D-a.c./ if for any ! 2 ˝, any bD 2 D , and any sequences tn ! C1, xn 2D.�tn!/; the sequence �.tn; �tn!; xn/ has a convergent subsequence.

Definition 11.3. A parameterized set bB D fB.!/g!2˝ 2 S is said to be pullbackD-absorbing if for each ! 2 ˝ and bD 2 D , there exists t0.!; bD/ � 0 such that

�.t; �t!;D.�t!// B.!/ for all t � t0.!; bD/:

We denote by distX.C1;C2/ the Hausdorff semi-distance between C1 and C2,defined as

distX.C1;C2/ D supx2C1

infy2C2

d.x; y/ for C1;C2 X:

Definition 11.4. A parameterized set bC D fC.!/g!2˝ 2 S is said to be pullbackD-attracting if

limt!C1 distX.�.t; �t!;D.�t!//;C.!// D 0 for all bD 2 D ; ! 2 ˝:

Definition 11.5. A parameterized set bA D fA.!/g!2˝ 2 S is said to be a globalpullback D-attractor if it satisfies

(i) A.!/ is compact for any ! 2 ˝;(ii) bA is pullback D-attracting,

(iii) bA is invariant, i.e.,

�.t; !;A.!// D A.t!/ for any .t; !/ 2 RC �˝:

Remark 11.2. Observe that Definition 11.5 does not guarantee the uniqueness of apullback D-attractor (see [36] for a discussion on this point). In order to ensure the


uniqueness one needs to impose additional conditions as, for instance, the conditionthat the attractor belongs to the same attraction universe D , or some kind ofminimality. However, we do not include this stronger assumptions in the definitionsince, as we will show in Theorem 11.4, under very general hypotheses, it is possibleto ensure the existence of a global pullback D-attractor which is minimal in anappropriate sense.

Definition 11.6. For each bD 2 S and ! 2 ˝; we define the !-limit set of bD at! as

�.bD; !/ D\

s�0

[

t�s

�.t; �t!;D.�t!//

!

:

Obviously, �.bD; !/ is a closed subset of X that can be possibly empty. It is easyto see that for each y 2 X; one has that y 2 �.bD; !/ if and only if there exist asequence tn ! C1 and a sequence xn 2 D.�tn!/ such that

limtn!C1 d.�.tn; �tn!; xn/; y/ D 0:

Our first aim is to prove the following result on the existence of global pullbackD-attractor.

Theorem 11.4. Suppose the -cocycle � is continuous and pullback D-asymptotically compact, and there exists bB 2 D which is pullback D-absorbing.Then, the parameterized set bA defined by

A.!/ D �.bB; !/ for ! 2 ˝;

is a global pullback D-attractor which is minimal in the sense that if bC 2 S is aparameterized set such that C.!/ is closed and

limt!C1 distX.�.t; �t!;B.�t!//;C.!// D 0;

then A.!/ C.!/.

In order to prove Theorem 11.4, we first need the following results.

Proposition 11.1. If bB 2 S is a pullback D-absorbing parameterized set, then

�.bD; !/ �.bB; !/ for all bD 2 D and ! 2 ˝:

If in addition bB 2 D , then

�.bD; !/ �.bB; !/ B.!/ for all bD 2 D and ! 2 ˝:


Proof. Let us fix bD 2 D , ! 2 ˝; and y 2 �.bD; !/. There exist a sequence tn !C1 and a sequence xn 2 D.�tn!/ such that

�.tn; �tn!; xn/ ! y: (11.20)

As bB is pullback D-absorbing, for each integer k � 1 there exists an index nk suchthat tnk � k and

�.tnk � k; �.tnk �k/.�k!/;D.�.tnk �k/.�k!/// B.�k!/: (11.21)

In particular, as xnk 2 D.�tnk!/ D D.�.tnk �k/.�k!//; we obtain from (11.21)

yk D �.tnk � k; �tnk!; xnk/ 2 B.�k!/:

But then

�.tnk ; �tnk!; xnk/ D �..tnk � k/C k; �tnk

!; xnk/

D �.k; �k!; �.tnk � k; �tnk!; xnk//

D �.k; �k!; yk/;

and consequently, by (11.20),

�.k; �k!; yk/ ! y with yk 2 B.�k!/:

Thus, y belongs to �.bB; !/.Finally, observe that if z belongs to �.bB; !/, then there exist a sequence �n !

C1 and a sequence wn 2 B.��n!/ such that �.�n; ��n!;wn/ ! z: But, as bB ispullback D-absorbing, there exists t0.bB; !/ � 0 such that �.�n; ��n!;wn/ 2 B.!/for all �n � t0.bB; !/; and consequently z belongs to B.!/: utRemark 11.3. As a straightforward consequence of Proposition 11.1, we have thatif bB 2 D is pullback D-absorbing, then

[

bD2D�.bD; !/ D �.bB; !/;

so that, under the assumptions in Theorem 11.4, we have

A.!/ D �.bB; !/ D[

bD2D�.bD; !/ for ! 2 ˝:


Proposition 11.2. If � is pullback D-asymptotically compact, then for bD 2 D and! 2 ˝; the set �.bD; !/ is nonempty, compact, and

limt!C1 distX.�.t; �t!;D.�t!//;�.bD; !// D 0: (11.22)

Proof. Let us fix bD 2 D and ! 2 ˝. If we consider a sequence tn ! C1, anda sequence xn 2 D.�tn!/; then from the sequence �.tn; �tn!; xn/ we can extracta convergent subsequence �.t; �t!; x/ ! y. By its construction, y belongs to

�.bD; !/, and thus this set is nonempty.We know that �.bD; !/ is closed, and consequently in order to prove its

compactness it is enough to see that for any given sequence fyng �.bD; !/, wecan extract a convergent subsequence. First, observe that as yn 2 �.bD; !/, thereexist tn � n and xn 2 D.�tn!/ such that

d.�.tn; �tn!; xn/; yn/ � 1

n: (11.23)

As � is pullback D-a.c., from the sequence �.tn; �tn!; xn/ we can extract aconvergent subsequence �.t; �t!; x/ ! y, and consequently, from (11.23), wealso obtain y ! y:

Finally, if (11.22) does not hold, there exist " > 0; a sequence tn ! C1, and asequence xn 2 D.�tn!/; such that

d.�.tn; �tn!; xn/; y/ � " for all y 2 �.bD; !/: (11.24)

But from the sequence �.tn; �tn!; xn/ we can extract a convergent subsequence�.t; �t!; x/ ! y 2 �.bD; !/; which contradicts (11.24). utProposition 11.3. If the -cocycle � is continuous and pullback D-asymptoticallycompact, then for any .t; !/ 2 RC �˝ and any bD 2 D ; one has

�.t; !;�.bD; !// D �.bD; t!/:

Proof. Let .t; !/ 2 RC � ˝ and bD 2 D be fixed, and let y 2 �.bD; !/. We showthat �.t; !; y/ 2 �.bD; t!/; and consequently we obtain

�.t; !;�.bD; !// �.bD; t!/:

We already know that there exist a sequence tn ! C1 and a sequence xn 2D.�tn!/ such that

�.tn; �tn!; xn/ ! y:


But

�.t; !; �.tn; �tn!; xn// D �.t C tn; �.tCtn/.t!/; xn/;

and consequently, as � is continuous,

�.t C tn; �.tCtn/.t!/; xn/ ! �.t; !; y/:

Since we have t C tn ! C1 and xn 2 D.�tn!/ D D.�.tCtn/.t!//; it follows that�.t; !; y/ 2 �.bD; t!/:

Now, we prove �.bD; t!/ �.t; !;�.bD; !//: For y 2 �.bD; t!/ there exist asequence tn ! C1 and a sequence xn 2 D.�tn.t!// such that

�.tn; �tn.t!/; xn/ ! y: (11.25)

If tn � t; we have

�.tn; �tn.t!/; xn/ D �.t C .tn � t/; �.tn�t/!; xn/

D �.t; !; �.tn � t; �.tn�t/!; xn//: (11.26)

But, as � is pullback D-a.c., tn � t ! C1; and xn 2 D.�tn.t!// D D.�.tn�t/!/,one can ensure that there exists a subsequence f.t; x/g f.tn; xn/g such that

�.t � t; �.t�t/!; x/ ! z 2 �.bD; !/;

and then, by (11.26) and (11.25),

y D �.t; !; z/ 2 �.t; !;�.bD; !//: ut

Now, we can prove the main Theorem.

Proof (of Theorem 11.4). The compactness of each A.!/ follows from Proposi-tion 11.2. Also, by this proposition and the fact that

S

bD2D �.bD; !/ A.!/; weobtain

limt!C1 distX.�.t; �t!;D.�t!//;A.!// D 0 for any bD 2 D :

Now, observe that, by Proposition 11.3,

A.t!/ D[

bD2D�.bD; t!/ D

[

bD2D�.t; !;�.bD; !// D �.t; !;

[

bD2D�.bD; !//;


and, as � is continuous andS

bD2D �.bD; !/ is compact,

�.t; !;[

bD2D�.bD; !// D �.t; !;

[

bD2D�.bD; !//:

Consequently, we have the invariance property �.t; !;A.!// D A.t!/:

Now, let bC 2 S be a parameterized set such that C.!/ is closed and

limt!C1 distX.�.t; �t!;B.�t!//;C.!// D 0:

Let y be an element of A.!/, then

y D limn!1�.tn; �tn!; xn/;

for some sequences tn ! C1, xn 2 B.�tn!/; and consequently y 2 C.!/ DC.!/: Thus, A.!/ C.!/. utRemark 11.4. Observe that if in Theorem 11.4 we assume that B.!/ is closed forall ! 2 ˝; and the attraction universe D is inclusion-closed (i.e., if bD0 2 S , bD 2 Dand D0.!/ D.!/ for all ! 2 ˝, then bD0 2 D), then it follows that the pullbackD-attractor bA belongs to D , and hence it is the unique pullback D-attractor whichbelongs to D . This situation appears very often in applications, in particular, in ourexample in Sect. 11.4.

Remark 11.5. Note that we do not assume any structure (neither measurable nortopological) on the set of parameters ˝. Consequently, this result can be applied toboth non-autonomous and random dynamical systems.

Although the cocycle formalism is very useful, in particular, in the case ofrandom dynamical systems or non-autonomous problems, we state below our mainresult (namely Theorem 11.4) in the language of evolutionary processes (see [225])and then apply it in the theory of Navier–Stokes equations.

Let us consider an evolutionary process (also called a two-parameter semigroupor just a process) U on X, i.e., a family fU.t; �/g�1<��t<C1 of continuousmappings U.t; �/ W X ! X, such that U.�; �/x D x; and

U.t; �/ D U.t; r/U.r; �/ for all � � r � t: (11.27)

The attraction universe D will now be a nonempty class of non-autonomous setsD D fD.t/gt2R such that D.t/ 2 P.X/ for all t 2 R.

Definition 11.7. The process U is said to be pullback D-asymptotically compact iffor any t 2 R, any D 2 D ; any sequence �n ! �1; and any sequence xn 2 D.�n/,the sequence fU.t; �n/xng is relatively compact in X.


Fig. 11.1 Illustration of apullback D-absorbingproperty. The sets D.ti/become larger as ti tend to�1 but still at time t alltrajectories originating fromthose sets are in B.t/

Definition 11.8. It is said that B D fB.t/gt2R 2 D is pullback D-absorbing for theprocess U if for any t 2 R and any D 2 D , there exists a �0.t;D/ � t such that

U.t; �/D.�/ B.t/ for all � � �0.t;D/:

The notion of pullback D�absorbing set is illustrated in Fig. 11.1.We have the following result.

Theorem 11.5. Suppose that the process U is pullback D-asymptotically compactand that B 2 D is a non-autonomous D-absorbing set for U.

Then, the non-autonomous set A D fA.t/gt2R with nonempty fibers A.t/ 2 P.X/for t 2 R defined by

A.t/ D �.B; t/ for t 2 R;

where for each D D fD.t/gt2R 2 D

�.D; t/ D\

s�t

[

��s

U.t; �/D.�/

!

;

has the following properties:

(i) the sets A.t/ are compact for t 2 R,(ii) the non-autonomous set A is pullback D-attracting, i.e.,

lim�!�1 distX.U.t; �/D.�/;A.t// D 0 for all D 2 D ;


(iii) the non-autonomous set A is invariant, i.e.,

U.t; �/A.�/ D A.t/ for � 1 < � � t < C1;

(iv) the sets A.t/ are given by

A.t/ D[

D2D�.D; t/ for t 2 R:

The non-autonomous set A, termed the global pullback D-attractor for theprocess U, is minimal in the sense that if C D fC.t/gt2R with C.t/ 2 P.X/ is anon-autonomous set such that C.t/ is closed and

lim�!�1 distX.U.t; �/B.�/;C.t// D 0;

then A.t/ C.t/.

Proof. It is enough to denote ˝ D R, and t� D � C t; and to apply Theorem 11.4to the non-autonomous dynamical system � defined by

�.t; �; x/ D U.t C �; �/x: ut

Remark 11.6. The comments contained in Remark 11.4 also hold true for evolu-tionary processes.

11.4 Application to Two-Dimensional Navier–StokesEquations in Unbounded Domains

Let ˝ R2 be an open set, not necessarily bounded, with the boundary @˝, and

suppose that ˝ satisfies the Poincaré inequality, i.e., there exists a constant �1 > 0

such that

�1

Z

˝

�2 dx �Z

˝

jr�j2 dx for all � 2 H10.˝/: (11.28)

Consider the following two-dimensional Navier–Stokes problem:

8ˆˆ<

ˆˆ:

@u

@t� ��u C

2X

iD1ui@u

@xiD f .t/ � rp in ˝ � .�;C1/;

div u D 0 in ˝ � .�;C1/;

u D 0 on @˝ � .�;C1/;

u.�; x/ D u0.x/ for x 2 ˝:

11.4 Application to Two-Dimensional Navier–Stokes Equations. . . 269

To set our problem in the abstract framework, we consider the following usualabstract space

QV Dnu 2 �C1

0 .˝/�2 W div u D 0

o:

We define H as the closure of QV in .L2.˝//2 with the norm j�j ; and the inner product.�; �/ where for u; v 2 .L2.˝//2 we set

.u; v/ D2X

jD1

Z

˝

uj.x/vj.x/ dx:

Moreover we define V as the closure of QV in .H10.˝//

2 with the norm k � k; and theassociated scalar product ..�; �//; where for u; v 2 .H1

0.˝//2 we have

..u; v// D2X

i;jD1

Z

˝

@uj

@xi

@vj

@xidx:

It follows that V H � H0 V 0; where the injections are continuous and dense.Finally, we will use k�kV0

for the norm in V 0 and h�; �i for the duality pairingbetween V and V 0:

Consider the trilinear form b on V � V � V given by

b.u; v;w/ D2X

i;jD1

Z

˝

ui@vj

@xiwj dx for u; v;w 2 V:

Define B W V � V ! V 0 by

hB.u; v/;wi D b.u; v;w/ for u; v;w 2 V;

and denote

B.u/ D B.u; u/:

Assume now that u0 2 H, f 2 L2loc.RI V 0/. For each � 2 R we consider the problem

8ˆˆ<

ˆˆ:

u 2 L2.�;TI V/ \ L1.�;TI H/ for all T > �;d

dt.u.t/; v/C �..u.t/; v//C hB.u.t//; vi D hf .t/; vi for all v 2 V

in the sense of scalar distributions on .�;C1/;

u.�/ D u0:(11.29)


It follows from the results of Chap. 7, cf. Theorem 7.2 and Remark 7.3, that prob-lem (11.29) has a unique solution, u.�I �; u0/, that moreover belongs to C.Œ�;T�I H/for all T � � .

Define

U.t; �/u0 D u.tI �; u0/ for � � t and u0 2 H: (11.30)

From the uniqueness of solution to problem (11.29), it follows that

U.t; �/u0 D U.t; r/U.r; �/u0 for all � � r � t and u0 2 H: (11.31)

One can prove (cf. Chap. 7) that for all � � t; the mapping U.�; t/ W H ! H definedby (11.30) is continuous. Consequently, the family fU.t; �/g��t defined by (11.30)is a process U in H.

Moreover, it can be proved that U is weakly continuous, and more exactly thefollowing result holds true. Its proof is identical to the proof of Lemma 8:1 in [164],and so we omit it.

Proposition 11.4. Let fu0ng H be a sequence converging weakly in H to anelement u0 2 H: Then

U.t; �/u0n ! U.t; �/u0 weakly in H for all � � t; (11.32)

U.�; �/u0n ! U.�; �/u0 weakly in L2.�;TI V/ for all � < T: (11.33)

From now on, we denote

� D ��1: (11.34)

Let R� be the set of all functions r W R ! .0;C1/ such that

limt!�1 e� tr2.t/ D 0; (11.35)

and denote by D� the class of all non-autonomous sets D D fD.t/gt2R such thatD.t/ 2 P.H/ for t 2 R, for which there exists rD 2 R� such that D.t/ B.0; rD.t//; where B.0; rD.t// denotes the closed ball in H centered at zero withradius rD.t/:

Now, we can prove the following result.

Theorem 11.6. Suppose that f 2 L2loc.RI V 0/ is such that

Z t

�1e��kf .�/k2V0

d� < C1 for all t 2 R: (11.36)


Then, there exists a unique global pullback D� -attractor for the process U definedby (11.30).

Proof. Let � 2 R and u0 2 H be fixed, and denote

u.t/ D u.tI �; u0/ D U.t; �/u0 for all t � �:

Taking into account that b.u; v; v/ D 0; it follows

d

dt

�e� tju.t/j2�C 2�e� tku.t/k2 D �e� tju.t/j2 C 2e� thf .t/; u.t/i

in�C10 .�;C1/

�0; and consequently, by (11.28),

e� tju.t/j2 � e�� ju0j2 C 1

�

Z t

�

e��kf .�/k2V0

d� for all � � t: (11.37)

Let D 2 D� be given. From (11.37), we easily obtain

jU.t; �/u0j2 � e��.t��/r2D.�/C e�� t

�

Z t

�1e��kf .�/k2V0

d� (11.38)

for all u0 2 D.�/; and all t � � .Denote by R� .t/ the nonnegative number given for each t 2 R by

.R� .t//2 D 2e�� t

�

Z t

�1e��kf .�/k2V0

d�;

and consider the non-autonomous set B� D fB� .t/gt2R of closed balls in Hdefined by

B� .t/ D fv 2 H W jvj � R� .t/g: (11.39)

It is straightforward to check that B� 2 D� , and moreover, by (11.35) and (11.38),the family B� is pullback D� -absorbing for the process U.

According to Theorem 11.5 and Remark 11.6, to finish the proof of the theoremwe only have to prove that U is pullback D� -asymptotically compact.

Let D 2 D� , a sequence �n ! �1, a sequence u0n 2 D.�n/ and t 2 R, be fixed.We must prove that from the sequence fU.t; �n/u0ng we can extract a subsequencethat converges in H.

As the non-autonomous set B� is pullback D� -absorbing, for each integer k � 0

there exists a �D.k/ � t � k such that

U.t � k; �/D.�/ B� .t � k/ for all � � �D.k/: (11.40)


It is not difficult to conclude from (11.40), by a diagonal procedure, the existenceof a subsequence f.�n0 ; u0n0/g f.�n; u0n/g; and a sequence fwkg1

kD0 H; such thatfor all k D 0; 1; 2; : : : we have wk 2 B� .t � k/ and

U.t � k; �n0/u0n0 ! wk weakly in H: (11.41)

Observe that, by Proposition 11.4,

w0 D weak � limn0!1 U.t; �n0/u0n0

D weak � limn0!1 U.t; t � k/U.t � k; �n0/u0n0

D U.t; t � k/.weak � limn0!1 U.t � k; �n0/u0n0/;

i.e.,

U.t; t � k/wk D w0 for all k D 0; 1; 2; : : : : (11.42)

Then, by the weak lower semicontinuity of the norm,

jw0j � lim infn0!1 jU.t; �n0/u0n0 j: (11.43)

If we now prove that also

lim supn0!1

jU.t; �n0/u0n0 j � jw0j; (11.44)

then we will have

limn0!1 jU.t; �n0/u0n0 j D jw0j;

and this, together with the weak convergence, will imply the strong convergencein H of U.t; �n0/u0n0 to w0.

In order to prove (11.44), we consider the Hilbert norm in V given by

Œu�2 D �kuk2 � �

2juj2;

which by (11.28) is equivalent to the norm kuk in V:It is immediate that for all � � t and all u0 2 H;

jU.t; �/u0j2 D ju0j2e�.��t/ (11.45)

C2Z t

�

e�.��t/�hf .�/;U.�; �/u0i � ŒU.�; �/u0�2

�d�;


and, thus, for all k � 0 and all �n0 � t � k;

jU.t; �n0/u0n0 j2 D jU.t; t � k/U.t � k; �n0/u0n0 j2 (11.46)

D e��kjU.t � k; �n0/u0n0 j2

C2Z t

t�ke�.��t/hf .�/;U.�; t � k/U.t � k; �n0/u0n0i d�

�2Z t

t�ke�.��t/ŒU.�; t � k/U.t � k; t � �n0/u0n0 �2 d�:

As, by (11.40),

U.t � k; �n0/u0n0 2 B� .t � k/ for all �n0 � �D.k/ and k � 0;

we have

lim supn0!1

�e��kjU.t � k; �n0/u0n0

j2� � e��kR2� .t � k/ for all k � 0: (11.47)

On the other hand, as U.t � k; �n0/u0n0 ! wk weakly in H, from Proposition 11.4we have

U.�; t � k/U.t � k; �n0/u0n0 ! U.�; t � k/wk weakly in L2.t � k; tI V/: (11.48)

Taking into account that, in particular, e�.��t/f .�/ 2 L2.t � k; tI V 0/; we obtainfrom (11.48),

limn0!1

Z t

t�ke�.��t/hf .�/;U.�; t � k/U.t � k; �n0/u0n0i d� (11.49)

DZ t

t�ke�.��t/hf .�/;U.�; t � k/wki d�:

Moreover, as�R t

t�k e�.��t/Œv.�/�2 d��1=2

defines a norm in L2.t � k; tI V/ which isequivalent to the usual one, we also obtain from (11.48),

Z t

t�ke�.��t/ŒU.�; t � k/wk/�

2 d� (11.50)

� lim infn0!1

Z t

t�ke�.��t/ŒU.�; t � k/U.t � k; �n0/u0n0 �2 d�:

Then, from (11.46), (11.47), (11.49), and (11.50), we easily obtain

lim supn0!1

jU.t; �n0/u0n0 j2 (11.51)

� e��kR2� .t � k/C 2

Z t

t�ke�.��t/

�hf .�/;U.�; t � k/wki � ŒU.�; t � k/wk/�2�

d�:


Now, from (11.42) and (11.45),

jw0j2 D jU.t; t � k/wkj2 (11.52)

D jwkj2e��k C 2

Z t

t�ke�.��t/

�hf .�/;U.�; t � k/wki � ŒU.�; t � k/wk�2�

d�:

From (11.51) and (11.53) we have

lim supn0!1

jU.t; �n0/u0n0 j2 � e��kR2� .t � k/C jw0j2 � jwkj2e��k

� e��kR2� .t � k/C jw0j2;

and thus, taking into account that

e��kR2� .t � k/ D 2e�� t

�

Z t�k

�1e��kf .�/k2V0

d� ! 0;

when k ! C1, we easily obtain (11.44) from the last inequality. utIn Chap. 13 we consider, in a similar framework, a problem from the theory of

lubrication and prove the finite dimensionality of the pullback attractor.


The presentation in Sects. 11.1 and 11.2 follows [99]. Estimates on the number ofdetermining modes and determining nodes can be found in [122]. We note here thatanother possibility of finite dimensional reduction of infinite dimensional dynamicalsystems is through the study of inertial manifolds [77] or of invariant manifolds[54]. For Navier–Stokes equations the approach by inertial manifolds leads to the so-called approximate inertial manifolds studied in [224], existence of inertial manifoldfor two-dimensional Navier–Stokes equations is still unresolved [103].

The results of Sects. 11.3 and 11.4 come from [45], where the notion of anasymptotically compact dynamical system as in Definition 11.2 was introduced.

The notion of pullback attractor is a direct generalization of that of globalattractor to the non-autonomous case.

The first attempts to extend the notion of global attractor to the non-autonomouscase led to the concept of the so-called uniform attractor (see [62]). The conditionsensuring the existence of the uniform attractor parallel those for autonomous sys-tems. In this theory non-autonomous systems are lifted in [225] to autonomousones by expanding the phase space. Then, the existence of uniform attractorsrelies on some compactness property of the solution operator associated to theresulting system. One disadvantage of uniform attractors is that they do not needto be invariant unlike the global attractor for autonomous systems. Moreover, theydemand rather strong conditions on the time-dependent data.


At the same time, the theory of pullback (or cocycle) attractors has beendeveloped for both the non-autonomous and random dynamical systems (see[80, 137, 152, 206, 232]), and has shown to be very useful in the understandingof the dynamics of non-autonomous dynamical systems. In this case, the conceptof pullback (or cocycle or non-autonomous) attractor provides a time-dependentfamily of compact sets which attracts families of sets in a certain universe (e.g.,the bounded sets in the phase space) and satisfying an invariance property, whatseems to be a natural set of conditions to be satisfied for an appropriate extensionof the autonomous concept of attractor. Moreover, this cocycle formulation allowsto handle more general time-dependent terms in the models—not only the periodic,quasi-periodic, or almost periodic ones (see, for instance, [38], and [43] for non-autonomous models containing hereditary characteristics). Note that, however, thenotion of pullback attractor has its limitations, namely it does not always capturethe asymptotic behavior of the non-autonomous system as time advances to C1,see [140]. To counteract this limitation the synthesis of the approach by pullbackattractors with the one by uniform attractors has been presented in [19, 53, 136]

In order to prove the existence of the attractor (in both the autonomous andnon-autonomous cases) the simplest, and therefore the strongest, assumption isthe compactness of the solution operator associated with the system, which isusually available for parabolic systems in bounded domains. However, this kind ofcompactness does not hold in general for parabolic equations in unbounded domainsand second order in time equations on either bounded or unbounded domains.Instead we often have some kind of asymptotic compactness. In this situation, thereare several approaches to prove the existence of the global (or non-autonomous)attractor. Roughly speaking, the first one ensures the existence of the global (resp.non-autonomous) attractor whenever a compact attracting set (resp. a family ofcompact attracting sets) exists. The second method consists in decomposing thesolution operator (resp. the cocycle or two-parameter semigroup) into two parts: acompact part and another one which decays to zero as time goes to infinity. However,as it is not always easy to find such decomposition, one can use a third approachwhich is based on the use of the energy equations which are in direct connectionwith the concept of asymptotic compactness. This third method has been used in[164] (and also [181]) to extend to the non-autonomous situation the correspondingone in the autonomous framework (see [201] and also [11]), but related to uniformasymptotic compactness. The method used in this chapter is a generalization of thelatter to the theory of pullback attractors.

More information on non-autonomous infinite dimensional dynamical systemscan be found in monographs [53, 136] and the review article [10], cf. also [208]. Forthe random case, see, e.g., [30, 31, 39, 95, 135, 153]. Two-dimensional problemswith delay have been studied in [37, 170].

12Pullback Attractors and Statistical Solutions

This chapter is devoted to constructions of invariant measures and statisticalsolutions for non-autonomous Navier–Stokes equations in bounded and certainunbounded domains in R

2.After introducing some basic notions and results concerning attractors in the

context of the Navier–Stokes equations, we construct the family of probabilitymeasures ftgt2R and prove the relations t.E/ D �.U.t; �/�1E/ for t; � 2 R,t � � and Borel sets E in H. Then we prove the Liouville and energy equations.Finally, we consider statistical solutions of the Navier–Stokes equations supportedon the pullback attractor.

12.1 Pullback Attractors and Two-DimensionalNavier–Stokes Equations

We begin this section by recalling the notion of pullback attractor introduced inChap. 11 in a more general context of cocycles.

Let us consider an evolutionary process U (a process U—for short) on a metricspace X, i.e., a family fU.t; �/g�1<��t<C1 of continuous mappings U.t; �/W X ! X,such that U.�; �/x D x; and

U.t; �/ D U.t; r/U.r; �/ for all � � r � t:

Let D be an attraction universe, i.e., a nonempty family of non-autonomous setsD D fD.t/gt2R such that D.t/ 2 P.X/ for all t 2 R, where P.X/ denotes thefamily of all nonempty subsets of X.


277

278 12 Pullback Attractors and Statistical Solutions

Definition 12.1. A process U.�; �/ is said to be pullback D-asymptotically compactif for any t 2 R, any D 2 D ; any sequence �n ! �1; and any sequence xn 2 D.�n/,the sequence fU.t; �n/xng is relatively compact in X.

Definition 12.2. It is said that B 2 D is pullback D-absorbing for the processU.�; �/ if for any t 2 R and any D 2 D , there exists a �0.t;D/ � t such that

U.t; �/D.�/ B.t/ for all � � �0.t;D/:

Definition 12.3. A non-autonomous set A D fA.t/gt2R with A.t/ 2 P.X/ for allt 2 R is said to be a pullback D-attractor for the process U.�; �/ if

(i) A.t/ is compact for all t 2 R,(ii) A is pullback D-attracting, i.e.,

lim�!�1 distX.U.t; �/D.�/;A.t// D 0 for all D 2 D and all t 2 R;

(iii) A is invariant, i.e., U.t; �/A.�/ D A.t/ for �1 < � � t < C1.

We consider the initial and boundary value problem for the Navier–Stokes equationsstudied already in Chap. 11. Namely, let ˝ R

2 be an open, bounded orunbounded, domain with smooth boundary @˝ such that there exists a constant�1 > 0 for which

�1

Z

˝

�2 dx �Z

˝

jr�j2 dx for all � 2 H10.˝/:

Consider the following problem:

8ˆˆ<

ˆˆ:

@u

@t� ��u C

2X

iD1ui@u

@xiD f .t/ � rp in ˝ � .�;C1/;

div u D 0 in ˝ � .�;C1/;

u D 0 on @˝ � .�;C1/;

u.�; x/ D u0.x/; for x 2 ˝:This problem is set in a suitable abstract framework by considering the usualfunction space

QV Dnu 2 �C1

0 .˝/�2 W div u D 0

o:

The space H is defined as the closure of QV in .L2.˝//2 with the norm j�j, and theassociated scalar product .�; �/ given by

.u; v/ D2X

jD1

Z

˝

uj.x/vj.x/ dx for u; v 2 .L2.˝//2:

12.1 Pullback Attractors and Two-Dimensional Navier–Stokes Equations 279

The space V is defined as the closure of QV in .H10.˝//

2 with the norm k�k ; and theassociated scalar product ..�; �//; given by

.ru;rv/ D ..u; v// D2X

i;jD1

Z

˝

@uj

@xi

@vj

@xidx for u; v 2 .H1

0.˝//2:

It follows that V H � H0 V 0; where the injections are continuous and dense.Finally, we will use k�kV0

for the norm in V 0 and h�; �i for the duality pairingbetween V and V 0:

Let A W V ! V 0 be given by hAu; vi D ..u; v// for u; v 2 V , and consider thetrilinear form b on V � V � V given by

b.u; v;w/ D2X

i;jD1

Z

˝

ui@vj

@xiwj dx for u; v;w 2 V:

Define B W V � V ! V 0 by hB.u; v/;wi D b.u; v;w/; for u; v;w 2 V; and denoteB.u/ D B.u; u/: Assume now that u0 2 H, f 2 L2loc.RI V 0/. For each � 2 R weconsider the problem

8ˆˆ<

ˆˆ:

u 2 L2.�;TI V/ \ L1.�;TI H/ for all T > �;d

dt.u.t/; v/C �..u.t/; v//C hB.u.t//; vi D hf .t/; vi for all v 2 V

in the sense of scalar distributions on .�;C1/;

u.�/ D u0:(12.1)

It follows from the results of Chap. 7, cf. Theorem 7.2 and Remark 7.3, that problem(12.1) has a unique solution, u.�I �; u0/, that moreover belongs to C.Œ�;T�I H/ forall T � � .

Now we formulate a result about the existence of the pullback attractor for theabove Navier–Stokes problem. We define the evolutionary process associated to(12.1) as follows. Let us consider the unique solution u.�I �; u0/ to (12.1). We set

U.t; �/u0 D u.tI �; u0/ for � � t and u0 2 H: (12.2)

By the uniqueness of the solution we have

U.t; �/u0 D U.t; r/U.r; �/u0 for all � � r � t and u0 2 H:

For all � � t; the mapping U.�; t/ W H ! H defined by (12.2) is continuous (cf.Theorem 7.2 and Remark 7.3). Thus, the family fU.t; �/g��t defined by (12.2) is aprocess U.�; �/ in H.

Let � D ��1 and R� be the set of all functions r W R ! .0;C1/ such that

limt!�1 e� tr2.t/ D 0;


and denote by D� the class of all non-autonomous sets D D fD.t/gt2R with D.t/ 2P.H/ for all t 2 R such that D.t/ B.0; rD.t//; for some rD 2 R� ; whereB.0; rD.t// denotes the closed ball in H centered at zero with radius rD.t/. Then wehave (see Chap. 11).

Theorem 12.1. Suppose that f 2 L2loc.RI V 0/ is such that

Z t

�1e��kf .�/k2V0

d� < C1 for all t 2 R: (12.3)

Then, the process U.�; �/ associated with problem (12.1) is D� -asymptoticallycompact and there exists B 2 D� which is pullback D� -absorbing for U.t; �/. Inconsequence, there exists a unique pullback D� -attractor A D fA.t/gt2R belongingto D� for the process U.�; �/ in H.

The choice of the attraction universe D� comes here naturally from the first energyinequality (cf. Chap. 11)

ju.t/j2 � e��.t��/ju.�/j2 C 1

�e�� t

Z t

�

e��kf .�/k2V0

d� for � � t:

If u.�/ 2 D.�/ B.0; rD.�//, rD 2 R� , then the first term on the right-hand sideconverges to zero as � ! �1, and

ju.t/j2 � 2

�e�� t

Z t

�1e��kf .�/k2V0

d� � R.t/ for all � � �0.t;D/: (12.4)

Thus U.t; �/D.�/ B.t/, where B.t/ is the ball B.0;R.t// in H. By (12.3), the non-autonomous set B D fB.t/gt2R belongs to D� and is D� -absorbing for the processU.t; �/. The uniqueness follows immediately from the fact that A 2 D� and fromthe attracting property.

Let us assume that the forcing f is time-independent, with f 2 V 0. Then, fromTheorem 12.1 there follows the existence of the global attractor A H for thesemigroup fS.t/gt�0 given by S.t/ D U.� C t; �/, associated with the relatedautonomous Navier–Stokes problem (cf. Chap. 7).

Theorem 12.2. There exists a unique global attractor for the semigroup fS.t/gt�0in H associated with the autonomous Navier–Stokes problem, i.e., problem (12.1)with f 2 V 0.

Actually, the family B.H/ of all bounded sets in H constitutes, for this case, theattraction universe of non-autonomous (and, in this case, constant in time) setsD from Definition 12.2. From (12.4) we conclude that the ball B.0;R/ in H withR D 2=.��/kf kV0 absorbs all bounded sets in H, i.e., S.t/B B.0;R/ for anybounded set B 2 B.H/ and all t � t0.B/. The semigroup fS.t/gt�0 is B.H/-asymptotically compact, that is, for any sequence tn ! 1 and any sequence xn

bounded in H the set fS.tn/xngn2N is relatively compact in H. Moreover, A D!.B.0;R//—the !-limit set of the ball B.0;R/.

12.2 Construction of the Family of Probability Measures 281

12.2 Construction of the Family of Probability Measures

Our main result of this section (Theorem 12.3) concerns the construction of afamily ftgt2R of time averaged probability measures in the phase space H andtheir relation to the pullback attractor fA.t/gt2R. Then (Lemma 12.3) we show howany two measures of the family are related to each other.

In Sects. 12.2–12.4 we work with somewhat stronger assumptions on the domainof the flow ˝ and on the forcing f . Namely, we assume that ˝ is an open boundedset in R

2 of class C2 and that f 2 L2loc.RI H/, with

Z t

�1e�� jf .�/j2d� < 1 (12.5)

for all t 2 R.

Lemma 12.1. Let ˝ be an open bounded set in R2 of class C2, f 2 L2loc.RI H/

satisfy (12.5), and let fU.t; �/g��t be the evolutionary process associated withproblem (12.1). Then, for every t 2 R there exists a compact set B.t/ absorbingfor fU.t; �/g��t.

Proof. The proof is based on the uniform Gronwall lemma (see Chap. 3) applied tothe second energy inequality (see Chap. 7 and also Chap. III, Sect. 2 in [220])

d

dtkuk2 C �kuk2 � 2

�jf j2 C c

�3juj2kuk4:

As usual, we use the first energy inequality (see Chap. 7)

d

dtjuj2 C �kuk2 � 1

�jf j2 (12.6)

and its consequence

ju.t/j2 � e��.t��/ju.�/j2 C e�� t

�

Z t

�1e�� jf .�/j2d�; (12.7)

together with (12.5) to obtain a bound on the norm ku.t/k. However, the bound onthe forcing (12.5) is not uniform in t 2 R, and this implies that our estimates arelocal in time. More precisely, we shall prove that for any u.�/ 2 D.�/ for D.�/in any non-autonomous set D D fD.t/gt2R 2 D , any fixed t 2 R, and T , r, with0 < r < T , there exists M.t;T; r/ such that

kU.�; �/u.�/k � M.t;T; r/ for all � 2 Œt � T C r; t� and � � �0.t;T;D/:(12.8)

Since ˝ is a bounded set, V is compactly embedded in H, and (12.8) implies thatthe closed ball B.t/ D BV.0;M.t;T; r// in V is a compact absorbing set for U.t; �/.


First, from (12.7) and (12.5) it follows that for all � 2 .t � T; t/,

jU.�; �/u.�/j2 � 2e�T

�

Z t

�1e�� jf .�/j2d� � M1.t;T/ for all � � �0.t;T;D/:

(12.9)

Since f 2 L2loc.RI H/, we have for all s 2 .t � T; t � r/ and some M2.t;T; r/ < 1,

1

�

Z sCr

sjf .�/j2d� � M2.t;T; r/: (12.10)

Now, from (12.6) together with (12.9) and (12.10) we have for all s 2 .t � T; t � r/and some M3.t;T; r/ < 1,

�

Z sCr

sku.�/k2d� � M3.t;T; r/: (12.11)

In the end, from (12.9) and (12.11) we obtain for all s 2 .t � T; t � r/ and someM4.t;T; r/ < 1,

c

�3

Z sCr

sju.�/j2ku.�/k2d� � M4.t;T; r/:

The last three estimates and the uniform Gronwall lemma give (12.8). utBelow we recall the notion of a Banach generalized limit introduced in Chap. 8.

Since the existence of a generalized limit follows from the Hahn–Banach theoremwhich does not guarantee uniqueness, in the sequel we assume implicitly that wework with any given generalized limit.

Definition 12.4. By a Banach generalized limit we name any linear functional,denoted LIMT!1, defined on the space of all bounded real-valued functions onŒ0;1/ and satisfying

(i) LIMT!1g.T/ � 0 for nonnegative functions g.(ii) LIMT!1g.T/ D limT!1g.T/ if the usual limit limT!1g.T/ exists.

Theorem 12.3. Let ˝ be an open bounded set in R2 of class C2 and f 2

L2loc.RI H/ satisfy (12.5). Let fU.t; �/gt�� be the evolutionary process associatedwith the Navier–Stokes system (12.1) and fA.t/gt2R be the pullback attractor fromTheorem 12.1. Let u0 be an arbitrary element of the phase space H. Then there existsa family of Borel probability measures ftgt2R in the phase space H such that thesupport of measure t is contained in A.t/ and

LIM�!�11

t � �Z t

�

'.U.t; s/u0/ds DZ

A.t/'.u/dt.u/ (12.12)

for all real-valued and continuous on H functionals ', (' 2 C.H/).

12.2 Construction of the Family of Probability Measures 283

Proof. First we prove a lemma.

Lemma 12.2. For u0 2 H and t 2 R the function

s ! '.U.t; s/u0/ (12.13)

is continuous and bounded on .�1; t�.

Proof. We first prove that for any � > 0 there exists a ı > 0 such that if jr � sj < ıthen jU.t; r/u0 � U.t; s/u0j < �. We may assume that r < s.

It is easy to check (cf. Chap. 7) that if u and v are two solutions of the Navier–Stokes equations in a domain satisfying the Poincaré inequality then

ju.t/ � v.t/j2 � ju.0/ � v.0/j2 exp

�

CZ t

0

ku.s/k2 ds

�

:

We have thus,

jU.t; r/u0 � U.t; s/u0j2 D jU.t; s/U.s; r/u0 � U.t; s/u0j2

� jU.s; r/u0 � u0j2 exp

�

CZ t

skU.�; s/u0k2 d�

�

:

From the continuity of the function r � � ! U.�; r/u0 with values in H it followsthat if jr � sj < ı is small enough then the right-hand side of the above inequalityis as small as needed. Since ' 2 C.H/ we conclude that the function in (12.13) iscontinuous.

As to its boundedness we know that there exists a compact absorbing set B1.t/for fU.t; �/g, in particular, there exists �0 such that for s � �0, U.t; s/u0 2 B1.t/.Hence, the function s ! '.U.t; s/u0/ is bounded on .�1; �0�. It is also boundedon the compact interval Œ�0; t�. utWe continue the proof of the theorem. Since the function in (12.13) is continuousand bounded, the left-hand side of (12.12) is well defined.

We have

LIM�!�11

t � �Z t

�

'.U.t; s/u0/ds D LIM�!�11

t � �Z �0

�

'.U.t; s/u0/ds:

From this equality it follows that the left-hand side of (12.12) depends only on thevalues of ' on the compact set B1.t/. By the Tietze extension theorem, the left-hand side of (12.12) defines a linear positive functional on C.B1.t// and from theKakutani–Riesz representation theorem it follows that there exists a measure 1t onB1.t/ defined on some � -algebra M in B1.t/ which contains all Borel sets in B1.t/and such that

LIM�!�11

t � �Z t

�

'.U.t; s/u0/ds DZ

B1.t/'.u/d1t .u/:

We can extend this measure easily to all Borel sets in H without enlarging itssupport.


We shall show that the support of the measure 1t is contained in A.t/. Let Bn.t/,n D 1; 2; : : :, be a sequence of compact absorbing sets such that Bn.t/ BnC1.t/ and\1

nD1Bn.t/ D A.t/. From the above construction, for every n there exists a measuren

t of support contained in Bn.t/ and such that

LIM�!�11

t � �Z t

�

'.U.t; s/u0/ds DZ

Bn.t/'.u/dn

t .u/

holds. As the left-hand side of this equation is independent of n, it follows that for' 2 C.H/,

Z

B1.t/'.u/d1t .u/ D

Z

Bn.t/'.u/dn

t .u/: (12.14)

By the Tietze extension theorem, every � 2 C.Bn.t// is of the form 'jBn.t/ for some' 2 C.B1.t//. For any 2 C0.B1.t/ n Bn.t// both ' and ' C are continuousextensions of �. It then follows that

Z

B1.t/nBn.t/ .u/d1t .u/ D 0:

For any compact K in B1.t/ n Bn.t/ there exist 2 C0.B1.t/ n Bn.t// such that � �K on B1.t/ n Bn.t/ where �K is the characteristic function of the set K. Wehave

0 DZ

B1.t/nBn.t/ .u/d1t .u/ �

Z

B1.t/nBn.t/�K.u/d

1t .u/ D 1t .K/ � 0;

whence 1t .K/ D 0 for every compact set K in B1.t/ n Bn.t/. From the regularity of1t it follows that 1t .B1.t/ n Bn.t// D 0. Therefore, the support of 1t is containedin Bn.t/. From (12.14) and the Tietze extension theorem we conclude that

Z

Bn.t/'.u/d1t .u/ D

Z

Bn.t/'.u/dn

t .u/

for all continuous functions ' on Bn.t/. By the regularity of measures 1t and nt , the

above equation holds for all1t - andnt -integrable functions. Thus1t D n

t on Bn.t/and on the whole phase space H. In the end, the support of 1t is contained in everyset Bn.t/ with n � 1 which implies that it is contained in \1

nD1Bn.t/ D A.t/. Theresulting measure with support in A.t/ is denoted by t. Setting ' D 1 in (12.12) weobtain t.A.t// D 1, and so t is a probability measure. The proof of the theoremis complete. ut

12.3 Liouville and Energy Equations 285

Lemma 12.3. The measures ftgt2R satisfy the following relations:

t.E/ D �.U.t; �/�1E/ for all t; � 2 R such that t � � (12.15)

for every Borel set E in the phase space H.

Proof. We have for t � � , and for all functionals ' 2 C.H/,

Z

H'.u/dt.u/ D LIMM!�1

1

t � M

Z t

M'.U.t; s/u0/ds

D LIMM!�11

t � M

Z �

M'.U.t; s/u0/ds

CLIMM!�11

t � M

Z t

�

'.U.t; s/u0/ds

D LIMM!�11

� � M

Z �

M'.U.t; �/U.�; s/u0/ds

DZ

H'.U.t; �/u/d�.u/;

whenceZ

H'.u/dt.u/ D

Z

H'.U.t; �/u/d�.u/:

Using the regularity of t, the Urysohn lemma, and a suitable approximationargument we can extend this relation to all t-integrable functions ' in H and thenit becomes equivalent to the generalized invariance property (12.15). ut

12.3 Liouville and Energy Equations

In this section we prove that the family of probability measures constructed inSect. 12.2 satisfies the Liouville equation for a suitable family of generalizedmoments and also the energy equation.

Definition 12.5 (Cf. [96, 99]). By T we denote the class of cylindrical testfunctionals consisting of the real-valued functionals ˚ D ˚.u/ depending onlyon a finite number k of components of u, namely,

˚.u/ D �..u; g1/; : : :; .u; gk//;

where � is a compactly supported C1 scalar function on Rk and where g1; : : :; gk

belong to V .


Let ˚ 0 denote the differential of ˚ in H, given as

˚ 0.u/ DkX

jD1@j�..u; g1/; : : :; .u; gk//gj; (12.16)

where @j�, j D 1; : : :; k, are partial derivatives of � on Rk. From (12.16) it follows

that ˚ 0.u/ 2 V .

Theorem 12.4. The family of measures ftgt2R satisfies the following Liouvilleequation:

Z

A.t/˚.u/dt.u/ �

Z

A.�/˚.u/d�.u/

DZ t

�

Z

A.s/.F.u; s/; ˚ 0.u//ds.u/ds; (12.17)

for all t � � , where F.u; s/ D f .s/ � �Au � B.u/, and ˚ 2 T .

Proof. Let u.t/ be a solution of the Navier–Stokes system

du

dtC �Au C B.u/ D f .t/ and u.t/jtDs D u0 where s 2 R:

Since u 2 L2.�; tI D.A// for s < � < t, it is not difficult to check that the functiont ! .f .t/ � �Au.t/ � B.u.t//; ˚ 0.u.t/// is integrable on .�; t/. From the Navier–Stokes equation it follows that ut 2 L1.�; t;H/. Thus for ˚ 2 T we have

d

dt˚.u.t// D .ut.t/; ˚

0.u.t///

D .f .t/ � �Au � B.u/; ˚ 0.u.t/// D .F.u.t/; t/; ˚ 0.u.t///

almost everywhere in the considered interval. Integrating in t we obtain

˚.u.t// � ˚.u.�// DZ t

�

.F.u.�/; �/; ˚ 0.u.�///d�:

For an arbitrary s < � , let u0 2 H and u.�/ D U.�; s/u0 for � � s. Then, for theabove solution we have

˚.U.t; s/u0/ � ˚.U.�; s/u0/ DZ t

�

.F.U.�; s/u0; �/; ˚0.U.�; s/u0//d�:

12.3 Liouville and Energy Equations 287

Applying the operator

LIMM!�11

� � M

Z �

M.: : :/ds

for � < � we obtain

LIMM!�11

� � M

Z �

M˚.U.t; s/u0/ds � LIMM!�1

1

� � M

Z �

M˚.U.�; s/u0/ds

D LIMM!�11

� � M

Z �

M

Z t

�

.F.U.�; s/u0; �/; ˚0.U.�; s/u0//d�ds: (12.18)

Now, in view of (12.15), (cf. also Chap. II in [228]) the left-hand side equals

LIMM!�11

� � M

�Z �

M˚.U.t; �/U.�; s/u0/ds �

Z �

M˚.U.�; �/U.�; s/u0/ds

�

DZ

˚.U.t; �/u/d�.u/ �Z

˚.U.�; �/u/d�.u/

DZ

˚.u/dt.u/ �Z

˚.u/d�.u/:

Similarly, the right-hand side of (12.18) equals

LIMM!�11

� � M

Z �

M

Z t

�

.F.U.�; s/u0; �/; ˚0.U.�; s/u0//d�ds

D LIMM!�11

� � M

Z t

�

�Z �

M.F.U.�; s/u0; �/; ˚

0.U.�; s/u0//ds

�

d�

DZ t

�

�

LIMM!�11

� � M

Z �

M.F.U.�; s/u0; �/; ˚

0.U.�; s/u0//ds

�

d�:

For � 2 .�; t/ we have

LIMM!�11

� � M

Z �

M.F.U.�; s/u0; �/; ˚

0.U.�; s/u0//ds

D LIMM!�11

� � M

Z �

M.F.U.�; �/U.�; s/u0; �/; ˚

0.U.�; �/U.�; s/u0//ds

DZ

H.F.U.�; �/u; �/; ˚ 0.U.�; �/u//d�.u/

DZ

H.F.u; �/; ˚ 0.u//d�.u/;


whence

LIMM!�11

� � M

Z �

M

Z t

�

.F.U.�; s/u0; �/; ˚0.U.�; s/u0//d�ds

DZ t

�

Z

H.F.u; �/; ˚ 0.u//d�.u/ d�:

We have used the integrability of the function

.t; �/ � .M; �/ 3 .�; s/ ! .F.U.�; s/u0; �/; ˚0.U.�; s/u0//

for � < � , the Fubini theorem, and the linearity of the LIMM!�1 operator.In the end, since for every t in R the support of the measure t is contained in

A.t/, we arrive at (12.17). utTheorem 12.5. The family of measures ftgt2R satisfies the following energyequation:

Z

A.t/juj2dt.u/ C 2�

Z t

�

Z

A.s/kuk2dsds

D 2

Z t

�

Z

A.s/.f .s/; u/ds.u/ds C

Z

A.�/juj2d�.u/ (12.19)

for all t; � 2 R, t � � .

Proof. The functionals u ! ˚.u/ D juj2 and u ! kuk2 are bounded (evencontinuous in the topology of H) on every A.t/, t 2 R. Moreover, ˚ 0.u/ D 2u.From (12.17) with this ˚ and F.u; s/ D f .s/� �Au � B.u/ we obtain (12.19) by anapproximation argument. Let the functionals ˚m 2 T , m D 1; 2; 3; : : :, be such that˚m.u/ D jPmuj2 on some ball B.0;R/ in H containing all A.�/ for � � � � t. Here,Pm is the usual Galerkin projection operator in H onto the space spanned by thefirst m eigenmodes of the Stokes operator. The identity (12.19) thus holds for Pmuin the place of u. Letting m ! 1 and using the Lebesgue dominated convergencetheorem we obtain (12.19). ut

12.4 Time-Dependent and Stationary Statistical Solutions

A simple corollary from the above considerations is, that assuming any measure� of the family ftgt2R to be the initial distribution, the family ftgt�� forms astatistical solutions of the two-dimensional Navier–Stokes system (Theorem 12.6,cf. Chap. 5 in [99]) for this particular initial data. This solution is unique and itreduces to the stationary time-average statistical solution in the case the forcing

12.4 Time-Dependent and Stationary Statistical Solutions 289

does not depend on time. The support of this stationary solution is contained in thecorresponding global attractor (Theorem 12.7, cf. Chap. 8 and also Chap. 4 in [99]).

Theorem 12.6. Let fU.t; �/gt�� be the evolutionary process associated with theNavier–Stokes system (12.1) and let u0 be an arbitrary element of the phase spaceH. Denote by Q0 the time-average initial measure given by the relation

LIM�!�11

0 � �Z 0

�

'.U.0; s/u0/ds DZ

H'.u/d Q0.u/

for all functionals ' 2 C.H/. Then the support of Q0 is contained in the pullbackattractor’s section A.0/ and the kinetic energy of Q0 is finite

Z

Hjuj2d Q0.u/ < C1:

Moreover, the family of measures ftgt�0 given by

t.E/ D Q0.U.t; 0/�1E/

for all Borel sets in H, is the unique statistical solution of the two-dimensionalNavier–Stokes system (12.1) with initial distribution Q0, namely, it satisfies thefollowing conditions:

(S1) the Liouville equation

Z

A.t/˚.u/dt.u/ D

Z

A.0/˚.u/d Q0.u/C

Z t

0

Z

A.s/.F.u; s/; ˚ 0.u//ds.u/ds;

holds for all t � 0, F.u; s/ D f .s/ � �Au � B.u/, and the family of moments˚ 2 T ,

(S2) the energy equation

Z

A.t/juj2dt.u/ C 2�

Z t

0

Z

A.s/kuk2ds.u/ds

DZ

A.0/juj2d Q0.u/C 2

Z t

0

Z

A.s/.f .s/; u/ds.u/ds

holds for all t � 0,(S3) the function

t !Z

A.t/˚.u/dt.u/


is measurable on RC for every bounded and continuous real-valued functional

˚ on H. The function

t !Z

A.t/juj2dt.u/

is in L1.RC/. The function

t !Z

A.t/kuk2dt.u/

is in L1loc.RC/.

If the system (12.1) is autonomous, that is, the forcing f does not depend on timethen Theorem 12.6 reduces to the following one.

Theorem 12.7. Let fS.t/gt�0 be the semigroup associated with the autonomousNavier–Stokes system

du

dtC �Au C B.u/ D f 2 H with u.t/jtD0 D u0;

and let u0 be an arbitrary element of the phase space H. Then the time-averagemeasure given by the relation

LIM�!�11

0 � �Z 0

�

'.S.0 � s/u0/ds D LIM�!C11

�

Z �

0

'.S.s/u0/ds

DZ

H'.u/d.u/

for all functionals ' 2 C.H/ has support contained in the global attractor A .Moreover, it is a stationary statistical solution of the considered Navier–Stokessystem, namely, the following conditions hold:

(ST1)

Z

Hkuk2d.u/ < 1;

(ST2)

Z

A.F.u; s/; ˚ 0.u//d.u/ D 0;

where F.u; s/ D f .s/ � �Au � B.u/, and ˚ belongs to T,

12.5 The Case of an Unbounded Domain 291

(ST3)

Z

E1�juj2<E2

�kuk2 � .f ; u/� d � 0

for all E1;E2 such that 0 � E1 < E2 � 1.

12.5 The Case of an Unbounded Domain

In this section we do not assume that the domain of the flow is bounded. Inconsequence, we do not assume the existence of compact absorbing sets for theprocess U.t; �/, the assumption we needed in Sect. 12.2 to construct a family ofinvariant measures. Moreover, we show that we can obtain time-average invariantmeasures for U.t; �/ using as an initial condition any continuous mapping that iscontained in the domain of attraction of the pullback attractor. Thus we obtain atwofold generalization of Theorem 12.3.

We need the following notion of continuity of the process U.t; �/.

Definition 12.6. A process U.�; �/ is said to be � -continuous if for every u0 in X andevery t 2 R, the X-valued function

� 7! U.t; �/u0 (12.20)

is continuous and bounded on .�1; t�.

Our results for the Navier–Stokes equations in certain unbounded domains are directconsequences of the following abstract theorem:

Theorem 12.8. Let U.�; �/ be a � -continuous evolutionary process in a completemetric space X that has a pullback D-attractor A D fA.t/gt2R. Fix a generalizedBanach limit LIMT!1 and let u W R ! X be a continuous map such that u.�/ 2 D .Then there exists a unique family of Borel probability measures ftgt2R in X suchthat the support of the measure t is contained in A.t/ and

LIM�!�11

t � �Z t

�

'.U.t; s/u.s// ds DZ

A.t/'.v/ dt.v/ (12.21)

for any real-valued continuous functional ' on X. In addition, t is invariant in thesense that

Z

A.t/'.v/ dt.v/ D

Z

A.�/'.U.t; �/v/ d�.v/ for t � �: (12.22)

Remark 12.1. Theorem 12.8 holds for any pullback attractor A D fA.t/gt2R.Assume that there exists a minimal pullback attractor Am D fAm.t/gt2R for the


process U.�; �/. Then the support of measure t from Theorem 12.8 is contained inAm.t/ and in formulas (12.21), (12.22) one can replace A.t/ by Am.t/.

In the proof of the theorem we will use the following lemma, cf. [197]:

Lemma 12.4. Let .X; �/ be a metric space, f W X ! R a continuous mapping, andlet K X be compact. Then for every � > 0 there exists a ı D ıf .�/ > 0 such thatif u 2 X, v 2 K, and �.u; v/ � ı then jf .u/ � f .v/j � �.

Proof (of Theorem 12.8). Fix u.�/ 2 D and ' 2 C.X/. In view of the existenceof the pullback attractor, the � -continuity of the process U.t; �/ and Lemma 12.4,the function � 7! 1

t��R t�'.U.t; s/u.s// ds is bounded on the interval .�1; t�. It

is continuous by an application of the dominated convergence theorem and theLebesgue differential theorem.

It suffices to prove that the function s 7! '.U.t; s/u.s// is bounded on .�1; t�:it is continuous, since U.�; �/ is � -continuous and u.�/ is continuous, and so it isbounded on every compact interval Œt0; t�. We now show that it is bounded on .1; t0�for t0 sufficiently large and negative.

If this is not the case then there is a sequence fsng with sn ! �1 such that

j'.U.t; sn/u.sn//j ! 1: (12.23)

However, from the existence of the pullback D-attractor it follows that the processU.�; �/ is pullback D-asymptotically compact, so there exists a subsequence of fsng(which we relabel) such that U.t; sk/u.sk/ ! ! for some ! 2 X (since u.�/ 2 D).From the continuity of ', we have '.U.t; sk/u.sk// ! '.!/, contradicting (12.23).

We now define

L.'/ D LIM�!�11

t � �Z t

�

'.U.t; s/u.s// ds; (12.24)

and claim that L.'/ depends only on the values of ' on A.t/. Indeed, take functions'; 2 C.X/ with ' D on A.t/, and given � > 0 use Lemma 12.4 to find a ı suchthat

�.u; v/ � ı ) j'.u/ � '.v/j C j .u/ � .v/j � � for every v 2 A.t/:(12.25)

Now let �0 be such that

distX.U.t; s/u.s/;A.t// � ı for all s � �0:

Then for every s � �0 there exists a vs 2 A.t/ such that

�.U.t; s/u.s/; vs/ � ı;

12.5 The Case of an Unbounded Domain 293

and so

j'.U.t; s/u.s// � .U.t; s/u.s//j� j'.U.t; s/u.s// � '.vs/j C j'.vs/ � .vs/j

C j .vs/ � .U.t; s/u.s//j � �;

using (12.25) and the fact that '.vs/ D .vs/. Thus

jL.' � /j DˇˇˇˇLIM�!�1

1

t � �Z t

�

.'.U.t; s/u.s// � .U.t; s/u.s/// ds

ˇˇˇˇ

DˇˇˇˇLIM�!�1

1

t � ��Z �0

�

CZ t

�0

�

.'.U.t; s/u.s// � .U.t; s/u.s/// ds

ˇˇˇˇ

� lim sup�!�1

.�0 � �/�t � �

C lim sup�!�1

.t � �0/t � � sup

s2Œ�0;t�fj'.U.t; s/u.s//j C j .U.t; s/u.s//jg � �:

Since � > 0 was arbitrary we conclude that L.'/ depends only on the values of 'on A.t/.

Define G.'/ D L.l.'// for ' 2 C.A.t//, where l.'/ is an extension of ' givenby the Tietze theorem. As G is a linear and positive functional on C.A.t//, we have,by the Kakutani–Riesz representation theorem,

G.'/ DZ

A.t/'.v/ dt.v/:

We extend t (by zero) to a Borel measure on X, which we denote again by t,whence for every ' 2 C.X/,

Z

A.t/'.v/ dt.v/ D

Z

X'.v/ dt.v/;

and thus (12.21) holds.To prove (12.22), observe that for t � � , and for all functionals ' 2 C.X/,

Z

X'.v/ dt.v/ D LIMM!�1

1

t � M

Z t

M'.U.t; s/u.s// ds

D LIMM!�11

t � M

Z �

M'.U.t; s/u.s// ds

C LIMM!�11

t � M

Z t

�

'.U.t; s/u.s// ds


D LIMM!�11

� � M

Z �

M'.U.t; �/U.�; s/u.s// ds

D LIMM!�11

� � M

Z �

MŒ' ı U.t; �/�fU.�; s/u.s/g ds

DZ

XŒ' ı U.t; �/�.v/ d�.v/ D

Z

X'.U.t; �/v/ d�.v/;

where we have used the fact that ' ı U.t; �/ 2 C.X/. utNow, we can come back to the Navier–Stokes equations. We define the evolu-

tionary process associated to (12.1) and the attraction universe D� exactly as inSect. 12.1. We have already proved the existence of a unique pullback D� -attractorA D fA.t/gt2R belonging to D� for the process U.t; �/ in H (cf. Theorem 12.1). ByLemma 12.2 the process U.t; �/ is � -continuous. Thus, the existence of a family ofinvariant measures t for U.t; �/ follows from Theorem 12.8. Namely, we have thefollowing result:

Theorem 12.9. Let˝ R2 be an open, bounded or unbounded, set such that there

exists a constant �1 > 0 for which

�1

Z

˝

�2 dx �Z

˝

jr�j2 dx for all � 2 H10.˝/:

Suppose that f 2 L2loc.RI V 0/ is such that

Z t

�1"��kf .�/k2V0

d� < C1 for all t 2 R:

Let fU.t; �/gt�� be the evolutionary process associated with the Navier–Stokessystem (12.1). Fix a generalized Banach limit LIMT!1 and let u W R ! X be acontinuous map such that u.�/ 2 D� . Then there exists a unique family of Borelprobability measures ftgt2R in the phase space H such that the support of measuret is contained in A.t/ and

LIM�!�11

t � �Z t

�

'.U.t; s/u.s//ds DZ

A.t/'.u/dt.u/

for any real-valued continuous functional ' on H. In addition, t is invariant in thesense that

Z

A.t/'.v/ dt.v/ D

Z

A.�/'.U.t; �/v/ d�.v/ for t � �:



Theorem 12.1 was proved in [45]. Theorem 12.2 was proved first in [201], cf.also [220]. Theorem 12.8 generalizes Theorem 3.3 from [160] (for the non-autonomous two-dimensional Navier–Stokes equations, using compactness prop-erty of those equations) and Theorem 5 from [165] (for autonomous problems,but under weaker assumptions on the dynamical system). Lemma 12.4 comes from[197] stated there as Exercise 10.1, see also Lemma 2 in [165] and Lemma 3.1 in[56]. Theorem 12.9 was proved in [163].

13Pullback Attractors and Shear Flows

In this chapter we consider the problem of existence and finite dimensionality of thepullback attractor for a class of two-dimensional turbulent boundary driven flowswhich naturally appear in lubrication theory. We generalize here the results fromChap. 9 to the non-autonomous problem.

The new element in our study with respect to that in Chap. 9 is the allowance ofthe speed of rotation of the cylinder to depend on time. Our aim is first to prove theexistence of the unique global in time solution of the problem, and then to study itstime asymptotics in the frame of the dynamical systems theory. We use the theory ofpullback attractors that allows us to pose quite general assumptions on the velocityof the boundary. Neither quasi-periodicity nor even boundedness is demanded toprove the existence and to estimate the fractal dimension of the correspondingpullback attractor. We shall apply the results from Chap. 11, reformulated here inthe language of evolutionary processes.

13.1 Preliminaries

In this section we recall a theorem about the existence of pullback attractors forevolutionary processes.

Let us consider an evolutionary process U on a metric space X, i.e., a familyfU.t; �/g�1<��t<C1 of continuous mappings U.t; �/ W X ! X, such that for x 2 Xand � 2 R U.�; �/x D x; and

U.t; �/ D U.t; r/U.r; �/ for all � � r � t:

By P.X/ we denote the family of all nonempty subsets of X. Let D be an attractionuniverse, i.e., a nonempty class of non-autonomous sets D D fD.t/gt2R such thatD.t/ 2 P.X/ for every t 2 R.

We have the following result, cf. Theorems 11.4 and 11.5 in Chap. 11.


297

298 13 Pullback Attractors and Shear Flows

Theorem 13.1. Suppose that the process U.t; �/ is pullback D-asymptoticallycompact, and that there exists a non-autonomous pullback D-absorbing set forU.�; �/ belonging to the universe D . Then there exists a minimal pullback D-attractor A D fA.t/gt2R for U.�; �/.Remark 13.1. Observe that there exists at most one pullback D-attractor A 2 D . Infact, if A1 and A2 are two global pullback D-attractors belonging to D , then by theinvariance property,

distX.A1.t/;A2.t// D distX.U.t; �/A1.�/;A2.t//:

But then, as A1 2 D and A2 is a global pullback D-attractor,

distX.A1.t/;A2.t// D lim�!�1 distX.U.t; �/A1.�/;A2.t// D 0;

and therefore

A1.t/ A2.t/ D A2.t/:

Analogously, one obtains A2.t/ A1.t/.

Remark 13.2. It is said that the attraction universe D is inclusion-closed if, giventhe non-autonomous set D 2 D , for any D

0 D fD0.t/gt2R with D0.t/ 2 P.X/ fort 2 R, such that D0.t/ D.t/ for all t, one also has D0 2 D :

Taking into account that �.B; t/ B.t/; and Remark 13.1, we can assert thatif the universe D is inclusion-closed and the sets B.t/ are closed, then the pullbackD-attractor A constructed in Theorem 13.1 belongs to the universe D , and is theunique pullback D-attractor for U.�; �/ belonging to this universe.

The rest of this chapter is organized as follows. In Sect. 13.2 we give a preciseformulation of the considered problem and write it in a weak form, suitable forour further considerations. In Sect. 13.3 we derive the first energy inequality andestablish the existence of unique global in time solution of the problem. In Sect. 13.4we prove the existence of a pullback attractor for the corresponding evolutionaryprocess by using the energy equation method. In Sect. 13.5 we obtain an upperbound of the pullback attractor dimension in terms of the data.

13.2 Formulation of the Problem

We consider two-dimensional Navier–Stokes equations,

ut � ��u C .u � r/u C rp D 0; (13.1)

div u D 0; (13.2)

13.2 Formulation of the Problem 299

in the channel

˝1 D fx D .x1; x2/ W �1 < x1 < 1; 0 < x2 < h.x1/g;

where h is a positive, smooth, and L-periodic function in x1.Let

˝ D fx D .x1; x2/ W 0 < x1 < L; 0 < x2 < h.x1/g;

and @˝ D �C [ �L [ �D, where �C and �D are the bottom and the top, and �L isthe lateral part of the boundary of ˝.

We are interested in solutions of (13.1) and (13.2) in ˝ which are L-periodicwith respect to x1, or, to be more precise, analogously to Chap. 10, both the velocityu and the Cauchy stress vector Tn on �L are L-periodic with respect to x1. Moreover,we assume

u D 0 on �D; (13.3)

u D U0.t/e1 D .U0.t/; 0/ on �C; (13.4)

where U0.t/ is a locally Lipschitz continuous function of time t. Finally, the initialcondition at some time � 2 R is given as

u.x; �/ D u0.x/ for x 2 ˝: (13.5)

The problem is motivated by a flow in an infinite (rectified) journal bearing˝ � .�1;C1/, where �D � .�1;C1/ represents the outer cylinder, and theinner, rotating cylinder is represented by �C � .�1;C1/. We can assume thatthe rectification does not change the equations as the gap between cylinders is verysmall with respect to their radii.

In our considerations we use the background flow method and transform theboundary condition (13.4) by defining a smooth background flow, a simple versionof the Hopf construction, described in detail in Sect. 13.3.

Let

u.x1; x2; t/ D U.x2; t/e1 C v.x1; x2; t/;

with

U.0; t/ D U0.t/ and U.h.x1/; t/ D 0 for x1 2 .0;L/ and t 2 R:

(13.6)


Then v is L-periodic in x1 and satisfies

vt � ��v C .v � r/v C rp D G on ˝ � .�;1/; (13.7)

div v D 0 on ˝ � .�;1/; (13.8)

v D 0 on .�D [ �C/ � .�;1/; (13.9)

where G D �U;x2x2 e1 � U;t e1 � Uv;x1 �.v/2U;x2 e1, and the initial condition

v.x; �/ D v0.x/ D u0.x/ � U.x2; �/e1 in ˝: (13.10)

By .v/2 we denote the second component of v.Now, we define a weak form of the above problem. To this end we need some

notations. Let

QV D fv 2 C1.˝/2 W v is L-periodic in x1; div v D 0; v D 0 on �D [ �Cg;

V D closure of QV in H1.˝/2;

H D closure of QV in L2.˝/2:

We define the scalar products

.u; v/ DZ

˝

u.x/ � v.x/dx;

and

..u; v// D .ru;rv/ DZ

˝

ru.x/W rv.x/ dx

in H and V , respectively, and the norms

jvj D .v; v/12 and kvk D ..v; v//

12 :

Let

b.u; v;w/ D ..u � r/v;w/;

and

a.u; v/ D .ru;rv/:

Then the natural weak formulation of the problem (13.7)–(13.10) is as follows.

13.3 Existence and Uniqueness of Global in Time Solutions 301

Problem 13.1. Find the velocity v such that for each T > � ,

v 2 C.Œ�;T�I H/ \ L2.�;TI V/;

and for all � 2 V and a.e. t > � ,

d

dt.v.t/;�/C �a.v.t/;�/C b.v.t/; v.t/;�/ D F.v.t/;�/; (13.11)

with

v.x; �/ D v0.x/ on ˝;

where

F.v;�/ D ��a.�;�/ � b.�; v;�/ � b.v; �;�/ � .�;t ; �/; (13.12)

and � D Ue1 is a suitable background flow.

In the next section we shall derive the first energy inequality and establish theexistence result for the above problem.

13.3 Existence and Uniqueness of Global in Time Solutions

To study the pullback attractor associated with a particular problem we first need toknow that the latter has a unique global in time solution. In our particular case wehave the following existence theorem.

Theorem 13.2. Let U0 be a locally Lipschitz continuous function on the real line.Then there exists a unique weak solution of Problem 13.1 such that for all �, T,� < � < T we have v 2 L2.�;T I H2.˝// and for each t > � the map v0 7! v.t/ iscontinuous as a map in H.

Proof. We shall start the proof from the derivation of the first energy estimate forthe solutions, namely, estimate (13.25) below.

Taking � D v in (13.11), we obtain

1

2

d

dtjvj2 C �a.v; v/C b.v; v; v/ D F.v; v/:

Since v D 0 on �D[�C, v is L-periodic in x1, and div v D 0, we have b.v; v; v/ D 0,and

1

2

d

dtjvj2 C �kvk2 D F.v; v/: (13.13)


Consider the term F.v; v/ on the right-hand side of (13.13). By (13.12) we have

F.v; v/ D ��a.�; v/ � b.v; �; v/ � .�;t ; v/: (13.14)

We have

j�a.�; v/j � �k�kkvk � 2�k�k2 C �

8kvk2: (13.15)

To estimate the second term in (13.14) we need the following lemma.

Lemma 13.1. For any t 2 R there exists a smooth extension

� D �.x2; t/ D U.x2; t/e1 D .U.x2; t/; 0/

of the boundary condition for u, such that

jb.v; �.t/; v/j � �

4kvk2 for all v 2 V: (13.16)

Proof. Let � W Œ0;1/ ! Œ0; 1� be a smooth function such that

�.0/ D 1 and supp � Œ0; 1=2�; and max j�0.s/j � p8:

Let h0 D min0�x1�L h.x1/, and

" D ".t/ D�2 if jU0.t/j � �=.8h0/;�=.4h0jU0.t/j/ if jU0.t/j � �=.8h0/:

(13.17)

Thus, 0 < ".t/ � 2. Define

U.x2; t/ D U0.t/�.x2=.h0".t///: (13.18)

Then U matches the boundary conditions (13.6), is L-periodic in x1, anddiv Ue1 D 0. Moreover, for Q" D .0;L/ � .0; h0"=2/,

jb.v;Ue1; v/j D jb.v; v;Ue1/j �ˇˇˇˇv

x2

ˇˇˇˇL2.Q"/

jrvjL2.Q"/jx2U.x2; t/jL1.Q"/:

Now,

jx2U.x2; t/jL1.Q"/ � h0"

2jU0j;

13.3 Existence and Uniqueness of Global in Time Solutions 303

and

ˇˇˇˇv

x2

ˇˇˇˇ

2

L2.Q"/

�Z L

0

Z h.x1/

0

ˇˇˇˇv.x1; x2/

x2

ˇˇˇˇ

2

dx2dx1 � 4

Z L

0

Z h.x1/

0

ˇˇˇˇ@v.x1; x2/

@x2

ˇˇˇˇ

2

dx2dx1

by the Hardy inequality. Thus, we obtain (13.16). utWe estimate the last term on the right-hand side of (13.14) using the Poincaré

inequality

jvj � 1p�1

kvk for v 2 V; (13.19)

where we can take 1=p�1 D hM=

p2 with hM D max0�x1�L h.x1/. We get

j.�;t ; v/j � j�;t jjvj � j�;t j 1p�1

kvk � 2

��1j�;t j2 C �

8kvk2: (13.20)

In view of estimates (13.15), (13.16), and (13.20),

jF.v; v/j � 2�k�k2 C 2

��1j�;t j2 C �

2kvk2;

and by (13.13),

1

2

d

dtjvj2 C �

2kvk2 � 2�k�k2 C 2

��1j�;t j2: (13.21)

To estimate the right-hand side in terms of the data, we use the following lemma.

Lemma 13.2. Let U be as in (13.18). Then for almost all t,

Z

˝

jU.x2; t/j2dx1dx2 � 1

2Lh0U

20.t/".t/; (13.22)

Z

˝

jU;x2 .x2; t/j2dx1dx2 � 4LU20.t/

h0

1

".t/; (13.23)

and

Z

˝

jU;t .x2; t/j2dx1dx2 ��

jU00.t/j C p

2jU0.t/j j"0.t/j".t/

�2 Lh0".t/

2: (13.24)


Proof. We have jU.x2; t/j � jU0.t/j and supp U Q", which gives (13.22).Moreover,

Z

˝

jU;x2 .x2; t/j2 dx D U20

h20"2

Z

˝

ˇˇˇˇ�

0�

x2h0"

�ˇˇˇˇ

2

dx

D U20

h20"2

LZ h0"

2

0

ˇˇˇˇ�

0�

x2h0"

�ˇˇˇˇ

2

dx2 � 4LU20

h0

1

";

as j�0j � p8, which gives (13.23). In order to obtain (13.24) we use (13.18) and the

definition of � to get

jU;t .x2; t/j � jU00.t/j�

�x2

h0".t/

�

C jU0.t/j 1

"2.t/

ˇˇˇˇx2"0.t/

h0

ˇˇˇˇ

ˇˇˇˇ�

0�

x2h0".t/

�ˇˇˇˇ

� jU00.t/j C p

2jU0.t/j j"0.t/j".t/

for 0 � x2 � h0"=2 and almost all t, from which the assertion follows easily. utApplying now Lemma 13.2 to inequality (13.21) with " as in (13.17) we obtain

the first energy estimate,

1

2

d

dtjv.t/j2 C �

2kv.t/k2 � f 2.t/ a.e. in t; (13.25)

where

f 2.t/ D c1.�;˝/.1C jU0.t/j3 C jU00.t/j2/: (13.26)

From (13.25) and (13.19) we conclude further that

1

2

d

dtjv.t/j2 C �

2jv.t/j2 � f 2.t/;

with � D ��1, and that

jv.t/j2 � e��.t��/jv.�/j2 C 2

Z t

�

e�.s�t/f 2.s/ds: (13.27)

To prove the uniqueness of solutions and their continuous dependence on the initialdata we assume that v and Nv are two solutions of Problem 13.1, write (13.11) fortheir difference, w D v � Nv, and set � D w, to get

1

2

d

dtjw.t/j2 C �

2kw.t/k2 C b.w; v;w/ D �b.w; �;w/:

13.4 Existence of the Pullback Attractor 305

Now we use the Ladyzhenskaya inequality (cf. Lemma 10.2),

kvkL4.˝/ � c2.˝/1=4jvj1=2kvk1=2

for all v 2 V (we can take c2.˝/ D 18maxf2; 1C .hM=L/2g), to get

jb.w; v;w/j � kwkL4.˝/kvkkwkL4.˝/

� c2.˝/1=2jwjkwkkjvk � 1

8�kwk2 C 2

�c2.˝/jwj2kvk2:

From this, together with (13.16), we get

d

dtjw.t/j2 C �

4kw.t/k2 � 4

�c2.˝/kvk2jwj2;

and

d

dtjw.t/j2 C �

4jw.t/j2 � 4

�c2.˝/kvk2jwj2:

By (13.25) and (13.27) the function s ! kv.s/k2 is locally integrable on the realline, whence the Gronwall lemma gives

jw.t/j2 � jw.�/j2 exp

�

�Z t

�

��

4� 4

�c2.˝/kv.s/k2

�

ds

�

: (13.28)

This means that the map v.�/ ! v.t/ for t > � is continuous in H, and, in particular,that the considered problem is uniquely solvable.

The existence part of the proof is based on inequality (13.25), the Galerkinapproximations, and the compactness method, and is very similar to the proof ofTheorem 9.6. ut

13.4 Existence of the Pullback Attractor

In this section we prove the existence of the pullback attractor for the evolutionaryprocess associated with the considered problem.

For t � � let us define the map U.t; �/ in H by

U.t; �/v0 D v.tI �; v0/ for t � � and v0 2 H; (13.29)

where v.tI �; v0/ is the solution of Problem 13.1. From the uniqueness of solution tothis problem, one immediately obtains that

U.t; �/v0 D U.t; r/U.r; �/v0; for all � � r � t and v0 2 H:


From Theorem 13.2 it follows that for all t � �; the mapping U.t; �/ W H !H defined by (13.29) is continuous. Consequently, the family fU.t; �/g��t definedby (13.29) is a process in H.

Moreover, it can be proved that U.�; t/ is sequentially weakly continuous, moreexactly, we have the following result.

Proposition 13.1. Let fv0ng H be a sequence converging weakly in H to anelement v0 2 H: Then

U.t; �/v0n ! U.t; �/v0 weakly in H for all t � �;

U.�; �/v0n ! U.�; �/v0 weakly in L2.�;TI V/ for all T > �:

Proof. The proof is almost identical to that of Lemma 2:1 in [201] and we omit it.ut

Now, we define the universe of the non-autonomous sets. For � D ��1, let

D� D fD W R ! P.H/ W limt!�1 e� tjD.t/j2 D 0g; (13.30)

where jD.t/j D supfjyj W y 2 D.t/g.We can prove the following result:

Theorem 13.3. Let U0 be a locally Lipschitz continuous function on the real linesuch that

Z t

�1e�s.jU0.s/j3 C jU0

0.s/j2/ ds < C1 for all t 2 R: (13.31)

Then, there exists a unique pullback D� -attractor A 2 D� for the process U.t; �/defined by (13.29).

Proof. Let D 2 D� be given. From the first energy estimate (13.27) andfrom (13.31) we obtain

jU.t; �/v0j2 � e��.t��/jD.�/j2 C 2e�� tZ t

�1e�sf .s/2ds < 1 (13.32)

for all v0 2 D.�/, and all t � � .Denote by R� .t/ the positive number given for each t 2 R by

.R� .t//2 D 3e�� t

Z t

�1e�sf .s/2ds;


and consider the non-autonomous set B� D fB� .t/gt2R where B� .t/ are closed ballsin H defined by

B� .t/ D fw 2 H W jwj � R� .t/g:

In view of (13.30) it is immediate to see that B� 2 D� . Moreover, by (13.30)and (13.32), the family B� is pullback D� -absorbing for the process U.t; �/.

Clearly, the attraction universe D� is inclusion-closed and the sets B� .t/ areclosed, and thus, according to Theorem 13.1 and Remark 13.2, to complete theproof of the theorem we have to prove that U.t; �/ is pullback D� -asymptoticallycompact.

Let D D fD.t/gt2R 2 D� , a sequence �n ! �1, a sequence v0n 2 D.�n/, andt 2 R be fixed. We must prove that we can extract a subsequence that converges inH from the sequence fU.t; �n/v0ng.

As the non-autonomous set B� D fB� .t/gt2R is pullback D� -absorbing, for eachinteger k � 0 there exists a �D.k/ � t � k such that

U.t � k; �/D.�/ B� .t � k/ for all � � �D.k/: (13.33)

It is not difficult to deduce from (13.33), by a diagonal procedure, the existence of asubsequence f.�n0 ; v0n0/g f.�n; v0n/g; and a sequence fwkg1

kD0 H; such that forall k D 0; 1; 2; : : : we have wk 2 B� .t � k/ and

U.t � k; �n0/v0n0 ! wk weakly in H:

Now, observe that, by Proposition 13.1,

w0 D weak� limn0!1 U.t; �n0/v00

n

D weak� limn0!1 U.t; t � k/U.t � k; �n0/v0n0

D U.t; t � k/.weak� limn0!1 U.t � k; �n0/v0n0/;

i.e.,

U.t; t � k/wk D w0 for all k D 0; 1; 2; : : : : (13.34)

Then, by the weak lower semicontinuity of the norm,

jw0j � lim infn0!1 jU.t; �n0/v0n0

j:

If we now prove that also

lim supn0!1

jU.t; �n0/v0n0

j � jw0j; (13.35)


then we will have

limn0!1 jU.t; �n0/v0n0

j D jw0j;

and this, together with the weak convergence, will imply the strong convergence inH of U.t; �n0/v0n0

to w0.Our aim is to prove (13.35). We shall use the notation,

hg.t/;wi D ��a.�.t/;w/ � .�;t ;w/ for w 2 V;

and

Q.t;w/ D �kwk2 � �

2jwj2 C b.w; �.t/;w/ for .t;w/ 2 R � V:

Then, the energy equation (13.13) can be written as

d

dtjvj2 C � jvj2 D 2.hg.t/; vi � Q.t; v//;

whence for all t � �; and all v0 2 H,

jU.t; �/v0j2 D jv0j2e�.��t/ (13.36)

C2Z t

�

e�.s�t/ .hg.s/;U.s; �/v0i � Q.s;U.s; �/v0// ds;

and, thus, for all k D 0; 1; 2; : : : and all �n0 � t � k;

jU.t; �n0/v0n0 j2 D jU.t; t � k/U.t � k; �n0/v0n0 j2 (13.37)

D e��kjU.t � k; �n0/v0n0 j2

C2Z t

t�ke�.s�t/hg.s/;U.s; t � k/U.t � k; �n0/v0n0i ds

�2Z t

t�ke�.s�t/Q.s;U.s; t � k/U.t � k; t � �n0/v0n0/ ds:

As, by (13.33),

U.t � k; �n0/v0n0

2 B� .t � k/ for all �n0 � �D.k/ and k D 0; 1; 2; : : : ;

we have

lim supn0!1

�e��kŒU.t � k; �n0/v0n0 �2

� � e��kR2� .t � k/ for k D 0; 1; 2; : : : :

(13.38)


On the other hand, as U.t � k; �n0/v0n0 ! wk weakly in H, from Proposition 13.1we have

U.�; t � k/U.t � k; �n0/v0n0 ! U.�; t � k/wk (13.39)

weakly in L2.t � k; tI V/:Taking into account that, in particular, e�.��t/g.�/ 2 L2.t � k; tI V 0/; we obtain

from (13.39),

limn0!1

Z t

t�ke�.s�t/hg.s/;U.s; t � k/U.t � k; �n0/v0n0i ds

DZ t

t�ke�.s�t/hg.s/;U.s; t � k/wki ds: (13.40)

Moreover, it is not difficult to see, using (13.16), the continuous injection ofH1.˝/ into L4.˝/; and the fact that by (13.23), r� 2 L1.t � k; tI L2.˝/2�2/; thatthe functional

�.w/ DZ t

t�ke�.s�t/Q.s;w.s// ds

is convex and continuous, and consequently weakly lower semicontinuous on thespace L2.t � k; tI V/: Thus, we also obtain from (13.39)

Z t

t�ke�.s�t/Q.s;U.s; t � k/wk/ ds (13.41)

� lim infn0!1

Z t

t�ke�.s�t/Q.s;U.s; t � k/U.t � k; �n0/v0n0/ ds:

Then, from (13.37), (13.38), (13.40), and (13.41), we obtain

lim supn0!1

jU.t; �n0/v0n0 j2 � e��kR2� .t � k/ (13.42)

C 2

Z t

t�ke�.s�t/ .hg.s/;U.s; t � k/wki � Q.s;U.s; t � k/wk// ds:

Now, from (13.34) and (13.36),

jw0j2 D jU.t; t � k/wkj2 D jwkj2e��k (13.43)

2

Z t

t�ke�.s�t/ .hg.s/;U.s; t � k/wki � Q.s;U.s; t � k/wk// ds:


From (13.42) and (13.43), we have

lim supn0!1

jU.t; �n0/v0n0 j2 � e��kR2� .t � k/C jw0j2 � jwkj2e��k

� e��kR2� .t � k/C jw0j2;

and thus, taking into account that

e��kR2� .t � k/ D 3e�� t

�

Z t�k

�1e�skg.s/k2V0

ds ! 0;

when k ! C1, we easily obtain (13.35) from the last inequality. ut

13.5 Fractal Dimension of the Pullback Attractor

In this section we prove

Theorem 13.4. Let U0 be a locally Lipschitz continuous function on the real linesuch that for some t? 2 R, r > 0, Mb > 0, M > 0, all t � t?, and all s � t? � r,

jU0.t/j � Mb andZ sCr

sjU0

0.�/j2d� � M: (13.44)

Then the attractor A D fA.t/gt2R from Theorem 13.3 has finite fractal dimension,namely,

dHf .A.t// � d (13.45)

for all t 2 R and some constant d.

To this end we shall use a result from [42] which in our notation can be expressedas follows.

Theorem 13.5. Suppose there exist constants K0;K1, > 0 such that

jA.t/j D supfjyj W y 2 A.t/g � K0jtj C K1 (13.46)

for all t 2 R. Assume that for any t 2 R there exist T D T.t/, l D l.t;T/ 2 Œ1;C1/,ı D ı.t;T/ 2 .0; 1=p2/, and N D N.t/ such that for any u; v 2 A.�/ and � � t�T,

jU.� C T; �/u � U.� C T; �/vj � lju � vj; (13.47)

jQN.U.� C T; �/u � U.� C T; �/v/j � ıju � vj; (13.48)

13.5 Fractal Dimension of the Pullback Attractor 311

where QN is the projector mapping from H onto some subspace H?N of codimension

N 2 N. Then, for any � D �.t/ > 0 such that � D �.t/ D .6p2l/N.

p2�/� < 1,

the fractal dimension of A.t/ is bounded, with

dHf .A.t// � N C �: (13.49)

Proof (of Theorem 13.4). First we show that for some M0 > 0 and all t � t?,

kA.t/k D supfkyk W y 2 A.t/g � M0: (13.50)

Here we need the second energy estimate. With � D Av in (13.11) we get

1

2

d

dtkv.t/k2 C �jAv.t/j2 C b.v.t/; v.t/;Av/ D F.v.t/;Av/; (13.51)

where

F.v;Av/ D �.��;Av/ � b.�; v;Av/ � b.v; �;Av/ � .�;t ;Av/:

To estimate the nonlinear term on the left-hand side of (13.51) we use the Agmoninequality

kvkL1.˝/ � cjvj1=2jAvj1=2;

that holds for v 2 D.A/. This estimate follows from the usual Agmon inequality

kwkL1.˝3/ � ckwk1=2L2.˝3/

kwk1=2H2.˝3/

;

where ˝3 is a smooth set such that ˝ ˝3 ˝1 and w D 'v, ' being thesuitable cut-off function, and the inequality kvkH2.˝/ � cjAvj for v 2 D.A/, cf. theproof of Theorem 9.6.

We have thus

jb.v; v;Av/j � kvkL1.˝/kvkjAvj � cjvj1=2kvkjAvj3=2 � �

8jAvj2 C c0

�3jvj2kvk4:

Moreover,

j.�;t ;Av/j � �

8jAvj2 C 2

�j�;t j2;

and

jb.�; v;Av/j � k�kL1.˝/kvkjAvj � �

8jAvj2 C 2

�k�k2L1.˝/kvk2:


Similarly,

jb.v; �;Av/j � jvjkr�kL1.˝/jAvj � �

8jAvj2 C 2

�kr�k2L1.˝/jvj2:

Finally,

j�.��;Av/j � �

8jAvj2 C 2�j��j2:

Therefore, from (13.51) we obtain

1

2

d

dtkv.t/k2 C 1

2�jAv.t/j2 � a.t/kv.t/k2 C b.t/; (13.52)

where

a.t/ D c0

�3jv.t/j2kv.t/k2 C 2

�k�.t/k2L1.˝/; (13.53)

and

b.t/ D 2

�j�;t .t/j2 C 2

�kr�.t/k2L1.˝/jv.t/j2 C 2�Œ��.t/�2: (13.54)

Now, recall that for all v0 2 A.�/ and all t � � , we have

jU.t; �/v0j2 � e��.t��/jA.�/j2 C 2

Z t

�1e�.s�t/f 2.s/ds: (13.55)

Let us fix some T > r and an arbitrary t � t�. Let v0 2 A.�/ for � near �1. Then,from (13.55) and (13.30) we deduce that for all � 2 .t � T; t/,

jv.�/j2 � 3e�TZ t

�1e�.s�t/f 2.s/ds (13.56)

D 3e�TC1.�;˝/

�1

�C e�� t

Z t

�1e�sŒjU0.s/j3 C jU0

0.s/j2�ds

�

:

From (13.54) we have

Z sCr

sb.�/d� � 2Lh0

�.3C 2

p2/

Z sCr

sjU0

0.�/�2d�C 8

�3

Z sCr

sjU0.�/j4/jv.�/j2d�

CL.4h0/3

�2

sup�2R

�00.�/!2 Z sCr

sjU0.�/j5d� if U0.�/ � �

8h0;


and

Z sCr

sb.�/d� � Lh0

Z sCr

sjU0

0.�/j2d�C �

83h40

Z sCr

sjv.�/j2d�

C L�3

83h20

sup�2R

�00.�/!2

if U0.�/ � �

8h0:

Thus, by (13.44) and (13.56) we conclude that there exists M2.r/ such that

Z sCr

sb.�/d� � M2.r/

for all s 2 .t � T; t � r/.From (13.25), (13.26), (13.31), (13.44), and (13.56) we deduce further that there

exists M3.r/ such that

Z sCr

skv.�/k2d� � M3.r/ (13.57)

for all s 2 .t � T; t � r/.Finally, by (13.53) together with (13.56), (13.57), and (13.44), there exists M1.r/

such that

Z sCr

sa.�/d� � M1.r/ (13.58)

for all s 2 .t � T; t � r/.With these estimates we apply now the uniform Gronwall lemma to inequal-

ity (13.52) to get, in particular,

kv.t/k2 ��

M3.r/

rC 2M2.r/

�

exp 2M1.r/ � QM.r/;

which gives (13.50).We shall prove now that in our case of the Navier–Stokes flow inequality (13.46)

holds true for t � t?, and inequalities (13.47) and (13.48) hold true for � C T �t � t?. This restriction for the time variable t in the hypotheses of Theorem 13.5produces a similar change in its assertion, namely, inequality (13.49) holds then truefor t � t?. Such restricted result is, however, sufficient in our case of the Navier–Stokes flow in order to obtain (13.45) for all t as it will be shown in the end of theproof.

Observe that in view of (13.50) inequality (13.46) is satisfied for t � t?, namely,

jA.t/j � 1p�1

M0 for t � t?:


Inequality (13.47) for � C T � t � t? follows from inequality (13.28). In fact, letv.�/ and Nv.�/ be in A.�/, and denote ! D v � Nv. Then from (13.28) and (13.50) itfollows that

j!.� C T/j2 � l.T/j!.�/j2; (13.59)

with l.T/ D expf.� �4

C 4�

c2.˝/M20/Tg (we may always assume that the constant

l.T/ in (13.59) is greater or equal than one).Now, we shall prove that inequality (13.48) holds true. Multiplying

d!

dtC �A! C B.!; Nv/C B.v; !/ D B.w; �/

by Qm! defined by Qm!.x; t/ D P1jDmC1 Q!.t/wj.x/ (Qm D I � Pm, Pm being the

projection on the space spanned by the first m eigenfuntions of the Stokes operatorwith the periodic boundary conditions on �L and Dirichlet boundary conditions on�C [ �D), we obtain

1

2

d

dtjQm!j2 C �kQm!k2 C b.!; v;Qm!/C b. Nv; !;Qm!/

D b.w; �;Qm!/: (13.60)

Our first step is to obtain a differential inequality for jQm!j2 of the form (13.65)below. To this end we have to estimate the nonlinear terms in (13.60).

We have,

b.!; v;Qm!/ D b.Pm!; v;Qm!/C b.Qm!; v;Qm!/:

Now,

jb.Pm!; v;Qm!/j D jb.Pm!;Qm!; v/j� c2.˝/

12 jPm!j 12 kPm!k 1

2 kQm!kjvj 12 kvk 12 ;

and kPm!k2 � �mjPm!j2, whence, from (13.50) we conclude

jb.Pm!; v;Qm!/j � 2c2.˝/M20

�p�1

�12mj!j2 C �

8kQm!k2: (13.61)

We have also

jb.Qm!; v;Qm!/j � 2c2.˝/M20

�jQm!j2 C �

8kQm!k2: (13.62)


We estimate the last term on the left-hand side of (13.60) as follows,

jb. Nv; !;Qm!/j D jb. Nv;Pm!;Qm!/j� c2.˝/

12 jvj 12 k Nvk 1

2 jPm!j 12 kPm!k 12 kQm!k

� 2c2.˝/M20

�p�1

�12mj!j2 C �

8kQm!k2: (13.63)

Using (13.44) we treat the term on the right-hand side of (13.60) as follows,

jb.!; �;Qm!/j D jb.!;Qm!; �/j � k�kL1.˝/j!jkQm!k

� 2

�M2

b j!j2 C �

8kQm!k2: (13.64)

From (13.60) together with (13.61)–(13.64), and in view of

�mC1jQm!j2 � kQm!k2;we obtain, in particular, for � � s � � C T � t � t?,

d

dsjQm!.s/j2 C �mjQm!.s/j2 � �mj!.s/j2; (13.65)

where

�m D�

��mC1 � 4c2.˝/M20

�

�

and �m D 8c2.˝/M20

�p�1

�12m C 4M2

b

�:

We can see that for m large enough we have �m > 0.Now, using the bound j!.s/j2 � l.T/j!.�/j2 for � � s � � C T , we obtain

d

dsjQm!.s/j2 C �mjQm!.s/j2 � �ml.T/j!.�/j2:

Multiplying both sides by expf�m.s � �/g we get

d

dsfjQm!.s/j2 expf�m.s � �/gg � �ml.T/j!.�/j2 expf�m.s � �/g:

Integrating with respect to s in the interval .�; � C T/ we obtain,

jQm!.� C T/j2 � ı2.T/j!.�/j2;

with

ı2 D expf��mTg C �ml.T/

�m<1

2


for m large enough, as �m is of the order of �mC1. This gives inequality (13.48) andthus completes the proof of a finite dimensionality of the sets A.t/, for t � t?, with

dHf .A.t// � m C �:

The same estimate holds also for t > t? in view of the property of the fractaldimension under Lipschitz continuous mappings, cf. Theorem 9.1. ut


To prove the existence of the pullback attractor we used, in Sect. 13.4, the energyequation method, developed in [44, 45] for the pullback attractor theory. Thismethod works also in the case of certain unbounded domains of the flow as itbypasses the usual compactness argument. In turn, to estimate the pullback attractordimension we used, in Sect. 13.5, the method proposed in [42], alternative to theusual one [149, 220]. It is also possible to study the finite dimensionality of pullbackattractors through so-called pullback exponential attractors, see [51, 52, 151].

When h D const and U0 D const, problem (13.1)–(13.5) was intensivelystudied in several contexts. Most results concern the autonomous Navier–Stokesequations (cf. Sect. 9.4). In [89], the domain of the flow is an elongated rectangle˝ D .0;L/ � .0; h/, L � h. In this case the attractor dimension can be estimatedfrom above by c L

h Re3=2, where c is a universal constant and Re D Uh�

is the Reynoldsnumber. In [241] optimal bounds of the attractor dimension are given for a flow ina rectangle .0; 2�L/ � .0; 2�L=˛/, with periodic boundary conditions and givenexternal forcing. The estimates are of the form c0=˛ � dH

f .A / � c1=˛, see also[180]. Some free boundary conditions are considered in [242], see also [222], andan upper bound on the attractor dimension established with the use of a suitableanisotropic version of the Lieb–Thirring inequality, similarly to [89]. Dirichlet-periodic and free-periodic boundary conditions and domains with more generalgeometry were considered in [22, 24] (cf. also [25]) where still other forms of theLieb–Thirring inequality were established to study the dependence of the attractordimension on the shape of the domain of the flow. Slip boundary condition andunbounded domain case were considered in [183].

The autonomous case with h ¤ const was considered in [22, 24].Boundary driven flows in smooth and bounded two-dimensional domains for

a non-autonomous Navier–Stokes system were considered in [178], by using anapproach of Chepyzhov and Vishik, cf. [62]. An extension to some unboundeddomains can be found in [181], cf. also [164].

14Trajectory Attractors and Feedback BoundaryControl in Contact Problems

I hail a semigroup when I see one and I seem to see themeverywhere. Friends have observed, however, that there aremathematical objects which are not semigroups.

– Einar Hille

In this chapter we consider two-dimensional nonstationary incompressibleNavier–Stokes shear flows with nonmonotone and multivalued leak boundaryconditions on a part of the boundary of the flow domain. Our considerationsare motivated by feedback control problems for fluid flows in domains withsemipermeable walls and membranes and by the theory of lubrication. Our aimis to prove the existence of global in time solutions of the considered problemwhich is governed by a partial differential inclusion, and then to prove the existenceof a trajectory attractor and a weak global attractor for the associated multivaluedsemiflow.

14.1 Setting of the Problem

The problem we consider is as follows. The flow of an incompressible fluid in atwo-dimensional domain ˝ is described by the equation of motion

ut � ��u C .u � r/u C rp D 0 for .x; t/ 2 ˝ � RC; (14.1)

where the viscosity coefficient � > 0, and the incompressibility condition

div u D 0 for .x; t/ 2 ˝ � RC: (14.2)

To define the domain ˝ of the flow let us consider the channel

˝1 D fx D .x1; x2/ W �1 < x1 < 1; 0 < x2 < h.x1/g;


317

318 14 Trajectory Attractors and Feedback Boundary Control in Contact Problems

where the function h W R ! R is positive, smooth, and L-periodic. Then we set

˝ D fx D .x1; x2/ W 0 < x1 < L; 0 < x2 < h.x1/g;

and @˝ D �C [ �L [ �D, where �C and �D are the bottom and the top, and �L isthe lateral part of the boundary of ˝. The domain ˝ is schematically presented inFig. 14.1. By n we will always denote the unit normal vector on @˝.

Fig. 14.1 Schematicoverview of the domain ˝and parts �C; �D; �L of itsboundary

We are interested in solutions of (14.1)–(14.2) such that the velocity u and theCauchy stress vector Tn, where T is the stress tensor given by the constitutive lawTij D �pıij C �.ui;j C uj;i/, are L-periodic with respect to x1 on �L. We assume that

u D 0 on �D � RC: (14.3)

On the bottom �C we impose the following conditions. The tangential componentu� of the velocity vector on �C is given, namely, for some s 2 R, we have

u� D u � unn D .s; 0/ on �C � RC; where un D u � n: (14.4)

Furthermore, we assume the following subdifferential boundary condition

Qp.x; t/ 2 @j.un.x; t// on �C � RC; (14.5)

where Qp D p C 12juj2 is the total pressure (called the Bernoulli pressure), j W R ! R

is a given locally Lipschitz potential, and @j is a Clarke subdifferential of j (seeSect. 3.7 and [72, 84] for the definition and properties of Clarke subdifferential).

Let, moreover,

u.0/ D u0 in ˝: (14.6)

The considered problem is motivated by the examination of a certain two-dimensional flow in an infinite (rectified) journal bearing˝�.�1;C1/, where the

14.2 Weak Formulation of the Problem 319

outer cylinder is represented by �D � .�1;C1/, and �C � .�1;C1/ representsthe inner, rotating cylinder. In the lubrication problems the gap h between cylindersis never constant. We can assume that the rectification does not change the equationsas the gap between cylinders is very small with respect to their radii.

A physical interpretation of the boundary condition (14.5) can be as follows. Thepotential j in our control problem is not convex as its subdifferential corresponds tothe nonmonotone relation between the normal velocity un and the total pressure Qp at�C. Assuming that, left uncontrolled, the total pressure at �C would increase withthe increase of the normal velocity of the fluid at �C, we control Qp by a hydraulicdevice which opens wider the boundary orifices at �C when un attains a certainvalue and thus Qp drops at this value of un. Particular examples of such relations areprovided in [175, 176].

14.2 Weak Formulation of the Problem

In this section we introduce the basic notations and define a notion of a weaksolution u of the initial boundary value problem (14.1)–(14.6). For convenience,we shall work with a homogeneous problem whose solution v has the tangentialcomponent v� at �C equal to zero, and then u D vC w for a suitable extension w ofthe boundary data.

In order to define a weak formulation of the homogeneous problem (14.1)–(14.6)we need to introduce some function spaces and operators. Let

W D fw 2 C1.˝/2 W div w D 0 in ˝; w is L-periodic in x1;

w D 0 on �D; w� D 0 on �Cg;

and let V and H be the closures of W in the norms of H1.˝/2 and L2.˝/2,respectively. In the sequel we will use the notation k � k; j � j to denote, respectively,the norms in V and H. We denote the trace operator V ! L2.�C/

2 by � . Bythe trace theorem, � is linear and bounded; we will denote its norm by k�k WDk�kL .VIL2.�C/2/. In the sequel we will write u instead of �u for the sake of notationsimplicity.

Let the operators A W H1.˝/2 ! V 0 and BŒ�� W H1.˝/2 ! V 0 be defined by

hAu; vi D �

Z

˝

rot u � rot v dx (14.7)

for all u 2 H1.˝/2, v 2 V , and BŒu� D B.u; u/, where

hB.u;w/; zi DZ

˝

.rot u � w/ � z dx (14.8)

for u;w 2 H1.˝/2, z 2 V .


According to the hydrodynamical interpretation of the considered problemgiven in Sect. 14.1, we can understand the rot operator in the following way. Foru.x1; x2/ D .u1.x1; x2/; u2.x1; x2// and Nu.Nx/ D .u1.x1; x2/; u2.x1; x2/; 0/, whereNx D .x1; x2; x3/, we set rot u.x1; x2/ D rot Nu.Nx/.

Let G D ˝ � .0; 1/ and f .Nx/ D f .x1; x2/ be a scalar function. Then

Z

˝

f .x1; x2/ dx DZ

Gf .Nx/ dNx: (14.9)

In particular, for u; v in V ,

hAu; vi D �

Z

˝

rot u.x/ � rot v.x/ dx D �

Z

Grot Nu.Nx/ � rot Nv.Nx/ dNx

D �

Z

Gr Nu.Nx/ � r Nv.Nx/ dNx D �

Z

˝

ru.x/ � rv.x/ dx: (14.10)

To work with the boundary condition (14.5) we rewrite equation of motion (14.1) inthe Lamb form,

ut C � rot rot u C rot u � u C r Qp D 0 in ˝ � RC: (14.11)

Further, to transform the problem into the homogeneous one, for u 2 H1.˝/2 letw 2 H1.˝/2 be such that div w D 0, w D 0 at �D, w is L-periodic on �L, andw� D u� D s and wn D 0 on �C, and let u D v C w. Then v 2 V as v� D 0 at �C.Moreover, vn D un on �C.

Multiplying (14.11) by z 2 V and using the Green formula we obtain

hv0.t/C Av.t/C BŒv.t/�; zi CZ

�C

Qpzn dS D hF; zi C hG.v/; zi; (14.12)

where

hF; zi D �

Z

˝

rot w � rot z dx � hB.w;w/; zi; (14.13)

and

hG.v/; zi D �hB.v;w/C B.w; v/; zi: (14.14)

Above we have used the formulaZ

˝

rot R � a dx DZ

˝

R � rot a dx CZ

@˝

.R � a/ � n dS; (14.15)

14.2 Weak Formulation of the Problem 321

with R D rot v or R D rot w. Formula (14.15) is easy to get using the three-dimensional vector calculus and (14.9). Observe that if a� D 0 on @˝, then wehave

.R � a/ � n D .R � ann/ � n D 0:

We need the following assumptions on the potential j:

.j1/ j W R ! R is locally Lipschitz,

.j2/ @j satisfies the growth condition j�j � c1 C c2juj for all u 2 R and all� 2 @j.u/, with c1 > 0 and c2 > 0,

.j3/ @j satisfies the dissipativity condition inf�[email protected]/ �u � d1 � d2juj2, for all

u 2 R, where d1 2 R and d2 2 0; �

k�k2�

.

From (14.12) we obtain the following weak formulation of the homogeneousproblem.

Problem 14.1. Let v0 2 H. Find

v 2 L2loc.RCI V/ \ L1

loc.RCI H/;

with v0 2 L43

loc.RCI V 0/, v.0/ D v0, and such that

hv0.t/C Av.t/C BŒv.t/�; zi C .�.t/; zn/L2.�C/ D hF; zi C hG.v.t//; zi; (14.16)

�.t/ 2 [email protected].�;t//;

for a.e. t 2 RC and for all z 2 V .

In the above definition we use the notation

S2U D fu 2 L2.�C/ W u.x/ 2 U.x/ for a.e. x 2 �Cg;

valid for a multifunction U W �C ! 2R. Note that from the fact that v 2 L2loc.RCI V/

and v0 2 L43

loc.RCI V 0/ it follows that v 2 C.RCI V 0/, and hence the initial condition

makes sense. We define Cw.Œ0;T�I H/ as the space of all functions u W Œ0;T� ! Hwhich are weakly continuous, i.e., such that for any � 2 H the scalar product.u.t/; �/ in H is a continuous function of t for t 2 Œ0;T�. Since v 2 L1

loc.RCI H/

then (see Lemma 3.18 or [227], Theorem II.1.7) v 2 Cw.RCI H/, and thus the initial

condition makes sense in the phase space H.One can see (cf. [176]) that if v 2 L2loc.R

CI V/ is a sufficiently smooth solutionof the partial differential inclusion (14.16) then there exists a distribution Qp suchthat the conditions (14.11) and (14.5) hold for u D v C w. In conclusion, thefunction u D vC w can be regarded as a weak solution of the initial boundary valueproblem (14.1)–(14.6), provided v is a solution of Problem 14.1 with v.0/ D u0�w,u0 2 H.


The trajectory space K C of Problem 14.1 is defined as the set of those of itssolutions with some v0 2 H that satisfy the following inequality,

1

2

d

dtjv.t/j2 C C1kv.t/k2 � C2.1C kFk2V0

/; (14.17)

where C1;C2 > 0 and inequality (14.17) is understood in the sense that for all0 � t1 < t2 and for all 2 C1

0 ..t1; t2//, � 0 we have

� 1

2

Z t2

t1

jv.t/j2 0.t/ dt C C1

Z t2

t1

kv.t/k2 .t/ dt � C2.1C kFk2V0

/

Z t2

t1

.t/ dt:

(14.18)

Note that since we cannot guarantee that for every solution of Problem 14.1hv0.t/; v.t/i D 1

2ddt jv.t/j2 holds, we cannot derive (14.17) for every solution of

Problem 14.1.In the next section we prove that the trajectory space is not empty.

14.3 Existence of Global in Time Solutions

In this section we give a proof of the existence of solutions of Problem 14.1 thatsatisfy inequality (14.17). The proof will be based on the standard technique thatuses the regularization of the multivalued term and in main points will follow theexistence proofs in Chap. 10 (see also [162, 176]).

The operators A and B defined in (14.7) and (14.8) and restricted to V have thefollowing properties:

(1) A W V ! V 0 is a linear, continuous, symmetric operator such that

hAv; vi D �kvk2 for v 2 V; (14.19)

(2) B W V � V ! V 0 is a bilinear, continuous operator such that

hB.u; v/; vi D 0 for v 2 V: (14.20)

Lemma 14.1. Given � > 0 and s 2 R there exists a smooth function w 2 H1.˝/2

such that div w D 0 in ˝, w D 0 on �D, w is L-periodic in x1, w� D .s; 0/, wn D 0

on �C, and for all v 2 V

jhB.v;w/; vij � �kvk2: (14.21)

Proof. Let w be of the form w.x2/ D .s�.x2=h0/; 0/, where � W Œ0;1/ ! Œ0; 1� is asmooth function such that �.0/ D 1, �0.0/ D 0, supp � Œ0;minf �

2jsj ; h0; 1/g�, and

14.3 Existence of Global in Time Solutions 323

h0 is the minimum value of h.x1/ on Œ0;L�. It is clear that all the stated properties ofw other than (14.21) hold. To prove (14.21) observe that under our assumptions

Z

˝

jrot wj2dx DZ

˝

jrwj2 dx; (14.22)

and then

jhB.v;w/; vij � jrvjkx2w.x2/kL1.Œ0;h0�/

ˇˇˇˇv

x2

ˇˇˇˇ (14.23)

� kvk�22kvk D �kvk2

in view of the Hardy inequality. utLemma 14.2. Let j W R ! R satisfy .j1/–.j3/. For any solution v of Problem 14.1,

kv0.t/kV0 � C3.1C kv.t/k C jv.t/j1=2kv.t/k 32 / for a.e. t 2 R

C; (14.24)

with a constant C3 > 0 independent of v.

Proof. Let v be a solution of Problem 14.1. For any test function z 2 V we have, fora.e. t 2 R

C,

jhv0.t/; zij � kFkV0kzk C k�.t/kL2.�C/k�kkzk C jhG.v.t// � Av.t/ � BŒv.t/�; zij:

From the growth condition .j2/ it follows that k�.t/kL2.�C/ � C.1C kv.t/k/ with aconstant C > 0. Moreover, jhAv.t/; zij C jhG.v.t//; zij � Ckv.t/kkzk. It remains toestimate the nonlinear term. For all w 2 V we have (14.22). Now, from the Hölderand Ladyzhenskaya inequalities we obtain

jhBŒv.t/�; zij �Z

˝

jrot v.t/jjv.t/jjzj dx � kv.t/kkv.t/kL4.˝/2kzkL4.˝/2

� Cjv.t/j1=2kv.t/k3=2kzk:

In this way we obtain (14.24). utTheorem 14.1. Let the potential j satisfy .j1/–.j3/, F 2 V 0. Then for every v0 2 Hthere exists v 2 K C such that v.0/ D v0.

Proof. Let % 2 C10 ..�1; 1// be a mollifier such that

R 1�1 %.s/ ds D 1 and %.s/ � 0.

We define %k W R ! R by %k.s/ D k%.ks/ for k 2 N and s 2 R. Then supp %k �� 1k ;

1k

�. We consider jk W R ! R defined by the convolution

jk.r/ DZ

R

%k.s/j.r � s/ ds for r 2 R:


By assertion .A/ of Theorem 3.21 we have jk 2 C1.R/. Let s 2 R. We calculate

j0k.s/s D limh!0C

jk.s C hs/ � jk.s/

hD lim

h!0C

Z

supp %k

%k.r/j.s C hs � r/ � j.s � r/

hdr:

Estimating the integrand from below, by the Lebourg mean value theorem (cf.Theorem 3.20) we get, for 2 .0; 1/ dependent on s; r; h, and � 2 @j.s � r C hs/,

%k.r/j.s C hs � r/ � j.s � r/

hD %k.r/�.s � r C hs C r � hs/

� %k.r/�d1 � d2js � r C hsj2 � .c1 C c2js � r C hsj/.jrj C hjsj/�

� %k.r/

d1 � d2 .jsj C 1C hjsj/2 � .c1 C c2 .jsj C 1C hjsj// .1C hjsj/�;

where we used the fact that jrj � 1k � 1. After integrating over r, passing with h to

zero and noting that k � 1 we get

j0k.s/s � d1 � d2.jsj C 1/2 � c1 � c2.jsj C 1/ � ed1 � ed2jsj2;

whence regularized functions jk still satisfy .j3/, where the constants d1; d2 are

different than the ones for j, but independent on k, and still we have ed2 2 0; �

k�k2�

.

Let us furthermore take the sequence Vk of finite dimensional spaces such that Vk

is spanned by the first k eigenfunctions of the Stokes operator with the Dirichlet andperiodic boundary conditions given in the definition of the space V . Then fVkg1

kD1approximate V from inside, i.e.,

S1kD1 Vk D V . Moreover we take the sequence

v0k ! v0 strongly in H such that v0k 2 Vk. We formulate the regularized Galerkinproblems for k 2 N:

Find a continuous function vk W RC ! Vk such that for a.e. t 2 R

C vk isdifferentiable and

hv0k.t/ C Avk.t/C BŒvk.t/�; zi C .j0k..vk/n.�; t//; zn/L2.�C/ (14.25)

D hF C G.vk.t//; zi;vk.0/ D v0k (14.26)

for a.e. t 2 RC and for all z 2 Vk.

We first show that if vk solves (14.25) then an estimate analogous to (14.17)holds.


We take z D vk.t/ in (14.25) and, using (14.19) and (14.20) as well as the factthat hB.v;w/; vi � �kvk2 for all v 2 V , where � can be made arbitrarily small(Lemma 14.1), we obtain

1

2

d

dtjvk.t/j2 C �kvk.t/k2 C .j0k..vk/n.�; t//; .vk/n.t//L2.�C/ � kFkV0kvk.t/k

C �kvk.t/k2

for a.e. t 2 RC. Using .j3/ and the Cauchy inequality with some " > 0 we obtain

1

2

d

dtjvk.t/j2 C �kvk.t/k2 C ed1m1.�C/ � ed2k.vk/n.t/k2L2.�C/

� C."/kFk2V0

C .�C "/kvk.t/k2

for a.e. t 2 RC, where " > 0 is arbitrary and the constant C."/ is positive. Note that

by the trace theorem k.vk/n.t/k2L2.�C/� kvk.t/k2L2.�C/2

� k�k2kvk.t/k2. It followsthat

1

2

d

dtjvk.t/j2 C .� � ed2k�k2/kvk.t/k2 � C."/kFk2V0

C .�C "/kvk.t/k2 C jed1jL

for a.e. t 2 RC. It is enough to take � D " D ��ed2k�k2

4. We get, with C1;C2 > 0

independent of t,

1

2

d

dtjvk.t/j2 C C1kvk.t/k2 � C2.1C kFk2V0

/: (14.27)

Note that, after integration,

1

2jvk.t/j2 C C1

Z t

0

kvk.s/k2 ds � 1

2jv0kj2 C C2.1C kFk2V0

/t for all t � 0:

(14.28)

Existence of the solution to the Galerkin problem (14.25)–(14.26) is standard andfollows by the Carathéodory theorem and estimate (14.28). Note that for all k 2 N

solutions of (14.25) satisfy the estimate of Lemma 14.2, where the constants do notdepend on initial conditions and the dimension k. We deduce from (14.27) that forall T > 0

vk is bounded in L2.0;TI V/ \ L1.0;TI H/: (14.29)

We denote by � W V ! L2.�C/ the linear and bounded map given as �v D .�v/n andby �� W L2.�C/ ! V 0 its adjoint operator. Now if Pk is the projection operator ontothe space Vk, (14.25) is equivalent to the following equation in V 0,

v0k.t/C Avk.t/C PkBŒvk.t/�C Pk�

�j0k..vk/k.�; t// D PkF C PkG.vk.t//:


From the choice of the space Vk it follows by Theorem 3.17 that kPkukV0 � kukV0

and by the argument of Lemma 14.2 it follows that for all T > 0

v0k is bounded in L

43 .0;TI V 0/: (14.30)

In view of (14.29) and (14.30), by diagonalization, we can construct a subsequencesuch that

vk ! v weakly in L2loc.RCI V/ (14.31)

and

v0k ! v0 weakly in L

43

loc.RCI V 0/: (14.32)

First we show that v.0/ D v0 in H. To this end choose T > 0. We have, for t 2.0;T/,

vk.t/ D vk0 CZ t

0

v0k.s/ ds; v.t/ D v.0/C

Z t

0

v0.s/ ds;

where the equalities hold in V 0. Take � 2 V . Then

Z T

0

hvk.t/ � v.t/; �i dt

DZ T

0

hvk0 � v.0/; �i dt CZ T

0

Z t

0

.v0k.s/ � v0.s// ds; �

dt

D Thvk0 � v.0/; �i CZ T

0

hv0k.t/ � v0.t/; .T � t/�i dt:

The left-hand side of this equality goes to zero with k ! 1 as vk ! v weakly inL2.0;TI V/. The last integral on the right-hand side also goes to zero with k ! 1as v0

k ! v0 weakly in L43 .0;TI V 0/ and .T � t/� 2 L4.0;TI V/. Hence, vk0 ! v0

weakly in V 0 and v.0/ D v0 in H.Now we pass with k ! 1 in Eq. (14.25). To this end let us choose T > 0. We

multiply (14.25) by � 2 C.Œ0;T�/ and integrate with respect to t in Œ0;T�. Passing tothe limit in linear terms A and G is standard. We focus on the multivalued term andthe convective term.

Let Z D H1�ı.˝/2 \ V for ı 2 �0; 1

2

�be endowed with H1�ı norm. Since

V Z and Z V 0, with compact and continuous embeddings, respectively, fromthe Aubin–Lions compactness theorem it follows that, for a subsequence, vk ! v

strongly in L2.0;TI Z/, and, by the continuity of the trace operator and also of thecorresponding Nemytskii operator, we have for all T > 0,

.vk/n ! vn strongly in L2.0;TI L2.�C//: (14.33)


By assertion .B/ of Theorem 3.21, the functions j0k satisfy the growth condition .j2/with the constants independent of k. It follows that the sequence j0k..vk/n.�; �// isbounded in L2.0;TI L2.�C//, and we can extract a subsequence, not renumbered,such that for all T > 0

j0k..vk/n.�; �// ! � weakly in L2.0;TI L2.�C//: (14.34)

We need to show that �.t/ 2 [email protected].�;t// for a.e. t 2 RC. We use assertion .C/ of

Theorem 3.21, whence we have, for all T > 0,

�.x; t/ 2 @j.vn.x; t// for a.e. .x; t/ 2 �C � .0;T/;and the desired result follows. Hence we can pass to the limit in the multivaluedterm.

Now we show the convergence in the nonlinear term, namely, that (for asubsequence)Z T

0

Z

˝

.rot vk.t/ � vk.t// � z.x/�.t/ dx dt !Z T

0

Z

˝

.rot v.t/ � v.t// � z.x/�.t/ dx dt:

(14.35)

First we prove that

Z T

0

Z

˝

.rot vk.t/ � vk.t// � z.x/�.t/ dx dt D 1

2

Z T

0

Z

�C

.vk/2n.x; t/zn.x/�.t/ dS dt

�Z T

0

Z

˝

.vk.x; t/ � r/z.x/vk.x; t/�.t/ dx dt: (14.36)

From (14.15), as well as the formulas .a � b/ � c D .b � c/ � a, and

r � F D 0;r � G D 0 H) rot .F � G/ D .G � r/F � .F � r/G (14.37)

we have

hBŒvk.t/�; zi DZ

˝

.vk.t/ � z/ � rot vk.t/ dx (14.38)

DZ

˝

rot.vk.t/ � z/ � vk.t/ dx CZ

@˝

.vk.t/ � .vk.t/ � z// � n dS:

The surface integral is equal to zero and hence, using (14.37) in the right-hand side,we obtain

hBŒvk.t/�; zi DZ

˝

.z � r/vk.t/ � vk.t/dx �Z

˝

.vk.t/ � r/z � vk.t/ dx: (14.39)

Integration by parts in the first integral on the right-hand side and then integrationin the time variable give the result.


Since we can deal with the second term on the right-hand side of (14.36) as inthe usual Navier–Stokes theory [219], we consider only the surface integral. Takingthe difference of the corresponding terms and setting zn.x/�.t/ D .x; t/ we obtain

???

Z T

0

Z

�C

..vk/n.x; t/ � vn.x; t//..vk/n.x; t/C vn.x; t// .x; t/ dS dt???

� k.vk/n � vnkL2.0;TIL2.�C//kvk C vkL2.0;TIL4.�C/2/ maxt2Œ0;T� k .t/kL4.�C/:

This proves (14.35) since the embedding H1=2.�C/2 Lr.�C/

2 is continuous forr � 1, vk is bounded in L2.0;TI L4.�C/

2/, and k.vk/n � vnkL2.0;TIL2.�C// ! 0

by (14.33).Hence, the limit v solves Problem 14.1. It remains to show (14.18). The proof

follows the lines of that of Theorem 8.1 in [61]. Let us fix 0 � t1 < t2 and choose 2 C1

0 ..t1; t2// with � 0. We multiply (14.27) by and integrate by parts. Wehave

� 1

2

Z t2

t1

jvk.t/j2 0.t/ dt C C1

Z t2

t1

kvk.t/k2 .t/ dt � C2.1C kFk2V0

/

Z t2

t1

.t/ dt:

(14.40)

We need to pass to the limit (k ! 1) in the last inequality. From (14.31) and (14.32)by the Aubin–Lions compactness theorem we conclude that vk ! v strongly inL2.t1; t2I H/. From inequality

Z t2

t1

.jvk.t/j � jv.t/j/2 dt �Z t2

t1

jvk.t/ � v.t/j2 ds

it follows that jvk.t/j ! jv.t/j strongly in L2.t1; t2/, and hence, for a subsequence,jvk.t/j2 ! jv.t/j2 for almost every t 2 .t1; t2/. Since from (14.28) it followsthat functions jvk.t/j2 0.t/ have an integrable majorant on .t1; t2/, we have, by theLebesgue dominated convergence theorem that

Z t2

t1

jvk.t/j2 0.t/ dt !Z t2

t1

jv.t/j2 0.t/ dt: (14.41)

Now, from (14.31) it follows that vk.t/. .t//12 ! v.t/. .t//

12 weakly in

L2.t1; t2I V/ and hence, by the sequential weak lower semicontinuity of the normwe obtain

Z t2

t1

kv.t/k2 .t/ dt � lim infk!1

Z t2

t1

kvk.t/k2 .t/ dt: (14.42)

Thereby, we can pass to the limit in (14.40) which gives us (14.18) and the proof iscomplete. ut

14.4 Existence of Attractors 329

14.4 Existence of Attractors

In this section we prove the existence of a trajectory attractor and a weak globalattractor for the considered problem.

We have shown that for any initial condition v0 2 H Problem 14.1 has at leastone solution v 2 KC. The key idea behind the trajectory attractor (see [61, 62,226, 227]) is, in contrast to the direct study of the solutions asymptotic behavior,the investigation of the family of shift operators fT.t/gt�0 defined on K C by theformula

.T.t/v/.s/ D v.s C t/:

Before we pass to a theorem on the existence of the trajectory attractor, which is themain result of this section, we recall some definitions and results (see [61, 62, 226,227]).

Let

F .0;T/ D fu 2 L2.0;TI V/ \ L1.0;TI H/ W u0 2 L43 .0;TI V 0/g;

and

F locC D fu 2 L2loc.RCI V/ \ L1

loc.RCI H/ W u0 2 L

43

loc.RCI V 0/g:

Note that F .0;T/ Cw.Œ0;T�I H/ and from the Lions–Magenes lemma (cf.Lemma 1.4, Chap. 3 in [219], Theorem II,1.7 in [62] or Lemma 2.1 in [227]) itfollows that for all u 2 F .0;T/ we have

ju.t/j � kukL1.0;TIH/ for all t 2 Œ0;T�: (14.43)

Furthermore, let us define the Banach space

F bC D fu 2 F locC W kukF bC

< 1g;

where the norm in F bC is given by

kukF bC

D suph�0

k˘0;1u.� C h/kF .0;1/:

In the above definition ˘0;1u.�/ is a restriction of u.�/ to the interval .0; 1/, and

jjvjjF .0;1/ D kvkL2.0;1IV/ C jjvjjL1.0;1IH/ C kv0kL43 .0;1IV0/

:

Finally, we define the topology by �locC in the space F locC in the following way: thesequence fukg1

kD1 F locC is said to converge to u 2 F locC in the sense of �locC if


uk ! u weakly in L2loc.RCI V/;

uk ! u weakly-� in L1loc.R

CI H/;

u0k ! u0 weakly in L

43

loc.RCI V 0/:

Note (see, for example, [227]) that the topology �locC is stronger than the topologyof Cw.Œ0;T�I H/ for all T � 0 and hence from the fact that vk ! v in �locC it followsthat vk.t/ ! v.t/ weakly in H for all t � 0.

Definition 14.1. A set P K C is said to be absorbing for the shift semigroupfT.t/gt�0 if for any set B K C bounded in F bC there exists � D �.B/ > 0 suchthat T.t/B P for all t � � .

Definition 14.2. A set P K C is said to be attracting for the shift semigroupfT.t/gt�0 in the topology �locC if for any set B K C bounded in F bC and for anyneighborhood O.P/ of P in the topology �locC there exists � D �.B;O.P// > 0

such that T.t/B O.P/ for all t � � .

Definition 14.3. A set U K C is called a trajectory attractor of the shiftsemigroup fT.t/gt�0 on K C in the topology �locC if

(a) U is bounded in F bC and compact in the topology �locC ,(b) U is an attracting set in the topology �locC ,(c) U is invariant, i.e., T.t/U D U for any t � 0.

In order to show the existence of a trajectory attractor for Problem 14.1 we will usethe following theorem (see, for instance, Theorem 4.1 in [227]).

Theorem 14.2. Assume that the trajectory set K C is contained in the space F bCand that

T.t/K C K C: (14.44)

Suppose that the semigroup fT.t/gt�0 has an attracting set P that is bounded in thenorm of F bC and compact in the topology �locC . Then the shift semigroup fT.t/gt�0has a trajectory attractor U P .

We are in position to formulate the main theorem of this section.

Theorem 14.3. The shift semigroup fT.t/gt�0 defined on the set K C of solutionsof Problem 14.1 has a trajectory attractor U which is bounded in F bC and compactin the topology �locC .

Before we pass to the proof of this theorem we will need three auxiliary lemmas.

Lemma 14.3. Let v 2 K C. For all h � 0 the following estimates hold:

kvk2L2.h;hC1IV/ C kvk2L1.h;hC1IH/ � C4 C C5kvk2L1.0;1IH/e�C6h; (14.45)


kv0k 43

L43 .h;hC1IV0/

� C7 C C8kvk 83

L1.0;1IH/e�C9h; (14.46)

where C4;C5;C6;C7;C8, and C9 are positive constants independent of h; v.

Proof. Let us fix h � 0 and v 2 K C. Since for all u 2 V , �1juj2 � kuk2 withthe constant �1 > 0, it follows from (14.18), that for all 0 � t1 < t2 and 2C10 ..t1; t2// with � 0,

� 1

2

Z t2

t1

jv.t/j2 0.t/ dt C C1�1

Z t2

t1

jv.t/j2 .t/ dt � C2.1C kFk2V0

/

Z t2

t1

.t/ dt:

(14.47)

We are in position to use Lemma 9.2 from [61] to deduce that there exists a Lebesguenull set Q R

C such that for all t; � 2 RC n Q we have

jv.t/j2eC1�1.t��/ � jv.�/j2 � eC1�1.t��/ � 1C1�1

C2.1C kFk2V0

/: (14.48)

Now, since the function t ! v.t/ is weakly continuous, it follows that t ! jv.t/j2is lower semicontinuous and (14.48) holds for all t � � (but only for a.e. � 2 R

C).Hence, for all t > 0 we can find � 2 .0;minf1; tg/ such that

jv.t/j2eC1�1.t��/ � kvk2L1.0;1IH/ C eC1�1.t��/

C1�1C2.1C kFk2V0

/; (14.49)

and moreover, for all t > 0, we have

jv.t/j2 � eC1�1.1�t/kvk2L1.0;1IH/ C C2.1C kFk2V0

/

C1�1: (14.50)

By Corollary 9.2 in [61] it follows from (14.18) that for almost all h > 0,

1

2jv.h C 1/j2 C C1

Z hC1

hkv.t/k2 dt � C2.1C kFk2V0

/C 1

2jv.h/j2:

Using (14.50) we obtain, for a.e. h > 0,

Z hC1

hkv.t/k2 dt � C2

C1.1C kFk2V0

/C C2.1C kFk2V0

/

2C21�1

C kvk2L1.0;1IH/eC1�1.1�h/

2C1;

and the estimate (14.45) follows for all h � 0 since both left- and right-hand side ofthe above inequality are continuous functions of h. Now, using (14.24) we get

kv0.t/k 43

V0

� C10 1C

1C jv.t/j 23

�kv.t/k2

�for a.e. t 2 R

C;


where C10 > 0. Integrating this inequality from h to h C 1 we obtain

Z hC1

hkv0.t/k4=3V0

dt � C10 C C10

�

1C jjv.t/jj 23L1.h;hC1IH/�Z hC1

hkv.t/k2 dt:

Inequality (14.46) follows from an application of (14.45). utLemma 14.4. The set K C is closed in the topology �locC .

Proof. Assume that for a sequence fvkg1kD1 K C we have

vk ! v weakly in L2loc.RCI V/;

vk ! v weakly-� in L1loc.R

CI H/;

v0k ! v0 weakly in L

43

loc.RCI V�/:

We need to show that v satisfies (14.16) and (14.18). Since fvkg1kD1 K C, we

have, for all k 2 N and z 2 V ,

hv0k.t/C Avk.t/C BŒvk.t/�; zi C .�k.t/; zn/L2.�C/ D hF; zi C hG.vk.t//; zi;

�k.t/ 2 [email protected]/n.�;t//

for a.e. t 2 RC. Passing to the limit in terms with A;B;G is analogous to that in the

proof of Theorem 14.1 (see also [62, 176, 219, 220]).In order to pass to the limit in the multivalued term observe that from .j2/ it

follows that for a.e. t 2 RC

k�k.t/k2L2.�C/� 2c21m1.�C/C 2c2k�k2kvk.t/k2:

Hence, after integration in the time variable we get, for all T 2 RC,

k�kk2L2.0;TIL2.�C//� 2Tc21L C 2c2k�k2kvkk2L2.0;TIV/:

By a diagonal argument it follows that there exists � 2 L2loc.RCI L2.�C// such that,

for a subsequence,

�k ! � weakly in L2loc.RCI L2.�C//:

As in the proof of Theorem 14.1,

.vk/n ! vn strongly in L2loc.RCI L2.�C//:

Using assertion .C/ of Theorem 3.21 we deduce that

�.x; t/ 2 @j.vn.x; t// a.e. in �C � RC;


and moreover,

�.t/ 2 [email protected].�;t// a.e. in RC:

Hence we can pass to the limit in the term .�k.t/; zn/L2.�C/. It remains to show thatv satisfies (14.18). The argument is similar to that in the proof of Theorem 14.1.We choose 0 � t1 < t2 and 2 C1

0 ..t1; t2//. We have, after possible refining toa subsequence, jvk.t/j2 0.t/ ! jv.t/j2 0.t/ for a.e. t 2 .t1; t2/. Moreover, sinceweakly-* convergent sequences in L1.t1; t2I H/ are bounded in this space it followsthat there exists a majorant for jvk.t/j2 0.t/. Hence, by the Lebesgue dominatedconvergence theorem and since vk.t/. .t//

12 ! v.t/. .t//

12 weakly in L2.t1; t2I V/,

we can pass to the limit in (14.18) written for vk, which completes the proof. utProof (of Theorem 14.3). Observe that from Lemma 14.3 it follows that for everyv 2 K C, kvkF b

C

< 1. Note that since the problem is autonomous, for any v 2K C and any h 2 R

C, the function vh, defined as vh.t/ D v.t C h/ for all t � 0,belongs to K C and hence (14.44) holds.

Now let us estimate kT.s/vkF bC

for v 2 K C and s � 0. We have

kT.s/vkF bC

D suph�s

nkvkL2.h;hC1IV/ C jjv.t/jjL1.h;hC1IH/ C kv0k

L43 .h;hC1IV0/

o:

Using Lemma 14.3 we get

kT.s/vkF bC

� suph�s

C

(

2 1C kvk2L1.0;1IH/e�C6h

� 12 C

�

1C kvk 83

L1.0;1IH/e�C9h

� 34

)

;

with a constant C > 0 dependent only on C4;C5;C7;C8 of Lemma 14.3. By a simplecalculation we obtain, for all s 2 R

C,

kT.s/vkF bC

� R0 C CkvkˇL1.0;1IH/e�ıs � R0 C CkvkˇF .0;1/e

�ıs; (14.51)

where C;R0; ˇ; ı > 0 do not depend on s; v.Now we define the set P as

P D fv 2 K C W kvkF bC

� 2R0g:

We will show that P is absorbing (and hence also attracting) for fT.t/gt�0. Let Bbe bounded in F bC. Hence B is bounded in F .0; 1/. Let kvkF .0;1/ � R for v 2 B.Now we choose s0 > 0 such that CRˇe�ıs0 � R0. From (14.51) we have for s � s0that

kT.s/vkF bC

� 2R0;

and we deduce that P is absorbing.


It suffices to show that P is compact in the topology �locC . The fact that itis relatively compact follows from its boundedness and basic properties of weakcompactness in reflexive Banach spaces. It remains to show that it is closed. Let

vk

�locC! v and fvkg P . From the weak lower semicontinuity of the norm it follows

that

kvkF bC

� lim infk!1 kvkkF b

C

� 2R0:

Moreover from Lemma 14.4 it follows that v 2 K C. Thus v 2 P and the proof iscomplete. utNow we show that any section of the trajectory attractor is a weak global attractor.We will start from the definition of a weak global attractor (cf., e.g., [227]).

Definition 14.4. A set A H is called a weak global attractor of the set K C if itis weakly compact in H and the following properties hold:

.i/ for any set B K C bounded in the norm of F bC its section B.t/ is attracted toA in the weak topology of H as t ! 1, that is, for any neighborhood Ow.A /

of A in the weak topology of H there exists a time � D �.B;Ow.A // suchthat

B.t/ Ow.A / for all t � �;

.ii/ A is the minimal weakly closed set in H that attracts the sections of all boundedsets in K C in the weak topology of H as t ! 1.

Fig. 14.2 Schematic illustration of trajectory attractor for the semiflow governed by the solutionmap of the one-dimensional ODE Px D jxj.1 � x/. The trajectory attractor U consists of twoconstant trajectories located at the stationary points 0 and 1, and of the family of bounded completetrajectories that connect those points. The global attractor A is the section of the trajectory attractorA D U.0/ D U.t/ for all t 2 R


We prove the following Theorem.

Theorem 14.4. The set A D U.0/ H is a weak global attractor of K C.

Proof. The argument is standard in the theory of trajectory attractors and followsthe lines of the proof of Assertion 5.3 in [227]. Since U is bounded in F bC and thusin F .0; 1/, by (14.43) we deduce that U.0/ is bounded in H. Moreover, since thetopology�locC is stronger than the topology of Cw.Œ0;T�I H/ it follows that if uk ! uin the topology�locC then uk.0/ ! u.0/weakly in H. From the fact that U is compactit follows that U.0/ is weakly compact in H.

To show .i/ let us take B K C bounded in the norm of F bC. This set isattracted to U in the topology�locC . Since for every sequence uk ! u in the topology�locC we have uk.0/ ! u.0/ weakly in H, it follows that .T.t/B/.0/ is attracted toU.0/ in the weak topology of H, or, in other words, B.t/ is attracted to U.0/ in theweak topology of H.

To show .ii/ let E be an arbitrary weakly closed subset of H that weakly attractsthe sections of bounded sets B K C, that is, for any weak neighborhood Ow.E /,B.t/ Ow.E / for all t � � D �.B;Ow.E //. Let us take B D U. We haveU.t/ Ow.E / for all t � � D �.U;Ow.E //. From the invariance of the trajectoryattractor we obtain that A D U.0/ D U.t/ Ow.E /. Since E is weakly closed inH, it follows that A E and the proof is complete. ut

The example which illustrates the fact that the global attractor is the section ofthe trajectory attractor is presented in Fig. 14.2.


In this chapter we followed [124].The system of Eqs. (14.1)–(14.2) with no-slip boundary conditions: (14.3) at

�D for h D const and u D const on �C (instead of (14.4)–(14.5) on �C) wasintensively studied in several contexts, some of them mentioned in the introductionof Boukrouche and Łukaszewicz [26]. The autonomous case with h ¤ const andwith u D const on �C was considered in [22, 24] (also see Chap. 9) and the non-autonomous case h ¤ const, u D U.t/e1 on �C was considered in [28] (alsosee Chap. 13). Existence of exponential attractors for the Navier–Stokes fluids andglobal attractors for Bingham fluids, with the Tresca boundary condition on �C

was proved in [162] (also see Chap. 10) and [27], respectively. Recently, attractorsfor two-dimensional Navier–Stokes flows with Dirichlet boundary conditions werestudied in [79], where the time continuous problem has a unique solution but asolution of the semidiscrete problem constructed by the implicit Euler scheme canbe nonunique, and hence the theory of multivalued semiflows is needed to study thetime discretized systems.

Asymptotic behavior of solutions for the problems governed by partial differen-tial inclusions where the multivalued term has the form of Clarke subdifferential


was studied in [132, 133], where the reaction-diffusion problem with multivaluedsemilinear term was considered, and in [123], where the strongly damped waveequation with multivalued boundary conditions was analyzed.

For the problem considered in this chapter, existence of weak solutions for thecase u� D 0 in place of (14.4) was shown in [176]. Our assumptions .j1/–.j3/ aremore general than the corresponding assumptions of Theorem 1 in [176], save forthe fact that j is assumed there to depend on space and time variables directly.

To prove the existence of attractors we used the theory of trajectory attractors,which, instead of the direct analysis of the multivalued semiflow (i.e., map thatassigns to initial condition the set of states obtainable after some time t, see [171,239] or the review paper [10]), focuses on the shift operator defined on the spaceof trajectories for the studied problem. This approach was introduced in papers [60,167, 210] as a method to avoid the nonuniqueness of solutions, indeed, the shiftoperator is uniquely defined even if the dynamics of the problem is governed by themultivalued semiflow. Recent results and open problems in the theory of trajectoryattractors are discussed in the survey papers [10, 227].

15Evolutionary Systems and the Navier–StokesEquations

This chapter is devoted to the study of three-dimensional nonstationaryNavier–Stokes equations with the multivalued frictional boundary condition. Weuse the formalism of evolutionary systems to prove the existence of weak globalattractor for the studied problem.

15.1 Evolutionary Systems and Their Attractors

We remind the framework of evolutionary systems [65, 66]. Let .X; ds.�; �// be ametric space. The metric ds will be referred to as the strong metric. We denote bydw another metric on X, referred to as the weak metric, satisfying the followingconditions:

1. the space X is dw compact,2. if limk!1 ds.uk; vk/ D 0 for some sequences uk; vk 2 X, then also

limk!1 dw.uk; vk/ D 0.

Any strongly compact (i.e., ds compact) subset of X is also weakly compact (i.e.,dw compact). Any strongly closed (i.e., ds closed) subset of X is also weakly closed(i.e., dw closed) and, as a subset of the dw compact set X, also dw compact. To definethe evolutionary system, let

T D fI W I D ŒT;1/ R or I D .�1;1/g:

By F .I/ we denote the set of all X-valued functions having the domain I 2 T .


337

338 15 Evolutionary Systems and the Navier–Stokes Equations

Definition 15.1. A map E that associates to each I 2 T a subset E .I/ F .I/ iscalled an evolutionary system if the following conditions hold:

.i/ E .Œ0;1// ¤ ;,.ii/ E .I C s/ D fu.�/ W u.� � s/ 2 E .I/g for all s 2 R,.iii/ fu.�/jI2 W u.�/ 2 E .I1/g E .I2/ for all I1; I2 2 T such that I2 I1,.iv/ E ..�1;1// D fu.�/ W u.�/jŒT;1/ 2 E .ŒT;1// for all T 2 Rg.

For I D ŒT;1/ the set E .I/ is referred to as the set of trajectories on the interval I,and E ..�1;1// is called the set of complete trajectories. Having the evolutionarysystem we can define the map G W Œ0;1/ � X ! 2X by

G .t; x/ D fu.t/ W u.0/ D x for some u 2 E .Œ0;1//g:

We can naturally extend G to Œ0;1/ � 2X by defining G .t;A/ D Sx2A G .t; x/.

Remark 15.1. If we additionally assume the existence, i.e., that for any x0 2 X thereexists u 2 E .Œ0;1// such that u.0/ D x0, then the map G becomes an m-semiflowin the sense of Melnik and Valero [171], that is:

.i/ G .0; x/ D fxg for all x 2 X,.ii/ G .t C s;X/ G .t;G .s; x// for all x 2 X and s; t � 0.

The illustration of an evolutionary system is presented in Fig. 15.1.

Fig. 15.1 Illustration of an evolutionary system. If a semiflow (or m-semiflow) is dissipative, i.e.,has a bounded absorbing set X, then this set, endowed with appropriate metrics becomes themetric space for the evolutionary system. Moreover, the endings of all trajectories, which mustbe contained in the absorbing set (thickened line in the figure), constitute the family E .I/

Let A X and r � 0. We denote Bw.A; r/ D fx 2 X W distw.x;A/ < rg,where distw.x;A/ D infy2A dw.x; y/. Moreover, for sets A;B X we define the weakHausdorff semi-distance distw.A;B/ D supx2A distw.x;B/. We pass to the definitionof uniformly weakly attracting sets and weak attractors.

Definition 15.2. A set A X is uniformly weakly attracting (uniformly dw

attracting) if for any " > 0 there exists t0 > 0 such that G .t;X/ Bw.A; "/ forall t � t0, or, equivalently

limt!1 distw.G .t;X/;A/ D 0:

15.1 Evolutionary Systems and Their Attractors 339

Definition 15.3. A set A X is a weak global attractor (dw global attractor) if itis a minimal weakly closed uniformly weakly attracting set.

Of course, the weak global attractor must be weakly compact. The followingtheorem gives the existence of the weak global attractor.

Theorem 15.1 (cf. [65, Theorem 3.9]). Every evolutionary system possesses aweak global attractor A X.

Let a; b 2 R be such that a < b. Consider the Banach space Cw.Œa; b�I X/ of weaklycontinuous functions from the interval Œa; b� to the space X. This space is a metricone, equipped with the metric

dCw.Œa;b�IX/.u; v/ D supt2Œa;b�

dw.u.t/; v.t//:

This metric can be naturally extended to Cw.Œa;1/I X/ by the formula

dCw.Œa;1/IX/.u; v/ DX

k2N

1

2k

supt2Œa;aCk� dw.u.t/; v.t//

1C supt2Œa;aCk� dw.u.t/; v.t//:

We make the following assumption on the evolution system:

.A1/ the set E .Œ0;1// is compact in Cw.Œ0;1/I X/:

Assumption .A1/ will yield the invariance of the global attractor. To define thenotion of invariance we first set

QG .t;A/ D fu.t/ W u.0/ 2 A for u 2 E ..�1;1//g; where A X; t 2 R:

Definition 15.4. The set A X is said to be invariant (in the sense of Foias–Cheskidov) if QG .t;A/ D A for all t � 0.

Remark 15.2. Since QG .t;A/ G .t;A/, then the fact that A is invariant implies thatA G .t;A/, i.e., A is negatively semi-invariant in the sense of Melnik and Valero(see [171]).

Theorem 15.2 (cf. [65, Theorem 5.8]). Let E be an evolutionary system thatsatisfies .A1/. Then the weak global attractor A is the maximal invariant set.Moreover,

A D fx 2 X W x D u.0/ for some u 2 E ..�1;1//g;where, by invariance, we can also take values at any t � 0 instead of zero. Moreover,for any " > 0 there exists t0 > 0 such that for any t� > t0 and for any trajectoryu 2 E .Œ0;1// we have

dCw.Œt�;1/IX/.u; v/ < "

for a certain complete trajectory v 2 E ..�1;1//.

The last assertion of the above theorem is known as the tracking property.


15.2 Three-Dimensional Navier–Stokes Problemwith Multivalued Friction

We will consider a flow between two surfaces, the flat one at the bottom, and theperiodic one at the top. Define ˝1 D f.x1; x2; x3/ 2 R

3 W 0 < x3 < h.x1; x2/g,where the function h.�; x2/ is assumed to be L1-periodic, and h.x1; �/ is assumed tobe L2-periodic. Moreover we assume that h is Lipschitz and that

min.x1;x2/2Œ0;L1��Œ0;L2�

h.x1; x2/ > 0:

The periodicity of h implies that it suffices to consider the flow inside the domain

˝ D f.x1; x2; x3/ 2 R3 W .x1; x2/ 2 .0;L1/ � .0;L2/; 0 < x3 < h.x; y/g:

Let t0 2 R be the initial time. The momentum equation and the incompressibilitycondition are, respectively, given by

ut � ��u C .u � r/u C rp D f on ˝ � .t0;1/; (15.1)

div u D 0 on ˝ � .t0;1/; (15.2)

where we look for the velocity u W ˝�.t0;1/ ! R3 and pressure p W ˝�.t0;1/ !

R. The function f W ˝ ! R3 is the mass force density, and � > 0 is the viscosity

coefficient. The boundary of ˝ is divided into the following parts:

�D D .0;L1/ � .0;L2/ � f0g;�C D f.x1; x2; x3/ 2 R

3 W .x1; x2/ 2 .0;L1/ � .0;L2/; x3 D h.x1; x2/g;�L D �L1 [ �L2;

�L1 D f0;L1g � f.x2; x3/ 2 R2 W 0 < x2 < L2; 0 < x3 < h.0; x2/g;

�L2 D f.x1; x2; x3/ 2 R3 W 0 < x1 < L1; x2 2 f0;L2g; 0 < x3 < h.x1; 0/g:

The domain ˝ and three parts of its boundary are schematically presented inFig. 15.2. The stress tensor T is associated with the symmetrized displacementgradient D.u/ D 1

2.ru C ru|/ and the pressure by the linear constitutive law

T D �pI C 2�D.u/. On the boundary of ˝ the stress can be decomposed intonormal and tangential components: Tn D .Tn/ � n and T� D Tn � Tnn.

On �D we assume the homogeneous Dirichlet boundary condition

u D 0 on �D � .t0;1/: (15.3)

15.2 Three-Dimensional Navier–Stokes Problem with Multivalued Friction 341

Fig. 15.2 The example setupof the studied problem. Weassume the frictional contactbetween the flowing liquidand the foundation on �C,periodic boundary conditionson �L, and the homogeneousDirichlet boundary conditionson �D

We assume the periodicity of normal stresses and velocities on �L, i.e.,

.x1; x2; x3; t/ 2 �L1 � Œt0;1/ ) u.0; x2; x3; t/ D u.L1; x2; x3; t/ (15.4)

^ .Tn/.0; x2; x3; t/ D .Tn/.L1; x2; x3; t/;

.x1; x2; x3; t/ 2 �L2 � Œt0;1/ ) u.x1; 0; x3; t/ D u.x1;L2; x3; t/ (15.5)

^ .Tn/.x1; 0; x3; t/ D .Tn/.x1;L2; x3; t/:

Note that, due to the periodicity of u, the periodicity of normal stresses on �L1 meansthat, for .x1; x2; x3/ 2 �L1 we have

p.0; x2; x3; t/ D p.L1; x2; x3; t/ and ui;1.0; x2; x3; t/ D ui;1.L1; x2; x3; t/ for i ¤ 1;

and on �L2 we have

p.x1; 0; x3; t/ D p.x1;L2; x3; t/ and ui;2.x1; 0; x3; t/ D ui;2.x1;L2; x3; t/ for i ¤ 2:

Assume that the velocity u is sufficiently smooth. The fact that on �L1 the functionu.�; x2; x3/ is L1-periodic implies the same periodicity of partial derivatives ui;2 andui;3 for i D 1; 2; 3, and, by the incompressibility condition, also of u1;1. Similarly,from L2-periodicity of u.x1; �; x3/ on �L2 we obtain the same L2-periodicity ofthe partial derivatives ui;1 and ui;3 for i D 1; 2; 3, and, by the incompressibilitycondition, also of u2;2.

On �C we assume that

un D 0 on �C � .t0;1/; (15.6)

�T� 2 g.u� / on �C � .t0;1/; (15.7)

where g W �C �R ! 2R is a friction multifunction, un D u � n is the normal velocity,and u� D u � unn is the tangential velocity. Finally, we need the initial condition

u D u0 on ˝ for t D t0: (15.8)


We pass to the weak formulation for the studied problem. First, we define the spaces:

QV Dfv 2 C1.˝/3 W div v D 0 in ˝; v D 0 on �D; vn D 0 on �C;

v.�; x2; x3/ is L1-periodic on �L1; v.x1; �; x3/ is L2-periodic on �L2g;

V D closure of QV in H1.˝/3 and H D closure of QV in L2.˝/3:

The spaces V H V 0 constitute the evolution triple with dense and continuousembeddings. Moreover, the embedding V H is compact. As the Dirichletboundary is nontrivial, the norm in the space V equivalent with the standard H1 normis given by kvk2 D R

˝rv.x/W rv.x/ dx. We will also use the space W D L2.�C/

3

as well as the linear and continuous trace operator � W V ! W, whose norm wewill denote by k�kL .VIW/ D k�k. The calligraphic symbols will be used for time-dependent spaces: V D L2.t0;TI V/, V 0 D L2.t0;TI V 0/.

To motivate the definition of the weak solution assume that smooth functionsu; p satisfy (15.1)–(15.8). Multiplying the momentum equation (15.1) by v 2 QVand integrating over ˝, after application of the incompressibility condition (15.2),the boundary conditions (15.3)–(15.7), and integration by parts, we arrive at thefollowing weak form of the problem,

Z

˝

ut � v dx C �

Z

˝

ru � rv dx CZ

˝

.u � r/u � v dx CZ

�C

� � v� dS DZ

˝

f � v dx;

where �.x; t/ 2 g.x; u� .x; t// for .x; t/ 2 �C � Œt0;1/.

Exercise 15.1. Derive the weak formulation of the considered problem.

Hint. The derivation of the above weak form is similar to that in Chap. 5. Notethat due to the fact that the contact boundary is a flat surface we can consider thebilinear form

R˝

ru.x/W rv.x/ dx instead of more generalR˝

D.u/.x/W D.v/.x/ dx(see Remark 5.1). Moreover, the periodic conditions on u;Tn, and the definition ofthe space QV imply that all boundary integrals on �L cancel.

15.3 Existence of Leray–Hopf Weak Solution

We assume that .t0;T/, the time interval, is given. Using the Galerkin method wewill establish the existence of the weak solution for considered problem, understoodin the Leray–Hopf sense, on the interval .t0;T/. First, we define the operator A WV ! V 0 by the formula

hAu; vi DZ

˝

ru.x/W rv.x/ dx for u; v 2 V:

15.3 Existence of Leray–Hopf Weak Solution 343

Obviously, the operator A is linear and bounded, moreover hAu; ui D kuk2 foru 2 V . We define the Nemytskii operator A W V ! V 0 by the formula .A u/.t/ DA.u.t// for a.e. t 2 .t0;T/. This Nemytskii operator is also linear and bounded,and hence it is continuous and moreover also weakly sequentially continuous, i.e.,uk ! u weakly in V implies that A uk ! A u weakly in V 0. Next, we define thebilinear operator B W V � V ! V 0 as

hB.u;w/; vi DZ

˝

.u � r/w � v dx:

Similar as in Lemmata 5.4 and 5.5 we have for u; v;w 2 V ,

hB.u;w/; vi � cBkukkwkkvk; (15.9)

hB.u; v/; vi D 0; (15.10)

hB.u;w/; vi D �hB.u; v/;wi; (15.11)

and B is sequentially weakly continuous, i.e.,

wk ! w and uk ! u weakly in V (15.12)

) hB.wk; uk/; vi ! hB.w; u/; vi for all v 2 V:

The Nemytskii operator for B has values only in L43 .t0;TI V 0/ and not in V 0. Indeed,

let us consider the function .t0;T/ 3 t ! B.u.t/; v.t// 2 V 0, where u; v 2 V \L1.t0;TI H/. We have

kB.u.t/; v.t//kV0 D supkwkD1

Z

˝

.u.t/ � r/v.t/ � w dt D supkwkD1

Z

˝

.u.t/ � r/w � v.t/ dt

�ku.t/kL4.˝/3kv.t/kL4.˝/3 : (15.13)

We use the following Ladyzhenskaya interpolation inequality valid for all u 2H1.˝/3 (see Theorem 3.11)

kukL4.˝/3 � Ckuk 14

L2.˝/3kuk 3

4

H1.˝/3:

In our case we have

kukL4.˝/3 � Cjuj 14 kuk 34 ; (15.14)

and hence

kB.u.t/; v.t//k 43

V0

�C2ju.t/j 13 ku.t/kjv.t/j 13 kv.t/k;


whereas we can define

B W .V \ L1.t0;TI H// � .V \ L1.t0;TI H// ! L43 .t0;TI V 0/

by .B.u; v//.t/ D B.u.t/; v.t//. We have the following result.

Lemma 15.1. Let the sequences uk ! u and vk ! v both converge weakly inL2.t0;TI V/ and strongly in L2.t0;TI H/. Then we have B.uk; vk/ ! B.u; v/ weaklyin L

43 .t0;TI V 0/.

Proof. The proof follows the lines of the proof of Lemma III.3.2 in [219]. It isenough to prove that for smooth w W .t0;T/ �˝ ! R

3 we have

Z T

t0

hB.uk.t/; vk.t//;w.t/i dt !Z T

t0

hB.u.t/; v.t//;w.t/i dt:

The result follows by the density of smooth functions in L4.t0;TI V/, a predual spaceof L

43 .t0;TI V 0/. We have

Z T

t0

hB.uk.t/; vk.t//;w.t/i dt D �3X

i;jD1

Z T

t0

Z

˝

.uk.t//[email protected]/

@xi.vk.t//j dx dt:

The last integrals converge to

�3X

i;jD1

Z T

t0

Z

˝

.u.t//[email protected]/

@xi.v.t//j dx dt D

Z T

t0

hB.u.t/; v.t//;w.t/i dt;

and the proof is complete. utWe pick the initial condition u0 2 H and the source term f 2 V 0.We assume that the friction multifunction g W �C � R

3 ! 2R3

satisfies:

.g1/ for every s 2 R3 the multifunction x ! g.x; s/ has a measurable selection,

i.e., there exists gs W �C ! R3, a measurable function, such that gs.x/ 2 g.x; s/

for all x 2 �C,.g2/ the set g.x; s/ is nonempty, compact, and convex in R

3, for all s 2 R3 and

for a.e. x 2 �C,.g3/ graph of s ! g.x; s/ is closed in R

6 for a.e. x 2 �C,.g4/ we have j�j � c1 C c2jsj for all s 2 R

3, a.e. x 2 �C and all � 2 g.x; s/, withc2 � 0 and c1 2 R,

.g5/ we have the following dissipativity condition � � s � �d1 � d2jsj2 for all

s 2 R3, a.e. x 2 �C and all � 2 g.x; s/ with d1 � 0 and d2 2

0; �

k�k2�

.

Finally, we define the multivalued operators N W W ! 2W and N W L2.t0;TI W/ !2L2.t0;TIW/ as


N.v/ Df� 2 W W �.x/ 2 g.x; v.x// a.e. on �Cg;N .v/ Df� 2 L2.t0;TI W/ W �.x; t/ 2 g.x; v.x; t// a.e. on �C � .t0;T/g:

If v; � 2 L2.t0;TI W/, then we have

� 2 N .v/ , �.t/ 2 N.v.t// for a.e. t 2 .t0;T/, �.x; t/ 2 g.x; v.x; t// for a.e. .x; t/ 2 �C � .t0;T/:

Let 0 < �1 � �2 � : : : � �k � : : : be the sequence of the eigenvalues of theoperator A, with the corresponding eigenvectors fzkg1

kD1 being orthonormal in Hand orthogonal in V . Define the finite dimensional spaces Vk D span fz1; : : : ; zkg.These spaces approximate V from the inside, i.e.,

S1kD1 Vk is dense in V . By Pkv we

denote the projection onto Vk of the function v 2 H. Then jPkvj � jvj and Pkv ! v

strongly in H as k ! 1. We start with the following Galerkin problem.

Problem 15.1. Find u 2 AC.Œt0;T�I Vk/ such that u.t0/ D Pku0 and for all v 2 Vk

and a.e. t 2 .t0;T/ we have

Z

˝

u0 �v dxC�Z

˝

ruW rv dxCZ

˝

.u �r/u �v dxCZ

�C

� �v� dS D hf ; vi; (15.15)

with � 2 N .u� /.

Theorem 15.3. Let .g1/–.g4/ hold. Problem 15.1 has at least one solution.

Proof. We will prove the existence of solution to this ordinary differential inclusionby means of the Kakutani–Fan–Glicksberg fixed point theorem (see Theorem 3.8).Fix � 2 L2.t0;TI W/ and w 2 L2.t0;TI V/, and consider the following auxiliarylinear problem.

Problem 15.2. Find u 2 AC.Œt0;T�I Vk/ such that u.t0/ D Pku0 and for all v 2 Vk

and a.e. t 2 .t0;T/ we have

Z

˝

u0.t/�v dxC�Z

˝

ru.t/W rv dxCZ

˝

.w.t/�r/u.t/�v dxCZ

�C

�.t/�v� dS D hf ; vi:(15.16)

This linear system of ODEs has a unique solution u. We define a multifunction� W V � L2.t0;TI W/ ! V � 2L2.t0;TIW/ which assigns to the pair .w; �/ the setfug �N .u� /. First, we note that by assertion .H1/ of Theorem 3.23, the set N .u� /is nonempty and convex, so the multifunction � assumes nonempty and convexvalues. We will prove, by the Kakutani–Fan–Glicksberg fixed point theorem (seeTheorem 3.8), that this multifunction has a fixed point. This fixed point is a solutionof Problem 15.1. Taking v D u.t/ in (15.16), we get

1

2

d

dtju.t/j2 C �ku.t/k2 � k�.t/kWk�kku.t/k C kf kV0ku.t/k:


It follows that

d

dtju.t/j2 C �ku.t/k2 � 2k�k2

�k�.t/k2W C 2kf k2V0

�:

After integration on .t0; t/, for any t 2 .t0;T/ we get

ju.t/j2 C �

Z t

t0

ku.s/k2 ds � ju0j2 C 2k�k2�

k�k2L2.t0;tIW/ C 2kf k2V0

.t � t0/

�:

In particular,

kuk2V � 1

�ju0j2 C 2k�k2

�2k�k2L2.t0;TIW/ C 2kf k2V0

.T � t0/

�2; (15.17)

and

ku.t/k2 � C2k ju.t/j2 � C2

k ju0j2 C C2k2k�k2�

k�k2L2.t0;TIW/ C C2k2kf k2V0

.T � t0/

�;

(15.18)

where the constant Ck depends only on the dimension of the space Vk. Let � 2N .u/. We have

k�.t/kW D supkzkW D1

Z

�C

�.x; t/ � z.x/ dx � supkzkW D1

Z

�C

.c1 C c2ju.x; t/j/jz.x/j dx

�c1p

m2.�C/C c2k�u.t/kW � c1p

L1L2 C c2k�kku.t/k:

Hence

k�.t/k2W � 2c21L1L2 C 2c22k�k2ku.t/k2 (15.19)

� 2c21L1L2C2c22k�k2

C2k ju0j2CC2

k2k�k2�

k�k2L2.t0;TIW/CC2

k2kf k2V0

.T�t0/

�

!

;

and, after integration on .t0;T/,

k�k2L2.t0;TIW/ � 2.T � t0/.c21L1L2 C c22k�k2C2

k ju0j2/ (15.20)

C 4.T � t0/c22C2kk�k4

�k�k2L2.t0;TIW/ C 4c22k�k2C2

kkf k2V0

.T � t0/2

�:

Clearly, from (15.17) and (15.20), if only .T � t0/ <�

4c22C2k k�k4 , we can find R1 > 0

and R2 > 0 such that � is a map from S to 2S, where

S D f.u; �/ 2 V � L2.t0;TI W/ W kukV � R1; k�kL2.0;TIW/ � R2g:


Consequently, the fixed point argument will give us the existence of solution onthe interval Œt0;T� for .T � t0/ <

�

4c22C2k k�k4 , and by subsequent continuation this

solution can be elongated to arbitrarily long time interval. It remains to prove thatthe graph of � is weakly closed in S � S. In view of the fact that V � L2.t0;TI W/is a reflexive Banach space, and S is a bounded, closed, and convex set, it is enoughto show the sequential weak closedness. Let wk ! w weakly in V and �k ! �

weakly in L2.0;TI W/, with .wk; �k/ 2 S. Let moreover .uk; �k/ 2 �.wk; �k/ besuch that uk ! u weakly in V and �k ! � weakly in L2.t0;TI W/. We know that.uk; �k/ 2 S. Let us introduce an equivalent norm on the finite dimensional space Vk

by the formula

kzkV0

kD sup

v2Vk ;kvkD1

Z

˝

z � v dx:

We have

ku0k.t/kV0

kD sup

v2VkkvkD1

Z

˝

u0k.t/ � v dx D sup

v2VkkvkD1

�

hf ; vi � �Z

˝

ruk.t/W rv dx

�Z

˝

.wk.t/ � r/uk.t/ � v dx �Z

�C

�k.t/ � v� dS

�

� kf kV0 C �kuk.t/k C kwk.t/kkuk.t/k C k�k.t/kWk�k:

Using (15.18) and (15.19) it follows from the fact that wk is uniformly boundedin L2.t0;TI W/ that u0

k is uniformly bounded in L2.t0;TI V 0k/ and moreover in V ,

as the space Vk is finite dimensional. Hence, for a subsequence, not renumbered,u0

k ! u0 weakly in V and uk ! u strongly in V . Moreover, .uk/� ! u� stronglyin L2.t0;TI W/. As �k ! � weakly in L2.t0;TI W/ and �k 2 N ..uk/� /, by assertion.H2/ of Theorem 3.23 it follows that � 2 N .u� /. It remains to prove that u solvesProblem 15.2 with w and �, using the fact the uk solves this problem with wk and �k.To this end, it suffices to multiply Eq. (15.16) written for uk;wk; �k by the function� 2 L2.t0;T/ and integrate the resultant equation over the interval .t0;T/. Theassertion follows by passing to the limit in the obtained equation. As the detailsare technical, their verification is left to the reader as an exercise. utWe are in position to define the Leray–Hopf weak solution of the consideredproblem. Since we need two types of weak solutions, we call this class a type ILeray–Hopf weak solution.

Definition 15.5. By the type I Leray–Hopf weak solution of the three-dimensionalNavier–Stokes problem with multivalued friction we mean a function

u 2 L2loc.Œt0;1/I V/ \ L1loc.Œt0;1/I H/ with u0 2 L

43

loc.Œt0;1/I V 0/ such that:

• u.t0/ D u0;• for all v 2 V and a.e. t > t0 we have


hu0.t/C �Au.t/C B.u.t/; u.t// � f ; vi CZ

�C

�.t/ � v� dS D 0; (15.21)

with � 2 L2loc.Œt0;1/I W/ such that �.t/ 2 N.u� .t// for a.e. t > t0,• for a.e. t � t0 (including t0) and for all s > t we have

1

2ju.s/j2 C �

Z s

tku.r/k2 dr C

Z s

t.�.r/; u� .r//W � hf ; u.r/i dr � 1

2ju.t/j2:

(15.22)

Remark 15.3. Every type I Leray–Hopf weak solution defined above belongs to thespaces C.Œt0;1/I V 0/ and Cw.Œt0;1/I H/. Moreover, (15.22) implies that ju.t/j2 iscontinuous from the right at t0.

Theorem 15.4. Assume .g1/–.g5/. For any u0 2 H there exists at least one type ILeray–Hopf weak solution given by Definition 15.5.

Proof. The proof is standard in the theory of three-dimensional Navier–Stokesequations and it is based on the Galerkin method. It is enough to prove, for certaingiven T > t0, the existence of a function u 2 L2.t0;TI V/ \ L1.t0;TI H/ withu0 2 L

43 .t0;TI V 0/ and corresponding � 2 L2.t0;TI W/ which satisfies the initial

condition, (15.21) for a.e. t 2 .t0;T/, and (15.22) for a.e. t 2 .t0;T/ and for t D t0.The solution on the interval .t0;1/ can then be obtained by concatenating the weaksolutions obtained on the intervals of length T�t0 by taking the value of the previoussolution at the endpoint of the time interval as the initial condition for the subsequentproblem. We will proceed in the standard way obtaining the Leray–Hopf weaksolution by passing to the limit in the Galerkin problems. Let uk 2 AC.Œt0;T�I Vk/

be the solutions of Problem 15.1 with the initial condition Pku0. We denote by �k thecorresponding selections of N ..uk/� /. Taking v D uk.t/ in (15.15) we get for a.e.t 2 .t0;T/,

1

2

d

dtjuk.t/j2 C �kuk.t/k2 C

Z

�C

�k.t/ � .uk/� .t/ dS D hf ; uk.t/i: (15.23)

Using .g4/ we get

Z

�C

j�k.t/ � .uk/� .t/j dS �Z

�C

.c1 C c2j.uk/� .x; t/j/j.uk/� .x; t/j dS

� .c2 C 1/kuk.t/k2W C c21m2.�C/

4: (15.24)

Let ı 2 .0; 12/ and let Z D V \ H1�ı.˝IR3/ be endowed with H1�ı.˝IR3/

topology. The embedding V Z is compact and Z H is continuous. Moreover,the trace operator from Z to W is linear and continuous. Thus, we can use the Ehrlinglemma (cf. Lemma 3.17) to deduce that for any " > 0 there exists C."/ > 0 suchthat kvkW � "kvk C C."/jvj for all v 2 V . Thus, from (15.24) we get


Z

�C

j�k.t/ � .uk/� .t/j dS � "kuk.t/k2 C C1."/juk.t/j2 C C2;

with constants C1."/;C2 > 0. Coming back to (15.23) we get

1

2

d

dtjuk.t/j2C�kuk.t/k2 � 2"kuk.t/k2C 1

4"kf kV0 CC1."/juk.t/j2CC2: (15.25)

It suffices to take " D �4. Now, we use the Gronwall lemma (cf. Lemma 3.20) to

deduce that uk is uniformly bounded in V and L1.t0;TI H/. To prove that �k isbounded in L2.t0;TI W/ we observe that

k�k.t/k2W �Z

�C

.c1 C c2j.uk/� .x; t/j/2 dS � 2c21m2.�C/C 2c22k�k2kuk.t/k2;

and the bound of uk in V implies that, indeed, �k is bounded in L2.t0I TI W/. Toobtain the estimates of u0

k we will use the fact that we chose the eigenfunctions ofthe operator A as a basis of the space Vk. The projection operator Pk W V 0 ! Vk isgiven by Pkv D Pk

iD1hv; ziizi, where zi is the i-th eigenfunction of the operator A.Define � W V ! W by the tangential component of the trace, i.e., �v D .�v/� and�� W W ! V 0 its adjoint operator given by h��v;wi D .v;w� /W . Then we have

Z

�C

�k.t/ � v� dS D h��k.t/; vi:

The Galerkin equation (15.15) can be rewritten in the following way:

u0k.t/C �Auk.t/C PkB.uk.t/; uk.t//C Pk�

��k.t/ D Pkf :

As, by Theorem 3.17, we have kPkvkV0 � kvkV0 , it follows that

ku0k.t/kV0 ��kAkL .VIV0/kuk.t/k C kf kV0 (15.26)

C k��kL .WIV0/k�k.t/kW C kB.uk.t/; uk.t//kV0 :

Using (15.13) and (15.14) we get

kB.uk.t/; uk.t//kV0 � Cjuk.t/j 12 kuk.t/k 32 :

The last two bounds imply that u0k is bounded in L

43 .t0;TI V 0/. Thus, for a

subsequence, not renumbered, we have

uk ! u weakly in V ; (15.27)

uk ! u weakly- � in L1.t0;TI H/; (15.28)



43 .t0;TI H/; (15.29)

uk ! u strongly in L2.t0;TI H/; (15.30)

uk.t/ ! u.t/ strongly in H for almost all t 2 .t0;T/; (15.31)

uk ! u strongly in C.Œt0;T�I V 0/; (15.32)

uk.t/ ! u.t/ weakly in H for all t 2 Œt0;T�; (15.33)

�k ! � weakly in L2.t0;TI W/; (15.34)

.uk/� ! u� strongly in L2.t0;TI W/: (15.35)

The convergence (15.30) follows from the compact embedding

fu 2 V W u0 2 L43 .t0;TI V 0/g L2.t0;TI H/;

the convergence (15.32) follows from the compact embedding

fu 2 L1.t0;TI H/ W u0 2 L43 .t0;TI V 0/g C.Œt0;T�I V 0/;

while (15.34) follows from the compact embedding

fu 2 L2.t0;TI V/ W u0 2 L43 .t0;TI V 0/g L2.0;TI Z/;

and the continuity of the Nemytskii trace operator from L2.0;TI Z/ to L2.0;TI W/.The convergence (15.33) follows by the computation

.uk.t/; v/ D .Pku0; v/CZ t

t0

hu0k.s/; vi ds;

valid for v 2 H, the strong convergence Pku0 ! u0 in H, and (15.29).Multiplying (15.15) by ' 2 C1..t0;T// and integrating over .t0;T/ we have, forv 2 Vk,

Z T

t0

hu0k.t/C �Auk.t/C B.uk.t/; uk.t// � f ; v'.t/i dt C

Z T

t0

.�k.t/; v�'.t//W dt D 0:

Using (15.27), (15.29), (15.30), (15.33), and Lemma 15.1 we can pass to the limitin above equation, getting

Z T

t0

hu0.t/C �Au.t/C B.u.t/; u.t// � f ; v'.t/i dt CZ T

t0

.�.t/; v�'.t//W dt D 0;

15.4 Existence and Invariance of Weak Global Attractor, and Weak Tracking. . . 351

valid for v 2 S1kD1 Vk and ' 2 C1..t0;T//, whence (15.21) follows. Integrat-

ing (15.23) on .t; s/ we get, for all t; s with t0 � t < s � T ,

1

2juk.s/j2 C �

Z s

tkuk.r/k2 dr C

Z s

t.�k.r/; .uk/� .r//W � hf ; uk.r/i dr � 1

2juk.t/j2:

(15.36)

Using (15.27), (15.31), (15.34), and (15.35) as well as the sequential weak lowersemicontinuity of the norm of the space L2.t; sI V/ we get, for a.e. t 2 .t0;T/ anda.e. s 2 .t;T/,

1

2ju.s/j2 C �

Z s

tku.r/k2 dr C

Z s

t.�.r/; u� .r//W � hf ; u.r/i dr � 1

2ju.t/j2:

The convergence (15.33) and the sequential weak lower semicontinuity of the normj � j imply that the last inequality holds for all s 2 Œt;T�. Integrating (15.23) on .t0; s/we get, for all s 2 .t0;T�,1

2juk.s/j2 C �

Z s

t0

kuk.r/k2 dr CZ s

t0

.�k.r/; .uk/� .r//W � hf ; uk.r/i dr � 1

2jPku0j2:

As we did in (15.36), we can pass to the limit in the above inequality, to get

1

2ju.s/j2 C �

Z s

t0

ku.r/k2 dr CZ s

t0

.�.r/; u� .r//W � hf ; u.r/i dr � 1

2ju0j2:

We have proved (15.22). The fact that u.t0/ D u0 follows from (15.33) for t D t0 andthe fact that uk.0/ D Pku0 ! u0 strongly in H as k ! 1. Finally, (15.34), (15.35),and assertion .H2/ of Theorem 3.23 imply that � 2 N .u� /. The proof is complete.

utRemark 15.4. There are two “sources” of possible nonuniqueness of Leray–Hopfweak solution of the studied problem. First, the nonlinear term B which, inthree-dimensional case, makes the weak solution uniqueness unknown (cf., e.g.,[99, 219]), and, moreover, the nonmonotone friction multifunction g which also cancontribute to the lack of solution uniqueness.

15.4 Existence and Invariance of Weak Global Attractor,and Weak Tracking Property

First we discuss the dissipativity of the type I Leray–Hopf weak solutions. We recalla useful lemma.


Lemma 15.2 (cf. [11, Lemma 7.2]). Let W Œt0;1/ ! R be lower semicontinu-ous, continuous at t0, 2 L1.0;T/ for all T > 0 and let, for some constant c � 0

.s/C cZ s

t.�/ d� � .t/;

for all s � t, for a.e. t > t0 and for t D t0. Then

.t/ � .t0/e�c.t�t0/ for all t � t0:

We use the above lemma to get a dissipativity result for type I Leray–Hopf weaksolutions.

Lemma 15.3. If u is a type I Leray–Hopf weak solution given by Definition 15.5,then for all t � t0 we have

ju.t/j2 � e��1.��d2k�k2/.t�t0/ju0j2 C kf k2V0

C 2d1L1L2.� � d2k�k2/�1.� � d2k�k2/2 : (15.37)

Proof. Using .g5/ in (15.22) we get

1

2ju.s/j2 C �

Z s

tku.r/k2 dr C

Z s

t

Z

�C

�d1 � d2ju� .x; r/j2 dS dr

� 1

2ju.t/j2 C "

Z s

tku.r/k2 dr C kf k2V0

.s � t/

4";

for all s � t and for a.e. t > t0 and for t D t0, where " > 0 is arbitrary. Using thetrace inequality, we get

1

2ju.s/j2C.� � d2k�k2�"/

Z s

tku.r/k2dr�1

2ju.t/j2C

kf k2V0

4"Cd1L1L2

!

.s�t/:

We take " D ��d2k�k22

. Using the Poincaré inequality �1jvj2 � kvk2 valid for v 2 V ,we get

ju.s/j2 C �1.� � d2k�k2/Z s

tju.r/j2dr

� ju.t/j2 C

kf k2V0

� � d2k�k2 C 2d1L1L2

!

.s � t/:

Denote A D �1.� � d2k�k2/ and B D kf k2V0

��d2k�k2 C 2d1L1L2. We have

ju.s/j2 � B

AC A

Z s

tju.r/j2 � B

Adr � ju.t/j2 � B

A:


As u 2 Cw.Œt0;1/I H/ and ju.t/j2 is continuous from the right at t0 (cf.Remark 15.3), it follows that the function .r/ D ju.r/j2� B

A satisfies all assumptionsof Lemma 15.2. It follows that

�

ju.t/j2 � B

A

�

��

ju0j2 � B

A

�

e�A.t�t0/ for all t � t0;

whence we get the assertion of the lemma. The proof is complete. utDefine

X D fu 2 H W juj2 � Rg;

where R is any number strictly greater thankf k2

V0

C2d1L1L2.��d2k�k2/�1.��d2k�k2/2 . Directly

from (15.37) we get the following result.

Lemma 15.4. Let K H be a bounded set, and let t0 2 R. For every type I Leray–Hopf weak solution u with the initial condition u.t0/ D u0 2 K there exists a timet1 D t1.t0;K/ such that for all t � t1 u.t/ 2 X.

We endow X with two metrics. The strong metric ds W X � X ! Œ0;1/ is given byds.u; v/ D ju � vj. As the weak topology of H is metrizable on the closed ball X wecan define the metric dw W X � X ! Œ0;1/ such that dw.uk; u/ ! 0 if and only ifuk ! u weakly in H. Note that the set X is compact in dw.

We need to define a class of type II Leray–Hopf weak solutions (see [65, 66]).

Definition 15.6. By the type II Leray–Hopf solution of the three-dimensionalNavier–Stokes problem with multivalued friction we mean a function u 2L2loc.Œt0;1/I V/ \ L1

loc.Œt0;1/I H/ with u0 2 L43

loc.Œt0;1/I V 0/ such that:

• for all v 2 V and a.e. t > t0 we have

hu0.t/C �Au.t/C B.u.t/; u.t// � f ; vi CZ

�C

�.t/ � v� dS D 0; (15.38)

with � 2 L2loc.t0;1I W/ such that �.t/ 2 N.u� .t// for a.e. t > t0,• for a.e. t � t0 (not necessarily for t0) and for all s > t we have

1

2ju.s/j2 C �

Z s

tku.r/k2 dr C

Z s

t.�.r/; u� .r//W � hf ; u.r/i dr � 1

2ju.t/j2:

(15.39)

Note that every type I Leray–Hopf weak solution is a type II Leray–Hopf weaksolution. Moreover, a restriction of either type I or type II Leray–Hopf weak solutiongiven on interval Œt0;1/ to a smaller interval ŒT;1/ Œt0;1/ is always a typeII Leray–Hopf weak solution on ŒT;1/. It is, however, impossible to concatenate


type II Leray–Hopf weak solutions, as the energy inequality (15.39) does not haveto hold at the initial time t D t0. We can define

E .ŒT;1// Dfu W u is a type II Leray–Hopf weak solution on ŒT;1/;

and u.t/ 2 X for all t � Tg: (15.40)

Obviously, E .ŒT;1// satisfies the first three axioms of Definition 15.1, and henceit defines the evolutionary system with E ..�1;1// given by axiom .iv/ ofDefinition 15.1. We can use Theorem 15.1 to conclude the following result on theexistence of a weak global attractor.

Theorem 15.5. The evolutionary system defined by (15.40) has a weak globalattractor A .

To study the attractor invariance we need the next result.

Theorem 15.6. Let uk be the sequence of type II Leray–Hopf weak solutions onŒt0;1/ such that uk.t/ 2 X for all t � t0. Let T > t0 be given arbitrarily. Then thereexists a subsequence, not renumbered, which converges in Cw.Œt0;T�I H/ to sometype II Leray–Hopf weak solution u, i.e.,

.uk.t/; v/ ! .u.t/; v/ uniformly on Œt0;T� for all v 2 H as k ! 1:

Proof. Let the sequence tm ! t0 be such that (15.39) holds for t D tm. Proceedingin the same way as in the proof of Lemma 15.3 we get

Z T

tm

kuk.r/k2 dr � C;

from (15.39). Passing with m ! 1 we obtain that uk is bounded in L2.t0I TI V/.Proceeding as in (15.26) we find that from (15.38) it follows that u0

k is bounded

in L43 .t0;TI V 0/. In the similar way as we obtained (15.27)–(15.35) in the proof of

existence, we have

uk ! u weakly in V ;

uk ! u weakly- � in L1.t0;TI H/;


43 .t0;TI H/;

uk ! u strongly in L2.t0;TI H/;

uk.t/ ! u.t/ strongly in H for almost all t 2 .t0;T/;uk ! u strongly in C.Œt0;T�I V 0/; (15.41)

uk.t/ ! u.t/ weakly in H for all t 2 Œt0;T�;


�k ! � weakly in L2.t0;TI W/;

.uk/� ! u� strongly in L2.t0;TI W/:

Now, exactly the same as in the proof of Theorem 15.4, it follows that u satis-fies (15.38) and (15.39) for a.e. t 2 .t0;T/, whence u is a certain type II Leray–Hopfweak solution. To show the uniform convergence in Cw.Œt0;T�I H/, let v 2 H and let" > 0. We can find v" 2 V with jv" � vj � ". We have

supt2Œt0;T�

j.uk.t/ � u.t/; v/j D supt2Œt0;T�

jhuk.t/ � u.t/; v"i C .uk.t/ � u.t/; v � v"/j

� supt2Œt0;T�

kuk.t/ � u.t/kV0kv"k C 2p

R";

where we have used the fact that uk.t/; u.t/ 2 X. We pass with k ! 1 anduse (15.41), to get

lim supk!1

supt2Œt0;T�

j.uk.t/ � u.t/; v/j � 2p

R":

As " was arbitrary, it follows that limk!1 supt2Œt0;T� j.uk.t/ � u.t/; v/j D 0, and theproof is complete. utTheorem 15.7. E .Œ0;1// is compact in Cw.Œ0;1/I H/.

Proof. The proof follows the lines of the proof of Lemma 3.12 in [66]. Take anysequence uk 2 E .Œ0;1//. By Theorem 15.6 there exists a subsequence, still denotedby uk, which converges in Cw.Œ0; 1�I H/ to some u1, a type II Leray–Hopf solution.Passing again to subsequence, not renumbered, uk converges in Cw.Œ0; 2�I H/ tosome u2, a type II Leray–Hopf solution such that u1 D u2 on Œ0; 1�. Continuingthis diagonalization process, uk converges in Cw.Œ0;1/I H/ to some u, a type IILeray–Hopf solution. As ju.t/j � lim infk!1 juk.t/j for all t � 0, it follows thatu.t/ 2 X, and in consequence u 2 E .Œ0;1//. The proof is complete. ut

As we have shown that .A1/ holds, we can use Theorem 15.2 to conclude the lastresult of this section.

Theorem 15.8. The weak global attractor A is a maximal invariant set. Moreover,

A D fx 2 X W x D u.0/ for some u 2 E ..�1;1//g;

and the tracking property holds, i.e., for every " > 0 there exists t0 > 0 such thatfor any t� > t0 and for any trajectory u 2 E .Œ0;1// we have

dCw.Œt�;1/IX/.u; v/ < "

for a certain complete trajectory v 2 E ..�1;1//.


The tracking property established in Theorem15.8 is illustrated in Fig. 15.3.

Fig. 15.3 Illustration of thetracking property. Everytrajectory has a correspondingcomplete trajectory on theattractor to which it convergesas the time goes to infinity


There are several mathematical formalisms useful to study the global attractors forautonomous evolution problems without uniqueness of solutions. Among them wecan list:

• generalized semiflows introduced and studied by Ball [11],• multivalued semiflows (m-semiflows) anticipated by Babin and Vishik [9] and

introduced and studied by Melnik and Valero in [171] and [172],• trajectory attractors introduced independently by Chepyzhov and Vishik [59],

Málek and Necas [167], and Sell [210]. The application of this approach to two-dimensional incompressible Navier–Stokes problem with multivalued feedbackcontrol boundary condition is presented in Chap. 14,

• evolutionary systems introduced by Cheskidov and Foias and [66] and laterextended by Cheskidov in [65].

The formalism of evolutionary systems chosen by us in this chapter appears to beespecially well suited to the problems governed by the three-dimensional Navier–Stokes equations. Still, such problems can be and have been dealt with by the othermethods: trajectory attractors [59, 167, 210], multivalued semiflows [128, 141], andgeneralized semiflows [11]. The approach based on generalized semiflows has beencompared with the one based on multivalued semiflows in [40], and both theseapproached were related to the one based on the trajectory attractors in [129].More information on the structure of weak attractor for three-dimensional casewith periodic boundary conditions can be found in [102] and on the structure ofthe attainability sets of weak solutions in [138].

If we assume the periodic or homogeneous Dirichlet boundary conditions onwhole @˝ it is known that a weak solution for three-dimensional Navier–Stokesequations, so-called Leray–Hopf solution (see [115, 154]), exists for arbitrarily longtime but its uniqueness is still an open problem. On the other hand, the regularsolution is known to exist and be unique only locally in time (see [98] for local


regularity results in three-dimensional case and global regularity results in two-dimensional case for periodic boundary conditions). There are many additionalconditions under which the weak solution becomes strong and the associated weakglobal attractor is the strong one, too, but so far it remains an open problem whetherany of such conditions is in fact satisfied [34, 66–68, 73, 82, 143, 144, 189]. Onlysuch conditional results are available on the existence of strong global attractor inthe three-dimensional case [65, 66, 128, 202]. An interesting recent idea is to use theresults of numerical simulations to verify the existence of strong solutions [64]. A lotof work has been put in recent years to study the partial regularity of solutions. Thiswork includes the analysis of the so-called singular set of the weak solutions, i.e.,the set of space-time points such that the solution is unbounded in the neighborhoodof such points, this work started from the works of Scheffer [205] and Caffarelli etal. [32]. For the recent progress in such studies see the forthcoming book [200].

There are some approaches relying on the modifications or simplifications ofthree-dimensional equations to improve their properties, for example, one can studyglobally modified Navier–Stokes equations [47, 48, 139], Navier–Stokes equationswith fractional operators [86], or ˛-model [45, 63, 100].

16Attractors for Multivalued Processesin Contact Problems

In this chapter we consider further non-autonomous and multivalued evolutionproblems, this time in the frame of the theory of pullback attractors for multivaluedprocesses.

First we prove an abstract theorem on the existence of a pullback D-attractorand then apply it to study a two-dimensional incompressible Navier–Stokes flowwith a general form of multivalued frictional contact conditions. Such conditionsrepresent the frictional contact between the fluid and the wall, where the frictionforce depends in a nonmonotone and even discontinuous way on the slip rate, andare a generalization of the conditions considered in Chap. 10.

In contrast to Chap. 10 we are able to obtain only the attractor existence, while,for example, the question of its fractal dimension for the case without uniquenessremains an open problem.

16.1 Abstract Theory of Pullback D-Attractorsfor Multivalued Processes

In this section we recall basic definitions of the theory of pullback D-attractorsfor multivalued processes, prove a theorem which gives some useful criteria ofexistence of pullback D-attractors for such processes, and introduce the notion of“condition (NW).”

Let .H; %/ be a complete metric space, and P.H/ be the family of all nonemptysubsets of H. By distH.A;B/ we denote the Hausdorff semi-distance between thesets A;B H defined as

distH.A;B/ D supx2A

infy2B%.x; y/:


359

360 16 Attractors for Multivalued Processes in Contact Problems

If the set A H is bounded, than we define its Kuratowski measure of noncompact-ness .A/ (cf. [144]) as

.A/ D finf ı > 0 W A has finite open cover of sets of diameter less than ı g:For the properties of , see, for example, [111, 125]. Subsets of R�H will be callednon-autonomous sets. For a non-autonomous set D R� H we identify D with thefamily of its fibers, i.e.,

D D fD.t/gt2R; where x 2 D.t/ , .x; t/ 2 D:

A family D of non-autonomous sets such that for every D 2 D and all t 2 R thefiber D.t/ is nonempty (i.e., D.t/ 2 P.H/) is called an attraction universe over H.We denote Rd D f.t; �/ 2 R

2 W t � �g.

Definition 16.1. The map U W Rd � H ! P.H/ is called a multivalued process(m-process) if:

(1) U.�; �; z/ D z for all z 2 H,(2) U.t; �; z/ U.t; s;U.s; �; z// for all z 2 H and all t � s � � .

If in (2), in place of inclusion we have the equality U.t; �; z/ D U.t; s;U.s; �; z//,then the m-process is said to be strict.

Definition 16.2. Let D be an attraction universe over H. The m-process U in H ispullback !-D-limit compact if for every D D fD.�/g�2R 2 D and t 2 R we have

[

��s

U.t; �;D.�//

!

! 0 as s ! �1:

Definition 16.3. Let D be an attraction universe over H. The m-process U in His pullback D-asymptotically compact if for any t 2 R, every D 2 D , and allsequences �k ! �1 and �k 2 U.t; �k;D.�k//, there exists a subsequence f�kmg suchthat for some � 2 H, �km ! � in H.

Lemma 16.1. Let D be an attraction universe over H. The m-process U in H ispullback !-D-limit compact if and only if it is pullback D-asymptotically compact.

Proof. We first prove that pullback !-D-limit compactness implies pullbackD-asymptotic compactness. Fix t 2 R. Let D 2 D and let �k be such that

[

��k

U.t; �;D.�//

!

� 1

kand �k ! �1: (16.1)

Let ti ! �1 and �i 2 U.t; ti;D.ti//. We shall prove that .f�ig1iD1/ D 0. For every

m 2 N we have

.f�ig1iD1/ D .f�igm

iD1 [ f�ig1iDmC1/ � .f�ig1

iDmC1/:

16.1 Abstract Theory of Pullback D-Attractors for Multivalued Processes 361

For every k 2 N and m such that tmC1 � �k we have .f�ig1iDmC1/ � 1

k ,whence .f�ig1

iD1/ D 0. This proves, in turn, the precompactness of f�ig1iD1, and,

in consequence, the pullback D-asymptotic compactness of U.Now let U be pullback D-asymptotically compact. We shall prove first that for

every D 2 D and t 2 R the set

!.D; t/ D\

s�t

[

��s

U.t; �;D.�// (16.2)

is nonempty.Indeed, let tk ! �1 and �k 2 U.t; tk;D.tk// and let, for a subsequence, still

denoted by k, �k ! � . For every s � t and every index k such that tk � s, we have�k 2 S

��s U.t; �;D.�//. Since �k ! � , then � 2 S��s U.t; �;D.�// for every

s � t. Thus � 2 !.D; t/.Now we prove that for every D 2 D and every t 2 R,

distH.U.t; �;D.�//; !.D; t// ! 0 as � ! �1:

Assume, to the contrary, that there exists the non-autonomous set D 2 D , thenumber Nt 2 R, and the sequences tk ! �1 and �k 2 U.Nt; tk;D.tk// suchthat distH.�k; !.D; t// � � > 0. By the asymptotic compactness property, for asubsequence, still denoted by k, we have �k ! � in H. But � 2 !.D; t/, which givesa contradiction.

Let D 2 D , t 2 R and let xn be a sequence in !.D; t/. We prove that this sequencehas a subsequence that converges to some element in !.D; t/ and thus the set !.D; t/is compact. As

xn 2[

��s

U.t; �;D.�// for every s � t;

then, for any sequence tk ! �1 there exists �mk 2 U.t; tmk ;D.tmk// such that�.xk; �mk/ � 1

k . By pullback D-asymptotic compactness, there exists a subsequence�� of �mk converging to some � 2 !.D; t/. Thus, also x� ! � .

Now let us fix � > 0. We need to show that there exists s < t such that the setS��s U.t; �;D.�// can be covered by a finite number of sets with diameter �. From

compactness of !.D; t/ it follows that there exists a finite number of points fxigkiD1

such that !.D; t/ SkiD1 B.xi;

�2/: Now choose s < t such that

distH.U.t; �;D.�//; !.D; t// <"

2;

whenever � � s. It follows that for all � � s we have G.t; �;D.�// SkiD1 B.xi; �/

and the proof is complete. ut


Definition 16.4. Let D be an attraction universe over H. We say that an m-processU has a pullback D-absorbing non-autonomous set B D fB.t/gt2R 2 D if for everyt 2 R and D 2 D there exists �0 � t depending on t and D such that for � � �0,

U.t; �;D.�// B.t/:

Definition 16.5. Let D be an attraction universe over H and let U be an m-processon H. The non-autonomous set A D fA.t/gt2R R � H is called a pullback D-attractor for U if:

(1) A.t/ is compact in H and nonempty for every t 2 R,(2) A.t/ U.t; �;A.�// for every t 2 R and � � t (A is negatively semi-invariant),(3) A is pullback D-attracting, that is, for every t 2 R and D 2 D ,

distH.U.t; �;D.�//;A.t// ! 0 as � ! �1;

(4) if C is a non-autonomous set such that C.t/ are closed for t 2 R and C ispullback D-attracting then A.t/ C.t/ for every t 2 R (minimality property).

Remark 16.1. Condition (4) in the above definition implies that pullbackD-attractor is always defined uniquely.

Remark 16.2. If we assume that, by definition, A 2 D then condition (4)(minimality) follows from conditions (1)–(3). In such a case, by Remark 16.1,conditions (1)–(3) suffice to guarantee the uniqueness of A.

The minimality property can be proved as follows. Suppose that there exists anon-autonomous set C such that C.t/ are closed for all t 2 R, C is attracting,and for some t 2 R, C.t/ A.t/ with proper inclusion. As A 2 D , we havedistH.U.t; �;A.�//;C.t// ! 0 for � ! �1. But A.t/ U.t; �;A.�//, and wehave distH.A.t/;C.t// � distH.U.t; �;A.t//;C.t// ! 0 as � ! �1, whenceA.t/ C.t/, a contradiction.

Definition 16.6. The m-process U on H is closed if for all � � t the graph of themultivalued mapping x ! U.t; �; x/ is a closed set in the product topology on H�H.

Now we formulate a theorem about the existence of pullback D-attractors incomplete metric spaces. A similar result is proved in [4, 35, 169] (see also [171, 172]for the case of m-semiflows and [41, 239] for the case of the universe of boundedand constant in time sets).

Theorem 16.1. Let H be a complete metric space and let the m-process U on H beclosed. Assume that

(i) U has a pullback D-absorbing non-autonomous set B 2 D .(ii) U is pullback !-D-limit compact (equivalently, U is pullback D-asymptotically

compact).

Then there exists a pullback D-attractor for U.


Proof. The proof is similar to that of Theorem 2.1 in [125].

Step 1. We define a candidate set for a pullback D-attractor by setting

A.t/ D\

s�t

[

��s

U.t; �;B.�//; (16.3)

where B D fB.�/gt2R is a non-autonomous pullback D-absorbing set.Step 2. A.t/ is nonempty and compact. We have, by !-D-limit compactness of U,

[

��s

U.t; �;B.�//

!

! 0 as s ! �1: (16.4)

Since the sets Xs D S��s U.t; �;B.�// are nonempty, bounded, and closed in

H, and the family of sets fXsgs�t is nonincreasing as s ! �1, we can apply awell-known property of (cf., e.g., [111]) to get the claims.

Step 3. Attraction property. Since for every t 2 R, every non-autonomous setD 2 D , and every r � t it is U.t; �;D.�// U.t; r;U.r; �;D.�/// U.t; r;B.r// for all � � �0, for some �0.D; r/ � r, it suffices to prove thatdistH.U.t; �;B.�//;A.t// ! 0 as � ! �1. We argue by contradiction. Assume,to the contrary, that there exist sequences �k ! �1 and �k 2 U.t; �k;B.�k//

such that infy2A.t/ %.�k; y/ � � > 0. By the pullback D-asymptotic compactnessproperty, there exists a convergent subsequence, � ! � in H, and by thedefinition of the pullback D-attractor in (16.3), � 2 A.t/, which gives acontradiction.

Step 4. We prove that A.t/ U.t; �;A.�// for all � � t. Let x 2 A.t/ and � � t.We shall prove that x 2 U.t; �; p/ for some p 2 A.�/. Since x 2 A.t/ definedin (16.3), there exist tk ! �1 and �k 2 U.t; tk;B.tk// such that �k ! x in H. Wehave

�k 2 U.t; tk;B.tk// U.t; �;U.�; tk;B.tk///

for large k, whence there exist zk 2 B.tk/ such that �k 2 U.t; �;U.�; tk; zk//,and pk 2 U.�; tk; zk/ such that �k 2 U.t; �; pk/. By the pullback D-asymptoticcompactness it follows that for a subsequence of fpkg we have p ! p in H.From (16.3) it follows that p 2 A.�/. Since � ! x and p ! p with � 2U.t; �; p/, from the fact that U is closed it follows that x 2 U.t; �; p/. The proofis complete.

Step 5. To prove the minimality property assume that there exists a pullback D-attracting non-autonomous set C such that C.t/ are closed. Then due to the factthat B 2 D we have distH.U.t; �;B.�//;C.t// ! 0 as � ! �1 . Let x 2 A.t/.By (16.3) there exist sequences �k ! �1 and �k 2 U.t; �k;B.�k// such that�k ! x, whereas, as C.t/ is closed, we have x 2 C.t/ and the proof is complete.

ut


Remark 16.3. The above theorem still holds if we assume that, in definition of thepullback D-asymptotic compactness, any set D 2 D is replaced with the pullbackD-absorbing set B. In such a case this set does not have to belong to the universe D(see [4, 35, 169]).

Remark 16.4. The existence of the pullback D-attractor implies condition (ii) in theabove theorem, that is, the !-D-limit compactness of U. To prove it we follow theargument provided, e.g., in [240]. Denote by O".A.t// the closed "-neighborhood ofA.t/. Since the set A.t/ is compact, then .A.t// D 0, and .O".A.t/// � 2". Fromthe attraction property of A, for any D 2 D we have distH.U.t; �;D.�//;A.t// � 2"

for all � � �0 for some �0.D; t/. Thus

[

��0U.t; �;D.�//

!

� 2";

which implies (ii).

Remark 16.5. In general, the existence of a pullback D-absorbing non-autonomousset in D does not follow from the existence of a pullback D-attractor.

Indeed, consider the equation u0.t/ D 0 in R. Then U.t; �/ D S.t � �/ D id. Letthe universe D consist of only one non-autonomous set D, given by D.�/ D fsin �g.Then the attractor is given as A.t/ D A D Œ�1; 1� for t 2 R. There is no choice fora family of D-absorbing set but B.�/ D D.�/. But U.t; �;D.�// D D.�/ 6D B.t/as sin � 6D sin t. Thus, in this case there does not exist a pullback D-absorbingnon-autonomous set in D . The presented example is illustrated in Fig. 16.1.

Fig. 16.1 Illustration of the example in Remark 16.5. The identity semiflow has a pullbackD-attractor but it does not have a pullback D-absorbing non-autonomous set in D

It is natural to ask what assumptions on the universe D are required to guarantee thatexistence of pullback D-attractor implies (i). We make the following assumptions:

(HD1) The universe D is inclusion-closed, that is if D D fD.t/g t2R 2 D andD

0 D fD0.t/gt2R is such that D0.t/ D.t/ and D0.t/ 2 P.H/ for all t 2 R, thenD

0 2 D .(HD2) If D D fD.t/gt2R 2 D , then there exists " > 0 such that the non-

autonomous set fO".D.t//gt2R also belongs to D .

Corollary 16.1. Let the universe D satisfy (HD1) and let the m-process U satisfythe assumptions of Theorem 16.1. Then we have A 2 D . If we moreover assume thatU is strict then A is invariant, i.e., U.t; �;A.�// D A.t/ for all t 2 R and � � t.


Proof. The proof follows the lines of the proofs of item (6) in Theorem 3 in [169]and items (5) (6) in Theorem 3.10 in [4]. Since B is pullback D-absorbing we havedistH.U.t; �;D.�//;B.t// D 0 for all D 2 D and � � t0.D; t/. Hence as � ! �1we have distH.U.t; �;D.�//;B.t// ! 0 and fB.t/gt2R is pullback D-attracting. Thenfrom the attractor minimality we have A.t/ B.t/ and the assertion follows from(HD1).

For the proof of the positive invariance, let t � � be fixed and let r � 0. We have

U.t; �;A.�// U.t; �;U.�; � C r;A.� C r/// D U.t; � C r;A.� C r//:

Since A 2 D , it pullback attracts itself, and

limr!�1 distH.U.t; � C r;A.� C r//;A.t// D 0:

Hence

limr!�1 distH.U.t; �;A.�//;A.t// D 0

and distH.U.t; �;A.�//;A.t// D 0, whence, by the closedness of A.t/, it follows thatU.t; �;A.�// A.t/ and the assertion is proved. utCorollary 16.2. Let the universe D satisfy (HD2) and let the m-process U have thepullback D-attractor A 2 D . Then U has a pullback D-absorbing non-autonomousset B 2 D .

Proof. Define B.t/ D O".A.t//, where " is given in (HD2). Then B D fB.t/gt2Rbelongs to D . Suppose, on the contrary, that B.t/ is not pullback D-absorbing.This means that there exist sequences tk ! �1 and �k 2 U.t; tk;D.tk// suchthat for large k we have �k 62 B.t/. From Remark 16.4 it follows that U ispullback D-asymptotically compact, and, for a subsequence, � ! � . As A ispullback D-attracting and A.t/ is closed, we have � 2 A.t/, a contradiction with�k 62 O".A.t//. ut

Summarizing, we have a following theorem which is a consequence of Theo-rem 16.1, Remark 16.4, and Corollaries 16.1 and 16.2.

Theorem 16.2. Let D be a universe of non-autonomous sets satisfying (HD1) and(HD2) and let U be a closed m-process. Then U has a pullback D-attractor A

belonging to D if and only if U is pullback D-asymptotically compact and has apullback D-absorbing non-autonomous set B 2 D :

In the sequel of this section we confine ourselves to Banach spaces where we willconsider both strong and weak topologies. We introduce condition (NW), “norm-to-weak,” that generalizes to the non-autonomous multivalued case the norm-to-weakcontinuity assumed in [240] for semigroups (see Definition 3.4 in [240]) and in[125] for multivalued semiflows (see Definition 2.11 in [125]). A similar conditionis introduced for the non-autonomous multivalued case in [231] (see condition (3) in


Definition 2.6 in [231]), where only the strict case is considered and, instead of asubsequence, whole sequence is assumed to converge weakly.

Definition 16.7. The m-process U on a Banach space H satisfies condition (NW) iffor every t 2 R and � � t, from xk ! x in H and �k 2 U.t; �; xk/ it follows that thereexists a subsequence f�mk g, such that �mk ! � weakly in H with � 2 U.t; �; x/.

Lemma 16.2. If a multivalued process U on a Banach space H satisfies condition(NW) then it is closed.

Proof. An elementary proof follows directly from the definitions. utFrom Theorem 16.1 together with Lemma 16.2 we have the following

Corollary 16.3. If the m-process U on a Banach space H has a pullback D-absorbing non-autonomous set B 2 D , is pullback D-asymptotically compact(equivalently, !-D-limit compact), and satisfies condition (NW) then there existsa pullback D-attractor for U.

The above corollary provides useful criteria of existence of pullback D-attractorsfor multivalued processes. In the next section we apply them to study the timeasymptotic behavior of solutions of a problem originating from contact mechanics.

16.2 Application to a Contact Problem

Problem Formulation The flow of an incompressible fluid in a two-dimensionaldomain ˝ is described by the equation of motion

u0.t/ � ��u.t/C .u.t/ � r/u.t/C rp.t/ D f .t/ in ˝ � .t0;1/; (16.5)


div u.t/ D 0 in ˝ � .t0;1/; (16.6)

where the unknowns are the velocity u W ˝ � .t0;1/ ! R2 and pressure p W

˝ � .t0;1/ ! R, � > 0 is the viscosity coefficient and f W ˝ � .t0;1/ ! R2 is

the volume mass force density. To define the domain ˝ of the flow, let ˝1 be theinfinite channel

˝1 D fx D .x1; x2/ 2 R2 W x2 2 .0; h.x1// g;

where h W R ! R is a positive function (we assume that h.x1/ � h0 > 0 for allx1 2 R), smooth, and L-periodic in x1. Then we set

˝ D fx D .x1; x2/ 2 R2 W x1 2 .0;L/; x2 2 .0; h.x1// g;

16.2 Application to a Contact Problem 367

with the boundary @˝ D �D [ �C [ �L, where �D D f.x1; h.x1// W x1 2 .0;L/g,�C D .0;L/ � f0g, and �L D f0;Lg � .0; h.0// are the top, bottom, and lateralparts of @˝, respectively. We will use the notation e1 D .1; 0/ and e2 D .0; 1/ forthe canonical basis of R2. Note that on �C the outer normal unit vector is given byn D �e2.

We are interested in the solutions of (16.5)–(16.6) periodic in the sense that forx2 2 Œ0; h.0/� and t 2 R

C we have u.0; x2; t/ D u.L; x2; t/,@u2.0;x2;t/

@x1D @u2.L;x2;t/

@x1,

and p.0; x2; t/ D p.L; x2; t/. The first condition represents the L-periodicity ofvelocities, while the latter two ones the L-periodicity of normal stresses in the spaceof divergence free functions. Moreover, we assume that

u.t/ D 0 on �D � .t0;1/: (16.7)

On the contact boundary �C we decompose the velocity into the normal componentun D u � n, where n is the unit outward normal vector, and the tangential one u� Du � e1. Note that since the domain˝ is two-dimensional it is possible to consider thetangential components as scalars, for the sake of the ease of notation. Likewise, wedecompose the stress on the boundary �C into its normal component Tn D Tn � nand the tangential one T� D Tn � e1. The stress tensor is related to the velocity andpressure through the linear constitutive law Tij D �pıij C �.ui;j C uj;i/.

We assume that there is no flux across �C,

un.t/ D 0 on �C � .t0;1/; (16.8)

and that the tangential component of the velocity u� on �C is in the followingrelation with the tangential stresses T� ,

� T� .x; t/ 2 @j.x; t; u� .x; t// on �C � .t0;1/: (16.9)

In the above formula, j W �C�R�R ! R is a potential which is locally Lipschitz andnot necessarily convex with respect to the last variable, and @ is the subdifferentialin the sense of Clarke (see Sect. 3.7) taken with respect to the last variable. Notethat the function j is assumed to depend on both space and time variables, however,sometimes the dependence on the space variable x 2 �C is skipped for the ease ofnotation. In the end, we have the initial condition

u.x; t0/ D u0.x/ in ˝: (16.10)

Relation (16.9) is a generalized version of the Tresca friction law. We assumethat the friction force is related by the multivalued and nonmonotone law with thetangential velocity. Our setup is more general than in the previous works [126, 162].In contract to [126] we do not assume any monotonicity type conditions on thepotential j and hence the solutions to the formulated problem can be nonuniquehere. Similar laws were used, for example, in [12, 213] for static problems in


elasticity, see Fig. 4.6 in [12] and Fig. 2(a) in [213] for examples of particularnonmonotone friction laws that satisfy the assumptions .j1/–.j4/ listed below.Complex nonmonotone behavior of the friction force density is related to thepresence of asperities on the surface [191]. Similar friction law is also used, forexample, in the study of the motion of tectonic plates, see [119, 190, 207] and thereferences therein.

Weak Formulation and Existence of Solutions We introduce the variationalformulation of the problem and, for the convenience of the readers, we describethe relations between the classical and the weak formulations.

We begin with some basic definitions. Let

QV D fu 2 C1.˝/2 W div u D 0 in ˝; u is L-periodic in x1;

u D 0 at �D; u � n D 0 at �Cg

and


We define scalar products in H and V , respectively, by

.u ; w/ DZ

˝

u.x/ � w.x/ dx and .ru ; rw/ DZ

˝

ru.x/W rw.x/ dx;


juj D .u; u/12 and kuk D .ru ; ru/

12 :

The duality in V 0 � V is denoted as h�; �i. Moreover, let, for u;w, and z in V ,

a.u ; w/ D .ru ; rw/ and b.u ; z ; w/ D ..u � r/z ; w/:

We define a linear operator A W V ! V 0 as hAu;wi D a.u;w/, and a nonlinearoperator hBu;wi D b.u; u;w/. For w 2 V we have the Poincaré inequality �1jwj2 �kwk2, where �1 is the first eigenvalue of the Stokes operator A in V .

The linear and continuous trace mapping leading from V to L2.�C/2 is denoted

by � , we will use the same symbol to denote the Nemytskii trace operator on thespaces of time-dependent functions. The norm of the trace operator will be denotedby k�k � k�kL .VIL2.�C/2/.

If u 2 L2.�C/ we will denote by [email protected];u/ the set of all L2 selections of @j.t; u/, i.e.,

all functions � 2 L2.�C/ such that �.x/ 2 @j.x; t; u.x// for a.e. x 2 �C (note thatdependence on x 2 �C is not written explicitly for the sake of notation brevity).


The variational formulation of the problem is as follows.

Problem 16.1. Given u0 2 H, find u W .t0;1/ ! H such that for all T > t0,

u 2 C.Œt0;T�I H/ \ L2.t0;TI V/ with u0 2 L2.t0;TI V 0/;

and

hu0.t/C �Au.t/C Bu.t/; zi C .�.t/; z� /L2.�C/ D hf .t/; zi; (16.11)

��.t/ 2 [email protected];u� .t// (16.12)

for a.e. t � t0 and for all z 2 V .

The assumptions on the problem data are the following. The functions f W R ! Hand j W �C � R � R ! R satisfy:

.f1/ f 2 L2loc.RI H/,

.f2/ for certain t 2 R (and thus for all t 2 R) we have

Z t

�1e.��dk�k2/�1sjf .s/j2 ds < 1;

.j1/ j.x; t; �/ is locally Lipschitz for a.e. .x; t/ 2 �C � R,

.j2/ j.�; �; s/ is measurable for all s 2 R,

.j3/ @j satisfies the growth condition j�j � aCbjsj for all s 2 R and � 2 @j.x; t; s/with a; b > 0 a.e. .x; t/ 2 �C � R,

.j4/ @j satisfies the condition �s � c � djsj2 for all s 2 R and � 2 @j.x; t; s/ with

c 2 R and d 2 0; �

k�k2�

a.e. .x; t/ 2 �C � R.

Note that by .j3/ it follows that for u 2 L2.�C/ the set [email protected];u/ is in fact the set of allmeasurable selections of @j.t; u/. Condition .j4/ is known as dissipativity condition.

We have the following relations between classical and weak formulations.

Proposition 16.1. Every classical solution of (16.5)–(16.10) is also a solution ofProblem 16.1. On the other hand, every solution of Problem 16.1 which is smoothenough is also a classical solution of (16.5)–(16.10).

Proof. The proof is only sketched here, the details are left to the reader. Let u be aclassical solution of (16.5)–(16.10). As it is (by assumption) sufficiently regular, wehave to check only (16.11). Remark first that (16.5) can be written as

u0.t/ � div T.u.t/; p.t//C .u.t/ � r/u.t/ D f .t/: (16.13)

Let z 2 V . Multiplying (16.13) by z, integrating by parts and using the Greenformula we obtain


Z

˝

u0.t/ � z dx CZ

˝

Tij.u.t/; p.t//zi;j dx C b.u.t/; u.t/; z/

DZ

@˝

Tij.u.t/; p.t//njzi dS CZ

˝

f .t/ � z dx (16.14)

for t 2 .t0;1/. As u.t/ and z are in V , after some calculations we obtain

Z

˝

Tij.u.t/; p.t//zi;j dx D �a.u.t/; z/: (16.15)

Taking into account the boundary conditions we get

Z

@˝

Tij.u.t/; p.t//njzi dS DZ

�C

T� .u.t/; p.t//z� dS CZ

�C

Tn.u.t/; p.t//zn dS;

and the last integral equals zero as z satisfies (16.8) a.e. on �C. Thus, (16.11) holds.Conversely, suppose that u is a sufficiently smooth solution to Problem 16.1. We

have immediately (16.6)–(16.8) and (16.10). Now, let z be in the space

.H10.div;˝//2 D fz 2 V W z D 0 on @˝ g:

We take such z in (16.11) to get

hu0.t/ � ��u.t/C .u.t/ � r/u.t/ � f .t/ ; zi D 0 for every z 2 .H10.div;˝//2:

Thus, there exists a distribution p.t/ on ˝ such that

u0.t/ � ��u.t/C .u.t/ � r/u.t/ � f .t/ D rp.t/ in ˝ � .t0;1/; (16.16)

so that (16.5) holds.Now, we shall derive the subdifferential boundary condition (16.9) from the weak

formulation. Take z 2 V . We haveZ

˝

Tij.u.t/; p.t//zi;j dx D �Z

˝

div T.u.t/; p.t// � z dx CZ

@˝

Tn � z dS: (16.17)

Since z 2 V , we have zn D 0 a.e. on �C and hence Tn � z D T� z� on �C.Applying (16.15) and (16.17) to (16.11) we get

Z

˝

.u0.t/ � div T.u.t/; p.t//C .u.t/ � r/u.t/ � f .t// � z dx

CZ

�L

T.u.t/; p.t//n � z dS C .�.t/; z� /L2.�C/ CZ

�C

T� .t/z� dS D 0:


By (16.16) we have (16.13) and so the first integral on the left-hand side vanishes.In the same way as in the proof of Proposition 10.1 we assume enough smoothnessof the classical solution, such that the Cauchy stress vector Tn on �L is L-periodicwith respect to x1. This means that the second integral on the left-hand side of thelast equality vanishes for any z 2 V . Thus we get

R�C.T� � �/z� dS. Analogously

to the proof of Proposition 10.1, we use the results of Chapter 1.2 in [146] on theextension of smooth functions on the boundary to divergence free vector fields todeduce that in the last inequality z� can be replaced by any sufficiently smoothfunction. It follows that �T� D � for a.e. x 2 �C and the proof is complete. utTheorem 16.3. Assuming .f1/, and .j1/–.j3/, Problem 16.1 has a solution.

The proof uses the standard approach by the mollification of the nonsmooth termand the Galerkin method. Since it is, on one hand, technical and quite involved, and,on the other hand the methodology is the same as in the proof of Theorem 10.8 inChap. 10 and Theorem 14.1 in Chap. 14, with the necessary modifications for thenon-autonomous terms, we skip the proof here.

Multivalued Process and Its Attractor We associate with Problem 16.1 themultifunction U W Rd � H ! P.H/, where U.t; t0; u0/ is the set of states attainableat time t from the initial condition u0 taken at time t0. Observe that U is a strictm-process.

Lemma 16.3. Assume .f1/, .j1/–.j4/. If the function u 2 L2loc.t0;C1I V/ withu0 2 L2loc.t0;C1I V 0/ solves Problem 16.1, then

d

dtju.t/j2 C .� � dk�k2/ku.t/k2 � C1.1C jf .t/j2/; (16.18)

ku0.t/kV0 � C2.1C jf .t/j/C C3.1C ju.t/j/ku.t/k (16.19)

for a.e. t 2 .t0;C1/ with C1;C2;C3 > 0 independent on t0; u0; t.

Proof. We take z D u.t/ in (16.11) and make use of

b.w;w;w/ D 0 for w 2 V (16.20)

to get

1

2

d

dtju.t/j2 C �ku.t/k2 C .�.t/; u� .t//L2.�C/ D .f .t/; u.t//;

where �.t/ 2 [email protected];u� .t//for a.e. t � t0. From .j4/ we obtain, with arbitrary " > 0,

1

2

d

dtju.t/j2 C �ku.t/k2 � "

2ku.t/k2 C 1

2"kf .t/k2V0

C dku� .t/k2L2.�C/� cL:


Taking " D � � dk�k2 we obtain, with a constant C > 0,

1

2

d

dtju.t/j2 C � � dk�k2

2ku.t/k2 � C.1C kf .t/k2V0

/;

which proves (16.18), as H V 0 is a continuous embedding.To show (16.19) let us observe that for z 2 V and a.e. t 2 R

C we have

hu0.t/; zi � kAkL .VIV�/ku.t/kkzkCkf .t/kV0kzkCk�.t/kL2.�C/k�kkzk C cju.t/jku.t/kkzk;

with �.t/ 2 [email protected]� .t//, where we used the fact that, in view of the Ladyzhenskaya

inequality (cf. Lemma 10.2), for w; z 2 V we have

jb.w;w; z/j � cjwjkwkkzk: (16.21)

Now (16.19) follows directly from the growth condition .j3/ and the trace inequality.ut

Denote � D .� � dk�k2/�1 and define R.t/ by

R.t/2 D C1�

C C1

Z t

�1e�sjf .s/j2 ds C 1 for t 2 R: (16.22)

By .f2/ the quantity R.t/ is well defined and finite. We denote by R� the family ofall functions r W R ! .0;1/ such that

limt!�1 e� tr2.t/ D 0:

By D� we denote the family of all non-autonomous sets D D fD.t/gt2R such thatD.t/ 2 P.H/ for all t 2 R and there exists rD 2 R� such that for all t 2 R and forall w 2 D.t/, jwj � rD.t/. The family D� will be our attraction universe.

Lemma 16.4. Let assumptions .f1/–.f2/ and .j1/–.j4/ hold. The non-autonomousset B D fB.t/gt2R defined as B.t/ D fv 2 H W jvj � R.t/g is D� -pullback absorbingand B 2 Rı .

Proof. Obviously R 2 R� and hence B 2 D� . Take t 2 R and D 2 D� . There existsrD 2 R� with jwj � rD.s/ for all w 2 D.s/ and s 2 R. We can choose t0 D t0.D; t/small enough such that rD.�/2e�� e� t for all � � t0. We will consider a solutionu of Problem 16.1 with the initial time � � t0 and the initial condition u0 2 D.�/.From (16.18) by the Poincaré inequality we obtain

d

dtju.s/j2 C .� � dk�k2/�1ju.s/j2 � C1.1C jf .s/j2/ for a.e. s 2 .�; t/:


From the Gronwall inequality we get

ju.t/j2 � ju.�/j2e��.t��/ C C1�

C C1e�� t

Z t

�

e�sjf .s/j2 ds: (16.23)

Hence

ju.t/j2 � ju.�/j2e��.t��/ C R2.t/ � 1:

Since u.�/ 2 D.�/, then

ju.t/j2 � rD.�/2e��e�� t � 1C R2.t/:

Due to the bound rD.�/2e�� e� t, we have

ju.t/j2 � R2.t/;

whence the assertion follows. utWe pass to the next lemma which states that the m-process U is D� -pullbackasymptotically compact.

Lemma 16.5. Assume .f1/–.f2/ and .j1/–.j4/. The m-process U is D� -asymptoti-cally compact.

Proof. The proof uses the technique of Proposition 7.4 in [11] (alternatively it isalso possible to obtain the same assertion by the method of [201]) and is based onthe energy equation. Let fD.t/gt2R D D 2 D� and let uk

0 2 D.tk0/ with tk

0 ! �1.Let moreover zk 2 U.t; tk

0; uk0/. We show that the sequence fzkg is relatively compact

in H. There exists a sequence of functions uk 2 L2.tk0; t C 1I V/ \ C.Œtk

0; t C 1�I H/with u0

n 2 L2.tk0; t C 1I V 0/, solutions of Problem 16.1, such that uk.tk

0/ D uk0 and

uk.t/ D zk. From Lemma 16.4 it follows that there exists N0 such that for all naturalk � N0 we have juk.t � 1/j � R.t � 1/. From now on we will consider the sequencefukg1

kDN0. The selections of [email protected];uk� .t//

given in (16.12) will be denoted by �k. Notethat the restrictions of uk; �k to the interval Œt � 1; t C 1� solve Problem 16.1 on thisinterval, with the initial conditions uk.t � 1/ taken at t � 1. Using (16.18) it followsthat the sequence uk is uniformly bounded in L2.t �1; t C1I V/\C.Œt �1; t C1�I H/,and, by (16.19) it follows that u0

k is uniformly bounded in L2.t � 1; t C 1I V 0/. Thegrowth condition .j3/ implies that �k are bounded in L2.t � 1; t C 1I L2.�C//. Thesebounds are sufficient to extract a subsequence, not renumbered, such that for certainu 2 L2.t � 1; t C 1I V/ with u0 2 .t � 1; t C 1I V 0/ and � 2 L2.t � 1; t C 1I L2.�C//

the following convergences hold:


uk ! u weakly in L2.t � 1; t C 1I V/;

u0k ! u0 weakly in L2.t � 1; t C 1I V 0/;

uk ! u strongly in L2.t � 1; t C 1I H/; (16.24)

uk.s/ ! u.s/ weakly in H for all s 2 Œt � 1; t C 1�; (16.25)

�uk� ! �u� strongly in L2.t � 1; t C 1I L2.�C//; (16.26)

�k ! � weakly in L2.t � 1; t C 1I L2.�C//: (16.27)

In view of (16.25) it is sufficient to show that juk.t/j ! ju.t/j, whence, as H isa Hilbert space, it follows that un.t/ ! u.t/ strongly in H and the assertion willbe proved. Note that by (16.24), for another subsequence, also not renumbered, thestrong convergence uk.s/ ! u.s/ holds in H for a.e. s 2 .t � 1; t C 1/. Taking thetest function uk.t/ in (16.11) written for uk, where the corresponding � is denoted by�k, and integrating from t � 1 to s 2 Œt � 1; t C 1�, we get

1

2juk.s/j2 C �

Z s

t�1a.uk.r/; uk.r// dr C

Z s

t�1.�k.r/; uk� .r//L2.�C/ dr

D 1

2juk.t � 1/j2 C

Z s

t�1.f .r/; uk.r// dr: (16.28)

We define the functions Vk W Œt � 1; t C 1� ! R as

Vk.s/ D 1

2juk.t � 1/j2 � �

Z s

t�1a.uk.r/; uk.r// dr:

Note that the coercivity of a implies that the functions Vk are nonincreasing. By theenergy equation (16.28) we have

Vk.s/ D 1

2juk.s/j2 C

Z s

t�1.�k.r/; uk� .r//L2.�C/ dr �

Z s

t�1.f .r/; uk.r// dr:

From (16.24), (16.26), and (16.27) it follows that

limk!1 Vk.s/ D 1

2ju.s/j2 C

Z s

t�1.�.r/; u� .r//L2.�C/ dr �

Z s

t�1.f .r/; u.r// dr

for a.e. s 2 .t � 1; t C 1/. We denote the right-hand side of the above equation byV.s/. We can choose sequences pm % t and qm & t such that Vk.pm/ ! V.pm/ andVk.qm/ ! V.qm/ as k ! 1 for all m 2 N. We have

Vk.pm/ � Vk.t/ � Vk.qm/;


and hence

V.pm/ D limk!1 Vk.pm/ � lim sup

k!1Vk.t/ � lim inf

k!1 Vk.t/ � limk!1 Vk.qm/ D V.qm/:

We pass with m to infinity and use the fact that V , by definition, is continuous.We get

V.t/ � lim supk!1

Vk.t/ � lim infk!1 Vk.t/ � V.t/;

whence Vk.t/ ! V.t/ as k ! 1. But we have

Z t

t�1.�k.r/; �uk� .r//L2.�C/ dr �

Z t

t�1.f .r/; uk.r// dr

�!Z t

t�1.�.r/; �u� .r//L2.�C/ dr �

Z t

t�1.f .r/; u.r// dr;

which gives us the convergence jun.t/j ! ju.t/j, and the proof is complete. utLemma 16.6. If for every integer k, uk solves Problem 16.1 with the initialcondition u0k taken at t0, and u0k ! u0 strongly in H then for any t > t0, for asubsequence, uk.t/ ! u.t/ weakly in H, where u is a solution of Problem 16.1 withthe initial condition u0 taken at t0. In consequence, the m-process U on H satisfiescondition (NW).

Proof. The proof is standard since the a priori estimates of Lemma 16.3 provideenough convergence to pass to the limit. Indeed, the sequence uk is bounded inL2.t0; tI V/ with u0

k bounded in L2.t0; tI V 0/. Hence, for a subsequence, uk ! uweakly in L2.t0; tI V/ with u0

k ! u0 weakly in L2.t0; tI V 0/. In consequence, for alls 2 Œ0; t�, uk.s/ ! u.s/ weakly in H, which means that u.t0/ D u0 and uk.t/ ! u.t/weakly in H. It also follows that uk� ! u� strongly in L2.t0; tI L2.�C// and inL2.�C �.t0; t//. We must show that u solves Problem 16.1 on .t0; t/. We only discusspassing to the limit in the multivalued term since for other terms it is standard. Let�k 2 L2.t0; tI L2.�C// be such that �k.s/ 2 [email protected];uk� .s//

a.e. s 2 .t0; t/ and (16.11) holds.

From the growth condition .j3/ it follows that �k is bounded in L2.t0; tI L2.�C//

and thus, for a subsequence, we have �k ! � weakly in L2.t0; tI L2.�C// andweakly in L2.�C � .t0; t//. Using assertion .H2/ in Theorem 3.24 it follows thatfor a.e. .x; s/ 2 �C � .t0; t/ �.x; s/ 2 @j.x; s; u� .x; s//, which, in turn, implies that�.s/ 2 [email protected];u� .s//

for a.e. s 2 .t0; t/, and the proof is complete. utFrom Lemmas 16.4–16.6 it follows that all assumptions of Corollary 16.3 hold

and hence we have shown the following Theorem.

Theorem 16.4. The m-process U on H associated with Problem 16.1 has a D� -pullback attractor. This attractor is moreover invariant and it belongs to D� .


In the above theorem the attractor invariance and the fact that it belongs to theuniverse D� follow from Corollary 16.1 as the m-process is strict and the attractionuniverse D� satisfies .HD1/.


In this chapter we were interested in the evolution problems for which the solutionsfor a given initial conditions are not unique. To deal with such problems, thetheory of m-semiflows and their global attractors was introduced in [171, 172] andlater extended to non-autonomous case [41]. Results on the existence of pullbackD-attractors for the problems without uniqueness of solutions were obtained in[4, 35, 169]. The abstract result on the existence of the pullback D-attractor shownin this chapter was based on these works, however, we showed some extensionto the results presented there: our alternative proof of the pullback D-attractorexistence was based on the notion of !-D-limit-compactness and seems particularlytransparent. We weakened the upper semicontinuity assumption from [4, 35, 169]replacing it by the graph closedness, and we additionally studied not only necessarybut also the necessary and sufficient conditions for the attractor existence.

We introduced the so-called condition (NW), earlier studied for autonomous casein [240] and recently in [125], and for non-autonomous case in [231] in context ofuniform attractors. This condition is convenient to verify for particular problems ofmathematical physics governed by PDEs.

For the notions of uniform attractors, pullback attractors, and skew-product flowswe refer to [10, 53, 55, 130, 131, 239]. For similar problems in contact mechanics, cf.[126, 162] and, in particular, [12, 213]. The results of this chapter come from [127].

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Index

Symbols!-limit set, 152

pullback, 262 -cocycle, 260

continuous, 261pullback D-asymptotically compact, 261

Aattraction universe, 261, 266, 277, 360

inclusion-closed, 266, 364attractor

(global) pullback D-, 261, 278for m-process, 362minimal, 262

exponential, 209global, 151, 208

weak, 334, 339trajectory, 330

BBanach contraction principle, 42Banach generalized limit, 170Bochner integral, 62boundary condition

Fourier, 112Tresca, 112

Boussinesq equations, 24

CCauchy

equation of motion, 17principle, 16

coercive bilinear form, 40

complex amplitude, 56condition (NW), 366contraction, 42coordinate

material, 12spatial, 12

Couette flow, 36

Dderivative

distributional, 48time, 63

generalized, 46material, 13

determining modes, 252determining nodes, 256, 257differential inclusion, 234dilation group, 33dimension

fractal, 183, 209Hausdorff, 193

Dirac delta, 47directional derivative

generalized Clarke, 71dissipation, function, 23distribution, 47

regular, 47Du Bois-Reymond, lemma, 45

EEhrling lemma, 65energy

dissipation rate, 196

© Springer International Publishing Switzerland 2016G. Łukaszewicz, P. Kalita, Navier–Stokes Equations, Advancesin Mechanics and Mathematics 34, DOI 10.1007/978-3-319-27760-8

387

388 Index

energy transferdirect, 165inverse, 166

energy, specific internal, 19enstrophy, 165entropy, balance of, 27equation

energy, 178, 288enstrophy, 156Liouville, 176, 286

stationary, 176Euler formula, 14evolutionary system, 338

Ffield equations, 22flow

large scale component, 164small scale component, 164

fluidincompressible, 14Newtonian, 21non-Newtonian, 21perfect, 18polar, 21Stokesian, 21

forcebody, 16contact, 16

Fourier coefficient, 56Fourier law, 20friction

coefficient, 112multifunction, 96, 341

functionBochner integrable, 62locally integrable, 45weakly continuous, 66

function spaceCk.˝/, 45Ck.˝/, 45C1.˝/, 45C1

0 .˝/, 45C1

p .Q/, 54Hs

p.Q/, 54H�m.˝/, 47L1loc.˝/, 45L1.˝/, 45Lp.˝/, 44Lp

loc.˝/, 45W�m;q.˝/, 47Wm;p.˝/, 46Wm;p0 .˝/, 46

PHsp.Q/, 56

PL2.Q/, 59D 0.˝/, 47D.˝/, 47

GGalerkin method, 40Galilean invariance, 31Grashof number, 159, 252gravity acceleration, 24

HHausdorff semi-distance, 151, 261, 359

weak, 338heat flux, 19

Iinequality

Agmon, 157first energy, 145Gronwall, 67

generalized, 70translated, 69uniform, 68

Hardy, 53Korn, 117Ladyzhenskaya, 51, 218, 343Lieb–Thirring, 200Poincaré, 50, 56, 120second energy, 155

Kkinematic viscosity coefficient, 23Kuratowski measure of noncompactness, 360Kuratowski–Painlevé upper limit, 74

Ll-trajectory, 209law

of thermodynamicsfirst, 19second, 27

Lax–Milgram, lemma, 40local variable, 25

Mmeasure

invariant, 169regular, 175

Index 389

NNavier–Stokes equations, 24nodal point, 256

Ooperator

completely continuous, 42

Ppoint

nonwandering, 172wandering, 173

Poiseuille flow, 35pressure, 18principle

Banach contraction, 42dimensional homogenity, 33of conservation of linear momentum, 16of conservation of mass, 15

processevolutionary, 266, 297

�-continuous, 291pullback D-asymptotically compact,

266multivalued (m-process), 360

closed, 362pullback !-D-limit compact, 360pullback D-asymptotically compact,

360strict, 360

RReynolds

equation, 141weak form, 137

number, 29rotation symmetry, 31

Sscaling, 31semiflow

uniformly quasidifferentiable, 186semigroup, 151

asymptotically compact, 154continuous, 151uniformly compact, 152

setabsorbing, 152

for shift semigroup, 330attracting

for shift semigroup, 330weakly, 338

invariant, 151, 339non-autonomous, 261, 266, 360

pullback D-absorbing, 267, 362parameterized, 261

pullback D-absorbing, 261pullback D-attracting, 261

prevalent, 185similarity

dynamical, 29variable, 33

smoothing estimate, 191Sobolev space, 46solution

approximate, 41, 144invariant, 32Leray–Hopf, 347, 353self-similar, 31statistical, 289

stationary, 178, 290squeezing property, 192Stokes

operator, 84, 86problem, 84

stresshydrostatic, 18normal, 16

subdifferentialgeneralized Clarke, 71

supportof function, 45of measure, 172

symmetry group, 31

Ttensor

deformation, 20stress, 17stress, viscous, 18

theoremAubin–Lions, 66Brouwer, 42Hölder–Mañé, 185Hahn–Banach, 175Kakutani–Fan–Glicksberg, 44Lebourg

mean value, 72Leray–Schauder, 43of stress means, 19Rellich, 49Riesz–Fréchet, 41Riesz–Kakutani, 175Schauder, 42Tietze, 176transport, 13

390 Index

theoryfalse, 37incomplete, 37overdetermined, 37well-set, 36

thermalconductivity, 20diffusion coefficient, 24expansion coefficient, 24

tracking property, 339Tresca condition, 211

generalized, 233, 367

Vvariable

hydromechanical, 12state, 26

variational inequality, 215velocity, 12vorticity, 25

Wwave number, 56

navier–stokes equations

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