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Abstract and Applied Analysis

Qualitative Theory of Functional Differential and Integral Equations 2016

Guest Editors: Cemil Tunç, Bingwen Liu, Luís R. Sanchez, Mohsen Alimohammady, Octavian G. Mustafa, and Samir H. Saker

Qualitative Theory ofFunctional Differential and Integral Equations2016

Abstract and Applied Analysis

Qualitative Theory ofFunctional Differential and Integral Equations2016

Guest Editors: Cemil Tunç, Bingwen Liu, Luís R. Sanchez,Mohsen Alimohammady, Octavian G. Mustafa,and Samir H. Saker

Copyright © 2017 Hindawi Publishing Corporation. All rights reserved.

This is a special issue published in “Abstract and Applied Analysis.” All articles are open access articles distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the originalwork is properly cited.

Editorial Board

Ravi P. Agarwal, USABashir Ahmad, KSAM. O. Ahmedou, GermanyNicholas D. Alikakos, GreeceDebora Amadori, ItalyDouglas R. Anderson, USAJan Andres, Czech RepublicGiovanni Anello, ItalyStanislav Antontsev, PortugalNarcisa C. Apreutesei, RomaniaNatig M. Atakishiyev, MexicoFerhan M. Atici, USAIvan Avramidi, USASoohyun Bae, Republic of KoreaChuanzhi Bai, ChinaZhanbing Bai, ChinaJozef Banas, PolandMartino Bardi, ItalyRoberto Barrio, SpainFeyzi Başar, TurkeyAbdelghani Bellouquid, MoroccoDaniele Bertaccini, ItalyLucio Boccardo, ItalyIgor Boglaev, New ZealandMartin Bohner, USAGeraldo Botelho, BrazilElena Braverman, CanadaRomeo Brunetti, ItalyJanusz Brzdek, PolandDetlev Buchholz, GermanySun-Sig Byun, Republic of KoreaFabio M. Camilli, ItalyJinde Cao, ChinaAnna Capietto, ItalyJianqing Chen, ChinaWing-Sum Cheung, Hong KongMichel Chipot, SwitzerlandC. Chun, Republic of KoreaS. Chung, Republic of KoreaSilvia Cingolani, ItalyJean M. Combes, FranceMonica Conti, ItalyJ.-C. Cortés, SpainGraziano Crasta, ItalyZhihua Cui, China

Bernard Dacorogna, SwitzerlandVladimir Danilov, RussiaMohammad T. Darvishi, IranLuis F. Pinheiro de Castro, PortugalToka Diagana, USAJesús I. Díaz, SpainJosef Diblik, Czech RepublicFasma Diele, ItalyTomas Dominguez, SpainAlexander Domoshnitsky, IsraelMarco Donatelli, ItalyBoQing Dong, ChinaWei-Shih Du, TaiwanLuiz Duarte, BrazilRoman Dwilewicz, USAPaul W. Eloe, USALuca Esposito, ItalyKhalil Ezzinbi, MoroccoJulian F. Bonder, ArgentinaDashan Fan, USAAngelo Favini, ItalyMárcia Federson, BrazilStathis Filippas, Equatorial GuineaAlberto Fiorenza, ItalyXianlong Fu, ChinaMassimo Furi, ItalyJesús G. Falset, SpainGiovanni P. Galdi, USAIsaac Garcia, SpainJosé A. García-Rodríguez, SpainLeszek Gasinski, PolandGyörgy Gát, HungaryVladimir Georgiev, ItalyLorenzo Giacomelli, ItalyJaume Giné, SpainValery Y. Glizer, IsraelMoshe Goldberg, IsraelJean P. Gossez, BelgiumJose L. Gracia, SpainMaurizio Grasselli, ItalyLuca Guerrini, ItalyYuxia Guo, ChinaUno Hämarik, EstoniaMaoan Han, ChinaFerenc Hartung, Hungary

Jiaxin Hu, ChinaChengming Huang, ChinaZhongyi Huang, ChinaGennaro Infante, ItalyIvan Ivanov, BulgariaHossein Jafari, South AfricaJaan Janno, EstoniaAref Jeribi, TunisiaUncig Ji, Republic of KoreaZhongxiao Jia, ChinaLUCAS JODAR, SpainJong S. Jung, Republic of KoreaHenrik Kalisch, NorwayChaudry M. Khalique, South AfricaSatyanad Kichenassamy, FranceTero Kilpeläinen, FinlandSung G. Kim, Republic of KoreaLjubisa Kocinac, SerbiaAndrei Korobeinikov, SpainPekka Koskela, FinlandVictor Kovtunenko, AustriaRen-Jieh Kuo, TaiwanPavel Kurasov, SwedenMilton C. L. Filho, BrazilMiroslaw Lachowicz, PolandKunquan Lan, CanadaRuediger Landes, USAIrena Lasiecka, USAMatti Lassas, FinlandChun-Kong Law, TaiwanMing-Yi Lee, TaiwanGongbao Li, ChinaShengqiang Liu, ChinaYansheng Liu, ChinaCarlos Lizama, ChileGuozhen Lu, USAJinhu Lü, ChinaGrzegorz Lukaszewicz, PolandWanbiao Ma, ChinaNazim I. Mahmudov, TurkeyEberhard Malkowsky, TurkeySalvatore A. Marano, ItalyCristina Marcelli, ItalyPaolo Marcellini, ItalyJesús Marín-Solano, Spain

Mieczysław Mastyło, PolandMing Mei, CanadaTaras Mel’nyk, UkraineAnna Mercaldo, ItalyStanislaw Migorski, PolandMihai Mihǎilescu, RomaniaFeliz Minhós, PortugalDumitru Motreanu, FranceGaston Mandata N’guérékata, USAMaria Grazia Naso, ItalyMicah Osilike, NigeriaMitsuharu Ôtani, JapanTurgut "Oziş, TurkeySehie Park, Republic of KoreaKailash C. Patidar, South AfricaKevin R. Payne, ItalyAdemir F. Pazoto, BrazilShuangjie Peng, ChinaAntonio M. Peralta, SpainSergei V. Pereverzyev, AustriaAllan Peterson, USAAndrew Pickering, SpainCristina Pignotti, ItalySomyot Plubtieng, ThailandMilan Pokorny, Czech RepublicSergio Polidoro, ItalyZiemowit Popowicz, PolandMaria M. Porzio, ItalyEnrico Priola, ItalyVladimir S. Rabinovich, MexicoIrena Rachunková, Czech RepublicMaria Alessandra Ragusa, ItalySimeon Reich, IsraelAbdelaziz Rhandi, ItalyHassan Riahi, Malaysia

Juan P. Rincón-Zapatero, SpainLuigi Rodino, ItalyYuriy Rogovchenko, NorwayJulio D. Rossi, ArgentinaWolfgang Ruess, GermanyBernhard Ruf, ItalySatit Saejung, ThailandStefan G. Samko, PortugalMartin Schechter, USAValery Serov, FinlandNaseer Shahzad, KSAAndrey Shishkov, UkraineStefan Siegmund, GermanyAbdel-Maksoud A. Soliman, EgyptPierpaolo Soravia, ItalyMarco Squassina, ItalySvatoslav Staněk, Czech RepublicAntonio Suárez, SpainWenchang Sun, ChinaWenyu Sun, ChinaRobert Szalai, UKChun-Lei Tang, ChinaSanyi Tang, ChinaGabriella Tarantello, ItalyNasser-Eddine Tatar, KSAGerd Teschke, GermanySergey Tikhonov, SpainClaudia Timofte, RomaniaThanh Tran, AustraliaJuan J. Trujillo, SpainMilan Tvrdy, Czech RepublicMehmet Ünal, TurkeyCsaba Varga, RomaniaCarlos Vazquez, SpainJesus Vigo-Aguiar, Spain

Jing P. Wang, UKQing-WenWang, ChinaShawn X. Wang, CanadaYouyu Wang, ChinaYushun Wang, ChinaPeixuan Weng, ChinaNoemi Wolanski, ArgentinaNgai-Ching Wong, TaiwanPatricia J. Y. Wong, SingaporeShanhe Wu, ChinaShi-Liang Wu, ChinaYonghong Wu, AustraliaZili Wu, ChinaTiecheng Xia, ChinaXu Xian, ChinaYanni Xiao, ChinaFuding Xie, ChinaGongnan Xie, ChinaDaoyi Xu, ChinaXiaodong Yan, USAZhenya Yan, ChinaBeong In Yun, Republic of KoreaAgacik Zafer, KuwaitJianming Zhan, ChinaChengjian Zhang, ChinaWeinian Zhang, ChinaZengqin Zhao, ChinaSining Zheng, ChinaTianshou Zhou, ChinaYong Zhou, ChinaChun-Gang Zhu, ChinaQiji J. Zhu, USAMalisa R. Zizovic, SerbiaWenming Zou, China

Contents

QualitativeTheory of Functional Differential and Integral Equations 2016Cemil Tunç, Bingwen Liu, Luís R. Sanchez, Mohsen Alimohammady, Octavian G. Mustafa,and Samir H. SakerVolume 2017, Article ID 3454192, 2 pages

Lie Group Solutions of Magnetohydrodynamics Equations andTheir Well-PosednessFu-zhi Li, Jia-li Yu, Yang-rong Li, and Gan-shan YangVolume 2016, Article ID 8183079, 8 pages

Resolvent for Non-Self-Adjoint Differential Operator with Block-Triangular Operator PotentialAleksandr Mikhailovich KholkinVolume 2016, Article ID 2964817, 6 pages

Global Existence of Weak Solutions to a Fractional Model in Magnetoelastic InteractionsIdriss Ellahiani, EL-Hassan Essoufi, and Mouhcine TiliouaVolume 2016, Article ID 9238948, 9 pages

Existence and Uniqueness Results for a Smooth Model of Periodic Infectious DiseasesGuy DeglaVolume 2016, Article ID 1708527, 4 pages

Existence of Solutions for Some Nonlinear Problems with Boundary Value ConditionsDionicio Pastor Dallos SantosVolume 2016, Article ID 5283263, 10 pages

EditorialQualitative Theory of Functional Differential andIntegral Equations 2016

Cemil Tunç,1 Bingwen Liu,2 Luís R. Sanchez,3 Mohsen Alimohammady,4

Octavian G. Mustafa,5 and Samir H. Saker6

1Department of Mathematics, Faculty of Science, Yuzuncu Yil University, 65080 Van, Turkey2College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing 314001, China3Faculty of Sciences, University of Lisbon, 1700 Lisbon, Portugal4Department of Mathematical Sciences, University of Mazandaran, Babolsar 47416-1467, Iran5University of Craiova, 200585 Craiova, Romania6Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Correspondence should be addressed to Cemil Tunç; [email protected]

Received 3 November 2016; Accepted 3 November 2016; Published 4 January 2017

Copyright © 2017 Cemil Tunç et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The study of qualitative theory of various kinds of differentialequations began with the birth of calculus, which datesto the 1660s. Part of Newton’s motivation in developingcalculus was to solve problems that could be attacked withdifferential equations. Now, with over 300 years of history,the subject of qualitative theory of differential equations,integral equations, and so on represents a huge body ofknowledge including many subfields and a vast array ofapplications in many disciplines. It is beyond exposition asa whole. Qualitative theory refers to the study of the behaviorof solutions without determining explicit formulas for thesolutions. In addition, it should be noted that if solutions ofan equation describing a dynamical system or of any kindof differential equations under consideration are known inclosed form, one can determine the qualitative properties ofthe system or the solutions of that equations, by applyingdirectly the definitions of relative mathematical concepts. Asis also well-known, in general, it is not possible to find thesolution of all linear and nonlinear differential equations,except numerically. Moreover, finding of solutions becomesvery difficult for functional differential equations, integralequations, partially differential equations, and fractionaldifferential equations rather than for ordinary differentialequations. Thus, indirect methods are needed. Therefore, itis very important to obtain information on the qualitative

behavior of solutions of differential equations when thereis no analytical expression for the solutions. So far, in therelevant literature, some methods have been improved toobtain information about qualitative behaviors of solutions ofdifferential equations without solving them. Here, we wouldnot like to give the details of methods.

It is worth mentioning that, in the last century, theoryof ordinary differential equations, functional differentialequations, partially differential equations, integral equations,and integrodifferential equations has developed quickly andplayed many important roles in qualitative theory and appli-cations of that equations. Some problems of considerableinterest in qualitative theory of ordinary differential equa-tions, functional differential and integral equations, integrod-ifferential equations, fractional differential equations, par-tially differential equations, and so forth include many topicssuch as stability and instability of solutions, boundednessof solutions, convergence of solutions, existence of periodicsolutions, almost periodic solutions, pseudo almost periodsolutions, existence and uniqueness of solutions, globalexistence of solutions, global stability, bifurcation analysis,control of chaos, boundary value problems, oscillation andnonoscillation of solutions, and global existence of solutions.Functional differential equations, which include ordinary anddelay differential equations, partially differential equations,

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2017, Article ID 3454192, 2 pageshttp://dx.doi.org/10.1155/2017/3454192

http://dx.doi.org/10.1155/2017/3454192

2 Abstract and Applied Analysis

and integral equations, have very important roles in manyscientific areas such as mechanics, engineering, economy,control theory, physics, chemistry, biology, medicine, atomicenergy, and information theory.

Over the years many scientific works have been dedicatedto the mentioned problems for various differential equations,fractional differential equations, partially differential equa-tions, and so forth. In particular, we can findmany interestingresults related to qualitative behaviors of solutions in thebooks or papers in [1–15] and in their references.

In response to the call for papers, 22 papers were received.After a rigorous refereeing process, 5 papers were acceptedfor publication in this special issue. The articles includedin the issue cover novel contributions to qualitative theoryof functional differential and integral equations, magnetohy-drodynamics equations, partial differential equations.

The paper by G. Degla investigates the existence of acurve (with respect to the scalar delay) of periodic positivesolutions for a smooth model of Cooke-Kaplan’s integralequation by using the implicit function theorem undersuitable conditions. The author also shows a situation inwhich any bounded solution with a sufficiently small delay isisolated, clearing an asymptotic stability result of Cooke andKaplan.

In the paper by A. M. Kholkin, a resolvent for the Sturm-Liouville operator with a block triangular operator potentialincreasing at infinite is constructed. The structure of thespectrum of such an operator is obtained.

The paper by I. Ellahiani et al. deals with global exis-tence of weak solutions to a one-dimensional mathematicalmodel describing magnetoelastic interactions. The model isdescribed by a fractional Landau-Lifshitz-Gilbert equationfor the magnetization field coupled to an evolution equa-tion for the displacement. They prove global existence byusing Faedo-Galerkin/Penalty method. Some commutatorestimates are used to prove the convergence of nonlinearterms.

In the paper by Y. Li et al., based on classical Lie Groupmethod, a class of explicit solutions of two-dimensional idealincompressible magnetohydrodynamics (MHD) equation byits infinitesimal generator is constructed. Via these explicitsolutions, the authors study the uniqueness and stability ofinitial-boundary problem on MHD.

In the paper by D. P. D. Santos, the existence of solutionsfor certain nonlinear boundary value problems is investi-gated. All the contemplated boundary value problems arereduced to find a fixed point for one operator defined ona space of functions, and Schauder fixed point theorem orLeray-Schauder degree are used.

Cemil TunçBingwen Liu

Luı́s R. SanchezMohsen Alimohammady

Octavian G. MustafaSamir H. Saker

References

[1] S. Ahmad andM. RamaMohana Rao,Theory of Ordinary differ-ential Equations. With Applications in Biology and Engineering,Affiliated East-West Press, New Delhi, India, 1999.

[2] L. C. Becker, “Uniformly continuous solutions of Volterraequations and global asymptotic stability,” Cubo, vol. 11, no. 3,pp. 1–24, 2009.

[3] T. A. Burton, “Stability theory for Volterra equations,” Journalof Differential Equations, vol. 32, no. 1, pp. 101–118, 1979.

[4] T. A. Burton, “Construction of Liapunov functionals forVolterra equations,” Journal ofMathematical Analysis andAppli-cations, vol. 85, no. 1, pp. 90–105, 1982.

[5] T. A. Burton, Volterra Integral and Differential Equations,vol. 202 of Mathematics in Science and Engineering, Elsevier,Amsterdam, The Netherlands, 2nd edition, 2005.

[6] C. Corduneanu, Integral Equations and Stability of FeedbackSystems, vol. 104 of Mathematics in Science and Engineering,Academic Press, London, UK, 1973.

[7] J. R. Graef and C. Tunç, “Continuability and boundednessof multi-delay functional integro-differential equations of thesecond order,” Revista de la Real Academia de Ciencias Exactas,Fı́sicas y Naturales, Serie A: Matematicas, vol. 109, no. 1, pp. 169–173, 2015.

[8] J. R. Graef, C. Tunc, and S. Sevgin, “Behavior of solutionsof nonlinear functional Volterra integro-differential equationswith multiple delays,” Dynamic Systems and Applications, vol.25, no. 1-2, pp. 39–46, 2016.

[9] G. Gripenberg, S.-O. Londen, and O. Staffans, Volterra Integraland Functional Equations, Encyclopedia of Mathematics andits Applications, Cambridge University Press, Cambridge, UK,1990.

[10] J. E. Napoles Valdes, “A note on the qualitative behavior of somesecond order nonlinear equation,” Applications and AppliedMathematics, vol. 8, no. 2, pp. 767–776, 2013.

[11] C. Tunç, “Stability to vector Liénard equation with constantdeviating argument,” Nonlinear Dynamics. An InternationalJournal of Nonlinear Dynamics and Chaos in Engineering Sys-tems, vol. 73, no. 3, pp. 1245–1251, 2013.

[12] C. Tunç, “New stability and boundedness results to Volterraintegro-differential equations with delay,” Journal of the Egyp-tian Mathematical Society, vol. 24, no. 2, pp. 210–213, 2016.

[13] C. Tunc, “Properties of solutions to Volterra integro-differentialequations with delay,” Applied Mathematics & InformationSciences, vol. 10, no. 5, pp. 1775–1780, 2016.

[14] C. Tunc, “On the existence of periodic solutions of functionaldifferential equations of third order,” Applied and Computa-tional Mathematics, vol. 15, no. 2, pp. 189–199, 2016.

[15] T. Yoshizawa, Stability theory by Liapunov’s second method,Publications of the Mathematical Society of Japan, No. 9, TheMathematical Society of Japan, Tokyo, Japan, 1966.

Research ArticleLie Group Solutions of MagnetohydrodynamicsEquations and Their Well-Posedness

Fu-zhi Li,1 Jia-li Yu,2 Yang-rong Li,1 and Gan-shan Yang3

1School of Mathematics and Statistics, Southwest University, Chongqing 400715, China2School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China3School of Mathematics and Computer Science, Yunnan Minzu University, Kunming, Yunnan 650500, China

Correspondence should be addressed to Gan-shan Yang; [email protected]

Received 27 June 2016; Accepted 20 September 2016

Academic Editor: Bingwen Liu

Copyright © 2016 Fu-zhi Li et al.This is an open access article distributed under the Creative CommonsAttribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Based on classical Lie Group method, we construct a class of explicit solutions of two-dimensional ideal incompressiblemagnetohydrodynamics (MHD) equation by its infinitesimal generator. Via these explicit solutions we study the uniqueness andstability of initial-boundary problem on MHD.

1. Introduction

Two-dimensional ideal incompressible magnetohydrody-namics (MHD) equation can be described by a set of twoscalar equations for the vorticity 𝑤 and the magnetic streamfunction 𝜓; namely [1],

(𝜕𝑡 + 𝑢 ⋅ ∇)𝑤 = 𝐵 ⋅ ∇𝑗, (1)(𝜕𝑡 + 𝑢 ⋅ ∇) 𝜓 = 0. (2)

Due to the divergence freedom of the magnetic field 𝐵, it ispossible to define amagnetic stream function𝜓 via 𝐵 = ∇⊥𝜓.In the incompressible case, ∇ ⋅ 𝑢 = 0, the velocity streamfunction 𝜑 and velocity 𝑢 are connected in the same way,and 𝑢 = ∇⊥𝜑. Vorticity and current density are defined asthe Laplacian of the stream functions, 𝑤 = Δ𝜑 and 𝑗 = Δ𝜑.Themagnetic stream function is convected with the flow field[2–4]; that means the ideal MHD equations do not allow formagnetic reconnection in contrast to the dissipative versionof the above equations [5].

Note that in contrast to the two-dimensional incompress-ible Euler equations case, there is not a production term onthe right-hand side of (1). The equations show a tendency todevelop fine structures, namely, current sheets. Analyticallythe problem about the regularity of solutions is still an openproblem [6, 7].

It is interesting to seek the solutions of MHD in mathe-matics and physics for a long time [8]. In order to constructsolutions of MHD, many effective methods have been putforward, such as the inverse scattering method, Backlundtransformation, Hirota method, and homogeneous balancemethod [3]. In the branches of mathematics and physics, LieGroup theory [9–11] was often used extensively. Ever since the1970s Bluman and Col proposed similarity theory for differ-ential equations, the Lie Group theory has been developed in-differential equations. The main idea of Lie Group method isto use the prolongation formulae, providing an effective com-putational procedure for finding the most general symmetrygroup of almost any system of partial differential equationsof interest. To the best of our knowledge, related classicalLie Group method has not been preformed to the MHDequation.

Many mathematicians are devoted to studying the MHDequations. For example, Duvant and Lions proved the exis-tence and uniqueness of the global strong solutions of two-dimensional MHD equations with initial-boundary valueproblem. They also proved existence and uniqueness oflocally strong solutions and the existence of the global weaksolutions of three-dimensionalMHD equations [6, 12]. As weall know, the studying of uniqueness and stability of MHDequations is based on some certain conditions or someassumptions. But, in this paper, we study a special class of


http://dx.doi.org/10.1155/2016/8183079


solutions—wave solutions. It does not need external hypothe-ses when proving the uniqueness, which is a novelty of thispaper.

2. Calculation of Symmetry Groups

In this section, we perform Lie symmetry analysis for (1) and(2) and obtain its infinitesimal generator. Now we transform(1) and (2) as follows:

𝜕3𝜑𝜕𝑥21𝜕𝑡 +𝜕3𝜑𝜕𝑥22𝜕𝑡 +

𝜕𝜑𝜕𝑥1𝜕3𝜑𝜕𝑥32 +

𝜕𝜑𝜕𝑥1𝜕3𝜑𝜕𝑥21𝜕𝑥2 −

𝜕𝜑𝜕𝑥2𝜕3𝜑𝜕𝑥31

− 𝜕𝜑𝜕𝑥2𝜕3𝜑𝜕𝑥22𝜕𝑥1 =

𝜕𝜓𝜕𝑥1𝜕3𝜓𝜕𝑥32 +

𝜕𝜓𝜕𝑥1𝜕3𝜓𝜕𝑥21𝜕𝑥2

− 𝜕𝜓𝜕𝑥2𝜕3𝜓𝜕𝑥31 −

𝜕𝜓𝜕𝑥2𝜕3𝜓𝜕𝑥22𝜕𝑥1

𝜕𝜓𝜕𝑡 + 𝜕𝜑𝜕𝑥1𝜕𝜓𝜕𝑥2 −

𝜕𝜓𝜕𝑥1𝜕𝜑𝜕𝑥2 = 0.

(3)

According to themethod of determining the infinitesimalgenerator of nonlinear partial differential equation, we takethe infinitesimal generator of equation as follows:

V = 𝑝∑𝑖=1

𝜉𝑖 (𝑥, 𝑢) 𝜕𝜕𝑥𝑖 +𝑞∑𝛼=1

𝜙𝛼 (𝑥, 𝑢) 𝜕𝜕𝑢𝛼 . (4)It is a vector field defined on an open subset𝑀 ⊂ 𝑋 × 𝑈; the𝑛th prolongation of V is the vector field

pr(𝑛)V = V + 𝑞∑𝛼=1

∑𝐽

𝜙𝐽𝛼 (𝑥, 𝑢(𝑛)) 𝜕𝜕𝑢𝛼𝐽 (5)defined on the corresponding jet space 𝑀(𝑛) ⊂ 𝑋 × 𝑈(𝑛),the second summation being overall multi-indices 𝐽 =(𝑗1, 𝑗2, 𝑗3, . . . , 𝑗𝑘), with 1 ≤ 𝑗𝑘 ≤ 𝑝, 1 ≤ 𝑘 ≤ 𝑛. The coefficientfunctions 𝜙𝐽𝛼 of pr(𝑛)V are given by the following formula:

𝜙𝐽𝛼 (𝑥, 𝑢(𝑛)) = 𝐷𝐽(𝜙𝛼 −𝑝∑𝑖=1

𝜉𝑖𝑢𝛼𝑖 ) +𝑝∑𝑖=1

𝜉𝑖𝑢𝛼𝐽,𝑖, (6)where 𝑢𝛼𝑖 = 𝜕𝑢𝛼/𝜕𝑥𝑖 and 𝑢𝛼𝐽,𝑖 = 𝜕𝑢𝛼𝐽 /𝜕𝑥𝑖 (in this paper 𝑥1 =𝑥1, 𝑥2 = 𝑥2, and 𝑥3 = 𝑡).

Firstly, we consider the circumstance of 𝑛 = 1. LetV = 𝜉1 𝜕𝜕𝑥1 + 𝜉2

𝜕𝜕𝑥2 + 𝜉3𝜕𝜕𝑡 + 𝜉4 𝜕𝜕𝜑 + 𝜉5 𝜕𝜕𝜓, (7)

where 𝜉𝑖 = 𝜉𝑖(𝑥1, 𝑥2, 𝑡, 𝜑, 𝜓) (𝑖 = 1, 2, 3, 4) are coefficientfunctions of the infinitesimal generator to be determined.And the first-order prolongation of V is as follows:

pr(1)V = V + 𝜂𝑥11 𝜕𝜕𝜑𝑥1 + 𝜂𝑥22

𝜕𝜕𝜑𝑥2 + 𝜂𝑥13

𝜕𝜕𝜓𝑥1 + 𝜂𝑥24

𝜕𝜕𝜓𝑥2+ 𝜂𝑡5 𝜕𝜕𝜑𝑡 + 𝜂

𝑡6

𝜕𝜕𝜓𝑡 .(8)

Applying pr(1)V to (3), we find

(𝜕3𝜑𝜕𝑥32 +𝜕3𝜑𝜕𝑥21𝜕𝑥2)𝜂

𝑥11 − (𝜕

3𝜑𝜕𝑥31 +

𝜕3𝜑𝜕𝑥22𝜕𝑥1)𝜂𝑥22

− (𝜕3𝜓𝜕𝑥32 +𝜕3𝜓𝜕𝑥21𝜕𝑥2)𝜂

𝑥13

− (𝜕3𝜓𝜕𝑥31 +𝜕3𝜓𝜕𝑥22𝜕𝑥1)𝜂

𝑥24 = 0,

𝜕𝜓𝜕𝑥2 𝜂𝑥11 − 𝜕𝜓𝜕𝑥1 𝜂

𝑥22 − 𝜕𝜑𝜕𝑥2 𝜂

𝑥13 + 𝜕𝜑𝜕𝑥1 𝜂

𝑥42 + 𝜂𝑡6 = 0.

(9)

Nowwe apply the third-order prolongation of V to (1) and (2).Let

V = 3∑𝑖=1

𝜉𝑖 (𝑥, 𝑢) 𝜕𝜕𝑥𝑖 +2∑𝛼=1

𝜙𝛼 (𝑥, 𝑢) 𝜕𝜕𝑢𝛼pr(3)V = V + 𝑞∑

𝛼=1

∑𝐽

𝜙𝐽𝛼 (𝑥, 𝑢(3)) 𝜕𝜕𝑢𝛼𝐽 ,(10)

where 𝑞 = 2, 𝑢1 = 𝜑, and 𝑢2 = 𝜓, the second summationbeing overall multi-indices 𝐽 = (𝑗1, 𝑗2, 𝑗3, . . . , 𝑗𝑘), with 1 ≤𝑗𝑘 ≤ 3, 1 ≤ 𝑘 ≤ 3.

The coefficient functions 𝜙𝐽𝛼 of pr(3)V are given by thefollowing formula:

𝜙𝐽𝛼 (𝑥, 𝑢(3)) = 𝐷𝐽(𝜙𝛼 −3∑𝑖=1

𝜉𝑖𝑢𝛼𝑖 ) +3∑𝑖=1

𝜉𝑖𝑢𝛼𝐽,𝑖, (11)

where 𝑢𝛼𝑖 = 𝜕𝑢𝛼/𝜕𝑥𝑖 and 𝑢𝛼𝐽,𝑖 = 𝜕𝑢𝛼𝐽 /𝜕𝑥𝑖 (𝑥1 = 𝑥1, 𝑥2 = 𝑥2,𝑥3 = 𝑡):pr(3)V = V + 𝜂𝑥11 𝜕𝜕𝜑𝑥1 + 𝜂

𝑥22

𝜕𝜕𝜑𝑥2 + 𝜂𝑥13

𝜕𝜕𝜓𝑥1+ 𝜂𝑥24 𝜕𝜕𝜓𝑥2 + 𝜂

𝑡5

𝜕𝜕𝜑𝑡 + 𝜂𝑡6

𝜕𝜕𝜓𝑡 + 𝜙𝑥1𝑥1𝑡1

𝜕𝜕𝜑𝑥1𝑥1𝑡+ 𝜙𝑥2𝑥2𝑡1 𝜕𝜕𝜑𝑥2𝑥2𝑡 + 𝜙

𝑥2𝑥2𝑥21

𝜕𝜕𝜑𝑥2𝑥2𝑥2+ 𝜙𝑥1𝑥2𝑥21 𝜕𝜕𝜑𝑥1𝑥2𝑥2+ 𝜙𝑥1𝑥1𝑥21 𝜕𝜕𝜑𝑥1𝑥1𝑥2 𝜙

𝑥1𝑥1𝑥11

𝜕𝜕𝜑𝑥1𝑥1𝑥1+ 𝜙𝑥1𝑥1𝑥12 𝜕𝜕𝜓𝑥1𝑥1𝑥2 + 𝜙

𝑥1𝑥1𝑥12

𝜕𝜕𝜓𝑥1𝑥1𝑥2+ 𝜙𝑥1𝑥2𝑥22 𝜕𝜕𝜓𝑥1𝑥2𝑥2 + 𝜙

𝑥2𝑥2𝑥22

𝜕𝜕𝜓𝑥2𝑥2𝑥2 .

(12)

Abstract and Applied Analysis 3

Applying pr(3)V to (3), we find

𝜙𝑥1𝑥1𝑡1 + 𝜙𝑥2𝑥2𝑡1 + 𝜙𝑥2𝑥2𝑥21 𝜕𝜑𝜕𝑥1 + 𝜙𝑥1𝑥1𝑥21

𝜕𝜑𝜕𝑥1− 𝜙𝑥1𝑥1𝑥11 𝜕𝜑𝜕𝑥2 − 𝜙

𝑥1𝑥2𝑥21

𝜕𝜑𝜕𝑥2 + 𝜂𝑥11

𝜕3𝜑𝜕𝑥32 − 𝜂

𝑥21

𝜕3𝜑𝜕𝑥31

+ 𝜂𝑥11 𝜕3𝜑𝜕𝑥21𝜕𝑥2 − 𝜂

𝑥22

𝜕3𝜑𝜕𝑥22𝜕𝑥1 = 𝜙𝑥2𝑥2𝑥22

𝜕𝜓𝜕𝑥1+ 𝜙𝑥1𝑥1𝑥22 𝜕𝜓𝜕𝑥1 − 𝜙

𝑥1𝑥1𝑥12

𝜕𝜓𝜕𝑥2 − 𝜙𝑥1𝑥2𝑥22

𝜕𝜓𝜕𝑥2+ 𝜂𝑥13 𝜕

3𝜓𝜕𝑥32 − 𝜂

𝑥24

𝜕3𝜓𝜕𝑥31 + 𝜂

𝑥13

𝜕3𝜓𝜕𝑥21𝜕𝑥2 − 𝜂𝑥24

𝜕3𝜓𝜕𝑥22𝜕𝑥1 ,𝜕𝜓𝜕𝑥2 𝜂𝑥11 − 𝜕𝜓𝜕𝑥1 𝜂

𝑥22 − 𝜕𝜑𝜕𝑥2 𝜂

𝑥13 + 𝜕𝜑𝜕𝑥1 𝜂

𝑥42 + 𝜂𝑡6 = 0.

(13)

Then merging similar terms coefficients, we have

𝜂𝑥11 = 𝜂𝑥22 = 𝜂𝑥13 = 𝜂𝑥24 = 𝜂𝑡6 = 0. (14)According to the formula 𝜙𝐽𝛼(𝑥, 𝑢(𝑛)) = 𝐷𝐽(𝜙𝛼 − ∑𝑝𝑖=1 𝜉𝑖𝑢𝛼𝑖 ) +∑𝑝𝑖=1 𝜉𝑖𝑢𝛼𝐽,𝑖, for example, we have

𝜂𝑡6 = 𝐷𝑡 (𝜉5 − 𝜉1𝜓𝑥1 − 𝜉2𝜓𝑥2 − 𝜉3𝜓𝑡) + 𝜉1𝜓𝑥1𝑡+ 𝜉2𝜓𝑥2𝑡 + 𝜉3𝜓𝑡𝑡

= 𝜉5𝑡 − 𝜉1𝑡𝜓𝑥1 − 𝜉1𝜓𝜓𝑥1𝜓𝑡 − 𝜉1𝜑𝜓𝑥1𝜓𝑡 − 𝜉2𝑡𝜓𝑥2− 𝜉2𝜓𝜓𝑥2𝜓𝑡 − 𝜉2𝜑𝜓𝑥2𝜑𝑡 − 𝜉3𝑡𝜓𝑡 − 𝜉3𝜑𝜑𝑡𝜓𝑡− 𝜉3𝜓𝜓𝑡𝜓𝑡 + 𝜉5𝜑𝜑𝑡 + 𝜉5𝜓𝜓𝑡

𝜙𝑥1𝑥1𝑥11 = 𝐷3𝑥1 (𝜉4 − 𝜉1𝜑𝑥1 − 𝜉2𝜑𝑥2 − 𝜉3𝜑𝑡)+ 𝜉1𝜑𝑥1𝑥1𝑥1𝑥1 + 𝜉2𝜑𝑥1𝑥1𝑥1𝑥2 + 𝜉3𝜑𝑥1𝑥1𝑥1𝑡

= 𝐷3𝑥1𝜉4 − 𝐷3𝑥1𝜉1𝜑𝑥1𝐷3𝑥1𝜉2𝜑𝑥2 − 𝐷3𝑥1𝜉3𝜑𝑡− 3𝐷2𝑥1𝜉1𝜑𝑥1𝑥1 − 3𝐷2𝑥1𝜉2𝜑𝑥1𝑥2 − 3𝐷2𝑥1𝜉3𝜑𝑥1𝑡− 3𝐷𝑥1𝜉1𝜑𝑥1𝑥1𝑥1 − 3𝐷𝑥1𝜉2𝜑𝑥1𝑥1𝑥2− 3𝐷𝑥1𝜉3𝜑𝑥1𝑥1𝑡.

(15)

So we have

𝜉5𝑡 = 0,𝜉5𝜓 − 𝜉3𝑡 = 0,

𝜉1𝑡 = 𝜉1𝜑 = 𝜉1𝜓 = 𝜉2𝑡 = 𝜉2𝜑 = 𝜉2𝜓 = 𝜉3𝜑 = 𝜉3𝜓= 𝜉5𝜑 = 0.

(16)

Similarly we can find the determining equations for thesymmetry group of the equations to be the following:

𝜉4𝑥1 = 0,𝜉4𝜑 − 𝜉1𝑥1 = 0,

𝜉1𝜑 = 𝜉1𝜓 = 𝜉2𝜑 = 𝜉2𝜓 = 𝜉2𝑥1 = 𝜉3𝑥1 = 𝜉3𝜑= 𝜉3𝜓 = 𝜉5𝜓 = 0.

𝜉4𝑥2 = 0,𝜉4𝜑 − 𝜉2𝑥2 = 0,

𝜉1𝜑 = 𝜉1𝜓 = 𝜉1𝑥2 = 𝜉2𝜑 = 𝜉2𝜓 = 𝜉3𝑥2 = 𝜉3𝜑= 𝜉3𝜓 = 𝜉4𝜓 = 0.

𝜉5𝑥1 = 0,𝜉5𝜓 − 𝜉1𝑥1 = 0,

𝜉1𝜑 = 𝜉1𝜓 = 𝜉2𝜑 = 𝜉2𝜓 = 𝜉2𝑥1 = 𝜉3𝑥1 = 𝜉3𝜑= 𝜉3𝜓 = 𝜉5𝜑 = 0.

𝜉5𝑥2 = 0,𝜉5𝜓 − 𝜉2𝑥2 = 0,

𝜉1𝜑 = 𝜉1𝜓 = 𝜉1𝑥2 = 𝜉2𝜑 = 𝜉2𝜓 = 𝜉3𝑥2 = 𝜉3𝜑= 𝜉3𝜓 = 𝜉5𝜑 = 0.

(17)

As usual, subscripts indicate derivatives. The solution of thedetermining equations is elementary. According to (16) and(17) we have

𝜉4 = 𝜉4 (𝑡, 𝜑) ,𝜉5 = 𝜉5 (𝜓) ,𝜉1 = 𝜉1 (𝑥1) ,𝜉2 = 𝜉2 (𝑥2) ,𝜉3 = 𝜉3 (𝑡) ,𝜉3𝑡 = 𝜉5𝜓,𝜉4𝜑 = 𝜉1𝑥1 ,𝜉4𝜑 = 𝜉2𝑥2 ,𝜉5𝜓 = 𝜉1𝑥1 ,𝜉5𝜓 = 𝜉2𝑥2 .

(18)

Finally, solving the above differential equations, we concludethat the most general infinitesimal symmetry of (3) hascoefficient functions of the form

𝜉1 = 𝑘𝑥1 + 𝑎𝜉2 = 𝑘𝑥2 + 𝑏


𝜉3 = 𝑘𝑡 + 𝑐𝜉4 = 𝑘𝜑 + 𝜎 (𝑡) + 𝑑𝜉5 = 𝑘𝜓 + 𝑒,

(19)

where 𝑎, 𝑏, 𝑐, 𝑑, 𝑒, and 𝑘 are arbitrary constants and 𝜎(𝑡)is an arbitrary function of 𝑡 only. Thus the Lie algebra ofinfinitesimal symmetries of the MHD equations is spannedby the six vector fields:

V1 = 𝑥1𝜕𝑥1 + 𝑥2𝜕𝑥2 + 𝑡𝜕𝑡 + 𝜑𝜕𝜑 + 𝜓𝜕𝜓V2 = 𝜕𝑥1V3 = 𝜕𝑥2V4 = 𝜕𝑡V5 = 𝜕𝜑V6 = 𝜕𝜓.

(20)

And the infinite-dimensional subalgebra

V𝛼 = 𝜎 (𝑡) 𝜕𝜑, (21)where 𝜎(𝑡) is an arbitrary function of 𝑡 only. So we have

V = 𝑐1V1 + 𝑐2V2 + 𝑐3V3 + 𝑐4V4 + 𝑐5V5 + 𝑐6V6 + V𝛼. (22)The entries give the transformed point exp (𝜖𝑖)(𝑥1, 𝑥2, 𝑡,𝜑, 𝜓) = (𝑥1, 𝑥2, �̃�, �̃�, �̃�):

𝐺1: (𝑒𝜖𝑥1, 𝑒𝜖𝑥2, 𝑒𝜖𝑡, 𝑒𝜖𝜑, 𝑒𝜖𝜓)𝐺2: (𝑥1 + 𝜖, 𝑥2, 𝑡, 𝜑, 𝜓)𝐺3: (𝑥1, 𝑥2 + 𝜖, 𝑡, 𝜑, 𝜓)𝐺4: (𝑥1, 𝑥2, 𝑡 + 𝜖, 𝜑, 𝜓)𝐺5: (𝑥1, 𝑥2, 𝑡, 𝜑 + 𝜖, 𝜓)𝐺6: (𝑥1, 𝑥2, 𝑡, 𝜑, 𝜓 + 𝜖)

𝐺𝛼: (𝑥1, 𝑥2, 𝑡, 𝜑 + 𝜖𝜎 (𝑡) , 𝜓) .

(23)

If 𝜑 = 𝑓(𝑥1, 𝑥2, 𝑡) and 𝜓 = 𝑔(𝑥1, 𝑥2, 𝑡) are known solutionsof (3), then using the above groups 𝐺𝑖, 𝑖 = (1, 2, . . . , 6),the corresponding new solutions 𝜑𝑖 and 𝜓𝑖 can be obtained,respectively, as follows:

𝜑1 = 𝑒𝜖𝑓 (𝑒−𝜖𝑥1, 𝑒−𝜖𝑥2, 𝑒−𝜖𝑡) ,𝜓1 = 𝑒𝜖𝑔 (𝑒−𝜖𝑥1, 𝑒−𝜖𝑥2, 𝑒−𝜖𝑡) ,𝜑2 = 𝑓 (𝑥1 − 𝜖, 𝑥2, 𝑡) ,𝜓2 = 𝑔 (𝑥1 − 𝜖, 𝑥2, 𝑡) ,𝜑3 = 𝑓 (𝑥1, 𝑥2 − 𝜖, 𝑡) ,𝜓3 = 𝑔 (𝑥1, 𝑥2 − 𝜖, 𝑡) ,

𝜑4 = 𝑓 (𝑥1, 𝑥2, 𝑡 − 𝜖) ,𝜓4 = 𝑔 (𝑥1, 𝑥2, 𝑡 − 𝜖) ,𝜑5 = 𝑓 (𝑥1, 𝑥2, 𝑡) + 𝜖,𝜓5 = 𝑔 (𝑥1, 𝑥2, 𝑡) ,𝜑6 = 𝑓 (𝑥1, 𝑥2, 𝑡) ,𝜓6 = 𝑔 (𝑥1, 𝑥2, 𝑡) + 𝜖,𝜑𝛼 = 𝑓 (𝑥1, 𝑥2, 𝑡) + 𝜖𝜎 (𝑡) ,𝜓𝛼 = 𝑔 (𝑥1, 𝑥2, 𝑡) ,

(24)

where 𝜖 is a real number and 𝜎(𝑡) is an arbitrary function of 𝑡only.

Theorem 1. For the known solutions 𝜑 = 𝑓(𝑥1, 𝑥2, 𝑡) and 𝜓 =𝑔(𝑥1, 𝑥2, 𝑡), by using one-parameter groups𝐺𝑖, 𝑖 = (1, 2, . . . , 6)continuously, one can obtain a new solution which can beexpressed in the following form:

𝜑 = 𝑒𝜖1𝑓 (𝑒−𝜖𝑥1 − 𝜖2, 𝑒−𝜖𝑥2 − 𝜖3, −𝑒−𝜖𝑡 − 𝜖4) + 𝜖5+ 𝜖6𝜎 (𝑡)

𝜓 = 𝑒𝜖1𝑓 (𝑒−𝜖𝑥1 − 𝜖2, 𝑒−𝜖𝑥2 − 𝜖3, −𝑒−𝜖𝑡 − 𝜖4) + 𝜖5,(25)

where 𝜖𝑖, 𝑖 = (1, 2, . . . , 6) are arbitrary constants.Theorem 2. One assumes that the solutions have forms of 𝜑 =𝜑(𝑎1𝑥1 + 𝑏1𝑥2 + 𝑐1𝑡 + 𝑑1), 𝜓 = 𝜓(𝑎2𝑥1 + 𝑏2𝑥2 + 𝑐2𝑡 + 𝑑2).

One has the following:(1) If 𝑐1 = 𝑐2 = 0, 𝑎1𝑏2 − 𝑎2𝑏1 = 0, then 𝜑, 𝜓 are arbitrary

functions.(2) If 𝑎1 = 𝑏1 = 𝑐2 = 0 and 𝑐1, 𝑎2 = 𝑏2 are arbitrary

constants, then 𝜑, 𝜓 are arbitrary functions.(3) If 𝜑 = −(𝑐2/(𝑎1𝑏2 − 𝑎2𝑏1))(𝑎1𝑥1 + 𝑏1𝑥2 + 𝑐1𝑡 + 𝑑1), then𝜓 is arbitrary function.

3. The Uniqueness of Wave Solutions

In this sectionwe give some nonzero solutions by consideringthe wave solutions to the two-dimensional MHD equations[13–15]. Firstly we could give the initial-boundary value ofthree conditions inTheorem 2.

(1) 𝜑(𝑥1, 𝑥2, 𝑡)|𝑡=0 = 𝜑(𝑎1𝑥1 + 𝑏1𝑥2 + 𝑑1) ∈ 𝐻10 (Ω),𝜑(𝑥1, 𝑥2, 𝑡)|𝑥∈𝜕Ω = 0𝜓(𝑥1, 𝑥2, 𝑡)|𝑡=0 = 𝜓(𝑘𝑎1𝑥1 + 𝑘𝑏1𝑥2 + 𝑑1) ∈ 𝐻10 (Ω),𝜓(𝑥1, 𝑥2, 𝑡)|𝑥∈𝜕Ω = 0.(2) 𝜑(𝑥1, 𝑥2, 𝑡)|𝑡=0 = 𝜑(𝑑1) ∈ 𝐻10 (Ω), 𝜑(𝑥1, 𝑥2, 𝑡)|𝑥∈𝜕Ω =0𝜓(𝑥1, 𝑥2, 𝑡)|𝑡=0 = 𝜓(𝑎2𝑥1 + 𝑏2𝑥2 + 𝑑1) ∈ 𝐻10 (Ω),𝜓(𝑥1, 𝑥2, 𝑡)|𝑥∈𝜕Ω = 0.(3) 𝜑(𝑥1, 𝑥2, 𝑡)|𝑡=0 = −(𝑐2/(𝑎1𝑏2−𝑎2𝑏1))(𝑎1𝑥1+𝑏1𝑥2+𝑑1) ∈𝐻10 (Ω), 𝜑(𝑥1, 𝑥2, 𝑡)|𝑥∈𝜕Ω = 0.𝜓(𝑥1, 𝑥2, 𝑡)|𝑡=0 ∈ 𝐻10 (Ω), 𝜓(𝑥1, 𝑥2, 𝑡)|𝑥∈𝜕Ω = 0.


Theorem 3. Assume that Ω ⊂ 𝑅2 is a bounded domain. Onecan make the solutions 𝜑1, 𝜓1 like the condition in Theorem 2and satisfying ‖Δ𝜓1‖𝐿∞ , ‖∇𝜓1‖𝐿∞ , ‖𝐷2𝜑1‖𝐿∞ ≤ 𝑀, respec-tively. Then the following initial-boundary value problem:

(𝜕𝑡 + 𝑢 ⋅ ∇)𝑤 = 𝐵 ⋅ ∇𝑗, 𝑖𝑛 Ω × (0,∞) ,(𝜕𝑡 + 𝑢 ⋅ ∇) 𝜓 = 0, 𝑖𝑛 Ω × (0,∞) ,

𝜑 = 𝜑0 (𝑎1𝑥1 + 𝑏1𝑥2 + 𝑑1) , 𝑖𝑛 Ω × 0,𝜓 = 𝜓0 (𝑎2𝑥1 + 𝑏2𝑥2 + 𝑑2) , 𝑖𝑛 Ω × 0

(26)

has a unique smooth solution 𝜑, 𝜓.Proof. To prove the uniqueness we consider two smoothsolution pairs, say 𝜑1, 𝜓1 and 𝜑, 𝜓. Let their difference be𝜑 = 𝜑 − 𝜑1, 𝜓 = 𝜓 − 𝜓1. Then subtracting the equations fromeach other in (26), we haveΔ𝜑𝑡 + ∇⊥𝜑 ⋅ ∇Δ𝜑 − ∇⊥𝜑1 ⋅ ∇Δ𝜑1= ∇⊥𝜓 ⋅ ∇Δ𝜓 − ∇⊥𝜓1 ⋅ ∇Δ𝜓1, in Ω × (0,∞) ,

𝜓𝑡 + ∇⊥𝜑 ⋅ ∇𝜓 − ∇⊥𝜑1 ⋅ ∇𝜓1 = 0, in Ω × (0,∞) ,𝜓 (𝑥1, 𝑥2, 0) = 0,𝜑 (𝑥1, 𝑥2, 0) = 0

in Ω × 0,𝜓 = 0,𝜑 = 0

on 𝜕Ω × (0,∞) .

(27)

Multiplying the first and second equations by 𝜑, 𝜓,respectively, integrating overΩ, we obtain− 12 𝑑𝑑𝑡 ∇𝜑2𝐿2 + ∫Ω ∇⊥𝜑 ⋅ ∇ (Δ𝜑) 𝜑𝑑𝑥+ ∫Ω∇⊥𝜑1 ⋅ ∇ (Δ𝜑) 𝜑𝑑𝑥

+ ∫Ω∇⊥𝜑 ⋅ ∇ (Δ𝜑1) 𝜑𝑑𝑥 = ∫

Ω∇⊥𝜓 ⋅ ∇ (Δ𝜓) 𝜑𝑑𝑥

+ ∫Ω∇⊥𝜓1 ⋅ ∇ (Δ𝜓) 𝜑𝑑𝑥 + ∫

Ω∇⊥𝜓 ⋅ ∇ (Δ𝜓1) 𝜑𝑑𝑥,

12 𝑑𝑑𝑡 𝜓2𝐿2 + ∫Ω (∇⊥𝜑 ⋅ ∇𝜓)𝜓𝑑𝑥+ ∫Ω(∇⊥𝜑 ⋅ ∇𝜓1) 𝜓𝑑𝑥 + ∫

Ω(∇⊥𝜑1 ⋅ ∇𝜓)𝜓𝑑𝑥 = 0.

(28)

It is easy to see that

∫Ω∇⊥𝜑 ⋅ ∇ (Δ𝜑) 𝜑𝑑𝑥 = ∫

𝜕ΩΔ𝜑 (𝜑 ⋅ ∇⊥𝜑) ⋅ �⃗� 𝑑𝑠

− ∫ΩΔ𝜑𝑑𝑖V (𝜑 ⋅ ∇⊥𝜑) 𝑑𝑥

= −∫ΩΔ𝜑 ⋅ 0 𝑑𝑥 = 0.

(29)

Similarly we have

∫Ω(∇⊥𝜑1 ⋅ ∇𝜓)𝜓 𝑑𝑥 = 0

∫Ω∇⊥𝜑 ⋅ ∇ (Δ𝜑1) 𝜑 𝑑𝑥 = 0∫Ω(∇⊥𝜑 ⋅ ∇𝜓)𝜓𝑑𝑥 = 0

∫Ω∇⊥𝜓 ⋅ ∇ (Δ𝜓) 𝜑 𝑑𝑥 = −∫

ΩΔ𝜓 (∇𝜑 ⋅ ∇⊥𝜓) 𝑑𝑥

∫Ω∇⊥𝜑1 ⋅ ∇ (Δ𝜑) 𝜑 𝑑𝑥 = −∫

ΩΔ𝜑 (∇𝜑 ⋅ ∇⊥𝜑1) 𝑑𝑥

∫Ω∇⊥𝜓1 ⋅ ∇ (Δ𝜓) 𝜑 𝑑𝑥 = −∫

ΩΔ𝜓∇𝜑 ⋅ ∇⊥𝜓1𝑑𝑥

∫Ω∇⊥𝜓 ⋅ ∇ (Δ𝜓1) 𝜑 𝑑𝑥 = −∫

ΩΔ𝜓1∇𝜑 ⋅ ∇⊥𝜓𝑑𝑥.

(30)

Notice that

𝜓𝑡 + ∇⊥𝜑 ⋅ ∇𝜓 − ∇⊥𝜑1 ⋅ ∇𝜓1 = 0. (31)Multiplying equation by Δ𝜓, integrating over Ω, we

obtain12 𝑑𝑑𝑡 ∇𝜓2𝐿2= ∫Ω(∇⊥𝜑 ⋅ ∇𝜓 + ∇⊥𝜑 ⋅ ∇𝜓1 + ∇⊥𝜑1 ⋅ ∇𝜓) Δ𝜓𝑑𝑥

(32)

so that12 𝑑𝑑𝑡 ∇𝜑2𝐿2 + 12 𝑑𝑑𝑡 𝜓2𝐿2 + 12 𝑑𝑑𝑡 ∇𝜓2𝐿2= −∫ΩΔ𝜑 (∇𝜑 ⋅ ∇⊥𝜑1) 𝑑𝑥 + ∫


+ ∫ΩΔ𝜓1∇𝜑 ⋅ ∇⊥𝜓𝑑𝑥 − ∫

Ω𝜓∇𝜓1 ⋅ ∇⊥𝜑𝑑𝑥

+ ∫Ω∇⊥𝜑1 ⋅ ∇𝜓Δ𝜓𝑑𝑥 + ∫

Ω∇⊥𝜑 ⋅ ∇𝜓1Δ𝜓𝑑𝑥

= 0.

(33)


12 𝑑𝑑𝑡 ∇𝜑2𝐿2 + 12 𝑑𝑑𝑡 𝜓2𝐿2+ 12 𝑑𝑑𝑡 ∇𝜓2𝐿2 𝐷2𝜑1𝐿∞ ∇𝜑2𝐿2+ 𝑐 𝐷2𝜑1𝐿∞ ∇𝜓2𝐿2+ 𝑐 Δ𝜓1𝐿∞ (∇𝜑2𝐿2 + ∇𝜓2𝐿2)+ 𝑐 ∇𝜓1𝐿∞ (𝜓2𝐿2 + ∇𝜑2𝐿2)

≤ 𝑐 (∇𝜓2𝐿2 + ∇𝜑2𝐿2 + 𝜓2𝐿2) .

(34)


Thanks to the Gronwall inequality, we have the following:

∇𝜓2𝐿2 + ∇𝜑2𝐿2 + 𝜓2𝐿2 ≤ 𝑒𝑐𝑡 (∇𝜓 (𝑥1, 𝑥2, 0)2𝐿2+ ∇𝜑 (𝑥1, 𝑥2, 0)2𝐿2 + 𝜓 (𝑥1, 𝑥2, 0)2𝐿2) = 0.

(35)

Therefore there exists a unique solution in the sense of𝐿2(((𝐿2(Ω))2; 0, 𝑇)), ∀𝑇 > 0.4. Analysis Stability of the Wave Solutions

In this section we discuss the stability of the solutions, in𝐿2(Ω) for problem (36) [14, 16–18]. Firstly we could give theinitial-boundary value of three conditions inTheorem 2.

(1) 𝜑(𝑥1, 𝑥2, 𝑡)|𝑡=0 = 𝜑(𝑎1𝑥1 + 𝑏1𝑥2 + 𝑑1) ∈ 𝐻1(Ω),𝜑(𝑥1, 𝑥2, 𝑡)|𝑥∈𝜕Ω = 𝜑(𝑎1𝑥1 + 𝑏1𝑥2 + 𝑑1)𝜓(𝑥1, 𝑥2, 𝑡)|𝑡=0 = 𝜓(𝑘𝑎1𝑥1 + 𝑘𝑏1𝑥2 + 𝑑2) ∈ 𝐻1(Ω),𝜓(𝑥1, 𝑥2, 𝑡)|𝑥∈𝜕Ω = 𝜓(𝑘𝑎1𝑥1 + 𝑘𝑏1𝑥2 + 𝑑2).(2) 𝜑(𝑥1, 𝑥2, 𝑡)|𝑡=0 = 𝜑(𝑑1) ∈ 𝐻1(Ω), 𝜑(𝑥1, 𝑥2, 𝑡)|𝑥∈𝜕Ω =𝜑(𝑐1𝑡 + 𝑑1)𝜓(𝑥1, 𝑥2, 𝑡)|𝑡=0 = 𝜓(𝑎2𝑥1 + 𝑏2𝑥2 + 𝑑2) ∈ 𝐻1(Ω),𝜓(𝑥1, 𝑥2, 𝑡)|𝑥∈𝜕Ω = 𝜓(𝑎2𝑥1 + 𝑎2𝑥2 + 𝑑2).(3) 𝜑(𝑥1, 𝑥2, 𝑡)|𝑡=0 = −(𝑐2/(𝑎1𝑏2−𝑎2𝑏1))(𝑎1𝑥1+𝑏1𝑥2+𝑑1) ∈𝐻1(Ω), 𝜑(𝑥1, 𝑥2, 𝑡)|𝑥∈𝜕Ω = 𝜑(𝑎1𝑥1 + 𝑏1𝑥2 + 𝑐1𝑡 + 𝑑1)𝜓(𝑥1, 𝑥2, 𝑡)|𝑡=0 ∈ 𝐻1(Ω), 𝜓(𝑥1, 𝑥2, 𝑡)|𝑥∈𝜕Ω = 𝜓(𝑎2𝑥1+𝑏2𝑥2 + 𝑐2𝑡 + 𝑑2).Assume that Ω ⊂ 𝑅2 is a bounded domain. We can make

the solutions 𝜑1, 𝜓1 like the condition in Theorem 2 andsatisfy ‖Δ𝜓1‖𝐿∞ , ‖∇𝜓1‖𝐿∞ , ‖𝐷2𝜑1‖𝐿∞ ≤ 𝑀, respectively. Thenthe following problem:

(𝜕𝑡 + 𝑢 ⋅ ∇)𝑤 = 𝐵 ⋅ ∇𝑗, in Ω × (0,∞) ,(𝜕𝑡 + 𝑢 ⋅ ∇) 𝜓 = 0 in Ω × (0,∞) (36)

has stable solution.Assume that 𝜑, 𝜓 are another solutions of the equations.

Let 𝜑, 𝜓 denote the solution pair of a little disturbance andlet 𝜑 = 𝜑 − 𝜑1, 𝜓 = 𝜓 − 𝜓1 be the difference of 𝜑, 𝜑1,𝜓, 𝜓1, with initial value ‖𝜑0(𝑥)‖𝐿2 → 0, ‖∇𝜑0(𝑥)‖𝐿2 → 0,‖𝜓0(𝑥)‖𝐿2 → 0, ‖∇𝜓0(𝑥)‖𝐿2 → 0 in the sense of 𝐿2(Ω) andagain assuming that boundary value 𝜑 → 𝜑1, 𝜓 → 𝜓1 in thesense of 𝐿2(𝜕Ω)⋂𝐿∞(𝜕Ω), ∀𝑡 ∈ (0, 𝑇]. Then subtracting oneequation from each other in (36), we get

Δ𝜑𝑡 + ∇⊥𝜑 ⋅ ∇Δ𝜑 − ∇⊥𝜑1 ⋅ ∇Δ𝜑1= ∇⊥𝜓 ⋅ ∇Δ𝜓 − ∇⊥𝜓1 ⋅ ∇Δ𝜓1, in Ω × (0,∞) ,

𝜓𝑡 + ∇⊥𝜑 ⋅ ∇𝜓 − ∇⊥𝜑1 ⋅ ∇𝜓1 = 0, in Ω × (0,∞) ,𝜓 (𝑥1, 𝑥2, 0) = 𝜓0 (𝑥1, 𝑥2, 0) ,𝜑 (𝑥1, 𝑥2, 0) = 𝜑0 (𝑥1, 𝑥2, 0)

in Ω × 0,𝜓 = 𝜑 = ℎ (𝑥, 𝑡) on 𝜕Ω × (0,∞) .

(37)

Similar to uniqueness, we can obtain

12 𝑑𝑑𝑡 ∇𝜑2𝐿2 + 12 𝑑𝑑𝑡 𝜓2𝐿2 + 12 𝑑𝑑𝑡 ∇𝜓2𝐿2= −∫ΩΔ𝜑 (∇𝜑 ⋅ ∇⊥𝜑1) 𝑑𝑥 + ∫


+ ∫ΩΔ𝜓1∇𝜑 ⋅ ∇⊥𝜓𝑑𝑥 − ∫


+ ∫Ω∇⊥𝜑1 ⋅ ∇𝜓Δ𝜓𝑑𝑥 + ∫


+ ∫𝜕Ω𝜑Δ𝜑∇⊥𝜑 ⋅ �⃗� 𝑑𝑠 + ∫

𝜕Ω𝜑Δ𝜑∇⊥𝜑1 ⋅ �⃗� 𝑑𝑠

+ ∫𝜕Ω𝜑Δ𝜑1∇⊥𝜑 ⋅ �⃗� 𝑑𝑠 − ∫

𝜕Ω𝜑Δ𝜓∇⊥𝜓 ⋅ �⃗� 𝑑𝑠

− ∫𝜕Ω𝜑Δ𝜓∇⊥𝜓1 ⋅ �⃗� 𝑑𝑠 − ∫

𝜕Ω𝜑Δ𝜓1∇⊥𝜓 ⋅ �⃗� 𝑑𝑠

− 12 ∫𝜕Ω 𝜓∇⊥𝜑 ⋅ �⃗� 𝑑𝑠.

(38)


12 𝑑𝑑𝑡 ∇𝜑2𝐿2 + 12 𝑑𝑑𝑡 𝜓2𝐿2 + 12 𝑑𝑑𝑡 ∇𝜓2𝐿2 ≤ 𝑐 (∇𝜓2𝐿2+ ∇𝜑2𝐿2 + 𝜓2𝐿2)+ 𝑐0 (𝜑𝐿2(𝜕Ω) ∇⊥𝜑𝐿2(𝜕Ω) Δ𝜑𝐿∞(𝜕Ω)+ 𝜑𝐿2(𝜕Ω) ∇⊥𝜑1𝐿2(𝜕Ω) Δ𝜑𝐿∞(𝜕Ω)+ 𝜑𝐿2(𝜕Ω) ∇⊥𝜑𝐿2(𝜕Ω) Δ𝜑1𝐿∞(𝜕Ω)+ 𝜑𝐿2(𝜕Ω) ∇⊥𝜓𝐿2(𝜕Ω) Δ𝜓𝐿∞(𝜕Ω)+ 𝜑𝐿2(𝜕Ω) ∇⊥𝜓1𝐿2(𝜕Ω) Δ𝜓𝐿∞(𝜕Ω)+ 𝜑𝐿2(𝜕Ω) ∇⊥𝜓𝐿2(𝜕Ω) Δ𝜓1𝐿∞(𝜕Ω)+ 𝜓𝐿2(𝜕Ω) ∇⊥𝜑𝐿2(𝜕Ω) 𝜓𝐿∞(𝜕Ω)+ 𝜓𝐿2(𝜕Ω) ∇⊥𝜑1𝐿2(𝜕Ω) 𝜓𝐿∞(𝜕Ω)) .

(∗)

Since 𝜑 → 𝜑1, 𝜓 → 𝜓1 in the sense of 𝐿2(𝜕Ω)⋂𝐿∞(𝜕Ω),we can make, for every 𝜖 > 0,

(𝜑𝐿2(𝜕Ω) ∇⊥𝜑𝐿2(𝜕Ω) Δ𝜑𝐿∞(𝜕Ω)+ 𝜑𝐿2(𝜕Ω) ∇⊥𝜑1𝐿2(𝜕Ω) Δ𝜑𝐿∞(𝜕Ω)+ 𝜑𝐿2(𝜕Ω) ∇⊥𝜑𝐿2(𝜕Ω) Δ𝜑1𝐿∞(𝜕Ω)+ 𝜑𝐿2(𝜕Ω) ∇⊥𝜓𝐿2(𝜕Ω) Δ𝜓𝐿∞(𝜕Ω)


+ 𝜑𝐿2(𝜕Ω) ∇⊥𝜓1𝐿2(𝜕Ω) Δ𝜓𝐿∞(𝜕Ω)+ 𝜑𝐿2(𝜕Ω) ∇⊥𝜓𝐿2(𝜕Ω) Δ𝜓1𝐿∞(𝜕Ω)+ 𝜓𝐿2(𝜕Ω) ∇⊥𝜑𝐿2(𝜕Ω) 𝜓𝐿∞(𝜕Ω)+ 𝜓𝐿2(𝜕Ω) ∇⊥𝜑1𝐿2(𝜕Ω) 𝜓𝐿∞(𝜕Ω)) ≤ 𝜖.

(39)

Using the Gronwall inequality in (∗), for every ∀𝑡 ∈(0, 𝑇],∇𝜓2𝐿2 + ∇𝜑2𝐿2 + 𝜓2𝐿2 ≤ 𝑒𝑐𝑡 (∇𝜓 (𝑥1, 𝑥2, 0)2𝐿2+ ∇𝜑 (𝑥1, 𝑥2, 0)2𝐿2 + 𝜓 (𝑥1, 𝑥2, 0)2𝐿2) + 2𝜖𝑒𝑐𝑡→ 0.

(40)

As 𝜑 → 𝜑1, 𝜓 → 𝜓1 in the sense of 𝐿2(𝜕Ω)⋂𝐿∞(𝜕Ω)and ‖𝜑0(𝑥)‖𝐿2 → 0, ‖∇𝜑0(𝑥)‖𝐿2 → 0, ‖𝜓0(𝑥)‖𝐿2 → 0,‖∇𝜓0(𝑥)‖𝐿2 → 0, in the sense of 𝐿2(Ω). So we reach thestability of the solution in the finite time.

5. The Lyapunov Stability ofSteady State Solution

In this section we discuss the stability of the steady statesolutions, in 𝐿2(Ω) for problem (41).Definition 4. A steady state solution 𝜐 is said to be stable ifand only if 𝜐 in any one of the neighborhood 𝑉, there is aneighborhood𝑊 of 𝜐, making any solutions 𝜐(𝑡, ⋅) with theinitial condition 𝜐(0, ⋅) ∈ 𝑊 satisfy 𝜐(𝑡, ⋅) ∈ 𝑉 (∀𝑡 ≥ 0).

Assume that Ω ⊂ 𝑅2 is a bounded domain. We can makethe solutions 𝜑1, 𝜓1 like the condition in Theorem 2 andsatisfy ‖Δ𝜓1‖𝐿∞ , ‖∇𝜓1‖𝐿∞ , ‖𝐷2𝜑1‖𝐿∞ ≤ 𝑀, respectively. Thenthe following problem solutions:

(𝜕𝑡 + 𝑢 ⋅ ∇)𝑤 = 𝐵 ⋅ ∇𝑗, in Ω × (0,∞) ,(𝜕𝑡 + 𝑢 ⋅ ∇) 𝜓 = 0 in Ω × (0,∞) (41)

are Lyapunov stable.Assume that 𝜑, 𝜓 are another solutions of the equations.

Let 𝜑, 𝜓 denote the solution pair of a little disturbance andlet 𝜑 = 𝜑 − 𝜑1, 𝜓 = 𝜓 − 𝜓1 be the difference of 𝜑, 𝜑1,𝜓, 𝜓1, with initial value ‖𝜑0(𝑥)‖𝐿2 → 0, ‖∇𝜑0(𝑥)‖𝐿2 → 0,‖𝜓0(𝑥)‖𝐿2 → 0, ‖∇𝜓0(𝑥)‖𝐿2 → 0, in the sense of 𝐿2(Ω)and assume that boundary value 𝜑 → 𝜑1, 𝜓 → 𝜓1 in thesense of 𝐿2(𝜕Ω)⋂𝐿∞(𝜕Ω), ∀𝑡 ∈ (0, 𝑇]. Then subtracting oneequation from each other in (41), we get

Δ𝜑𝑡 + ∇⊥𝜑 ⋅ ∇Δ𝜑 − ∇⊥𝜑1 ⋅ ∇Δ𝜑1= ∇⊥𝜓 ⋅ ∇Δ𝜓 − ∇⊥𝜓1 ⋅ ∇Δ𝜓1, in Ω × (0,∞) ,

𝜓𝑡 + ∇⊥𝜑 ⋅ ∇𝜓 − ∇⊥𝜑1 ⋅ ∇𝜓1 = 0, in Ω × (0,∞) ,𝜓 (𝑥1, 𝑥2, 0) = 𝜓0 (𝑥1, 𝑥2, 0) ,𝜑 (𝑥1, 𝑥2, 0) = 𝜑0 (𝑥1, 𝑥2, 0)

in Ω × 0.(42)

Similar to uniqueness, we can obtain12 𝑑𝑑𝑡 ∇𝜑2𝐿2 + 12 𝑑𝑑𝑡 𝜓2𝐿2 + 12 𝑑𝑑𝑡 ∇𝜓2𝐿2= −∫ΩΔ𝜑 (∇𝜑 ⋅ ∇⊥𝜑1) 𝑑𝑥 + ∫


+ ∫ΩΔ𝜓1∇𝜑 ⋅ ∇⊥𝜓𝑑𝑥 − ∫


+ ∫Ω∇⊥𝜑1 ⋅ ∇𝜓Δ𝜓𝑑𝑥 + ∫


+ ∫𝜕Ω𝜑Δ𝜑∇⊥𝜑 ⋅ �⃗� 𝑑𝑠 + ∫

𝜕Ω𝜑Δ𝜑∇⊥𝜑1 ⋅ �⃗� 𝑑𝑠

+ ∫𝜕Ω𝜑Δ𝜑1∇⊥𝜑 ⋅ �⃗� 𝑑𝑠 − ∫

𝜕Ω𝜑Δ𝜓∇⊥𝜓 ⋅ �⃗� 𝑑𝑠

− ∫𝜕Ω𝜑Δ𝜓∇⊥𝜓1 ⋅ �⃗� 𝑑𝑠 − ∫

𝜕Ω𝜑Δ𝜓1∇⊥𝜓 ⋅ �⃗� 𝑑𝑠

− 12 ∫𝜕Ω 𝜓∇⊥𝜑 ⋅ �⃗�𝜓 𝑑𝑠.

(43)

It is easy to see that12 𝑑𝑑𝑡 ∇𝜑2𝐿2 + 12 𝑑𝑑𝑡 𝜓2𝐿2 + 12 𝑑𝑑𝑡 ∇𝜓2𝐿2≤ 𝑐 (∇𝜓2𝐿2 + ∇𝜑2𝐿2 + 𝜓2𝐿2)+ 𝑐0 (𝜑𝐿2(𝜕Ω) ∇⊥𝜑𝐿2(𝜕Ω) Δ𝜑𝐿∞(𝜕Ω)+ 𝜑𝐿2(𝜕Ω) ∇⊥𝜑1𝐿2(𝜕Ω) Δ𝜑𝐿∞(𝜕Ω)+ 𝜑𝐿2(𝜕Ω) ∇⊥𝜑𝐿2(𝜕Ω) Δ𝜑1𝐿∞(𝜕Ω)+ 𝜑𝐿2(𝜕Ω) ∇⊥𝜓𝐿2(𝜕Ω) Δ𝜓𝐿∞(𝜕Ω)+ 𝜑𝐿2(𝜕Ω) ∇⊥𝜓1𝐿2(𝜕Ω) Δ𝜓𝐿∞(𝜕Ω)+ 𝜑𝐿2(𝜕Ω) ∇⊥𝜓𝐿2(𝜕Ω) Δ𝜓1𝐿∞(𝜕Ω)+ 𝜓𝐿2(𝜕Ω) ∇⊥𝜑𝐿2(𝜕Ω) 𝜓𝐿∞(𝜕Ω)+ 𝜓𝐿2(𝜕Ω) ∇⊥𝜑1𝐿2(𝜕Ω) 𝜓𝐿∞(𝜕Ω)) .

(∗∗)

Since 𝜑 → 𝜑1, 𝜓 → 𝜓1 in the sense of 𝐿2(𝜕Ω)⋂𝐿∞(𝜕Ω),we can make, for every 𝜖 > 0,

(𝜑𝐿2(𝜕Ω) ∇⊥𝜑𝐿2(𝜕Ω) Δ𝜑𝐿∞(𝜕Ω)+ 𝜑𝐿2(𝜕Ω) ∇⊥𝜑1𝐿2(𝜕Ω) Δ𝜑𝐿∞(𝜕Ω)


+ 𝜑𝐿2(𝜕Ω) ∇⊥𝜑𝐿2(𝜕Ω) Δ𝜑1𝐿∞(𝜕Ω)+ 𝜑𝐿2(𝜕Ω) ∇⊥𝜓𝐿2(𝜕Ω) Δ𝜓𝐿∞(𝜕Ω)+ 𝜑𝐿2(𝜕Ω) ∇⊥𝜓1𝐿2(𝜕Ω) Δ𝜓𝐿∞(𝜕Ω)+ 𝜑𝐿2(𝜕Ω) ∇⊥𝜓𝐿2(𝜕Ω) Δ𝜓1𝐿∞(𝜕Ω)+ 𝜓𝐿2(𝜕Ω) ∇⊥𝜑𝐿2(𝜕Ω) 𝜓𝐿∞(𝜕Ω)+ 𝜓𝐿2(𝜕Ω) ∇⊥𝜑1𝐿2(𝜕Ω) 𝜓𝐿∞(𝜕Ω)) ≤ 𝜖.

(44)

Using the Gronwall inequality in (∗∗), for every ∀𝑡 ∈(0, 𝑇],∇𝜓2𝐿2 + ∇𝜑2𝐿2 + 𝜓2𝐿2 ≤ 𝑒𝑐𝑡 (∇𝜓 (𝑥1, 𝑥2, 0)2𝐿2+ ∇𝜑 (𝑥1, 𝑥2, 0)2𝐿2 + 𝜓 (𝑥1, 𝑥2, 0)2𝐿2) + 2𝜖𝑒𝑐𝑡→ 0.

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As 𝜑 → 𝜑1, 𝜓 → 𝜓1 in the sense of 𝐿2(𝜕Ω)⋂𝐿∞(𝜕Ω)and ‖𝜑0(𝑥)‖𝐿2 → 0, ‖∇𝜑0(𝑥)‖𝐿2 → 0, ‖𝜓0(𝑥)‖𝐿2 → 0,‖∇𝜓0(𝑥)‖𝐿2 → 0, in the sense of 𝐿2(Ω). Definition 4 is satis-fied. So we reach the Lyapunov stability of the steady statesolutions in finite time.

6. Conclusions

In this paper, we studied the symmetry groups by using theclassical Lie Group method to structural equation solutions.First, we perform Lie symmetry analysis for the MHDequation and get its infinitesimal generator. Then, we obtainmany solutions by it. It can be seen by the results of this paperthat the LieGroupmethod is an effectivemethod for studyingnonlinear partial differential equations.

Competing Interests

The authors declare that they have no competing interests.

Authors’ Contributions

All authors contributed equally to the writing of this paper.All authors read and approved the final manuscript.

Acknowledgments

The first author is grateful to Professor Ganshan Yang andYangrong Li for their support and also thanks Jiali Yu forhelp in writing this paper. The work is supported by theNational Natural Science Foundation of China (nos. 11161057and 11561076).

References

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[2] D. Biskamp, Nonlinear Magnetohydrodynamics, vol. 1 of Cam-bridge Monographs on Plasma Physics, Cambridge UniversityPress, Cambridge, Uk, 1993.

[3] Q. L. Chen, C. X. Miao, and Z. F. Zhang, “The Beale-Kato-Majda criterion for the 3Dmagneto-hydrodynamics equations,”Communications in Mathematical Physics, vol. 275, no. 3, pp.861–872, 2007.

[4] J. Wu, “Viscous and inviscid magnetohydrodynamics equa-tions,” Journal d’Analyse Mathématique, vol. 73, pp. 251–265,1997.

[5] C. Cao and J. Wu, “Global regularity for the 2D MHD equa-tions with mixed partial dissipation and magnetic diffusion,”https://arxiv.org/abs/0901.2908.

[6] C.He andZ.Xin, “Partial regularity of suitableweak solutions tothe incompressible magnetohydrodynamic equations,” Journalof Functional Analysis, vol. 227, no. 1, pp. 113–152, 2005.

[7] V. Vyalov, “Partial regularity of solutions to the magnetohy-drodynamic equations,” Journal of Mathematical Sciences (NewYork), vol. 150, no. 1, pp. 1771–1786, 2008.

[8] M. Sermange and R. Temam, “Some mathematical questionsrelated to the MHD equations,” Communications on Pure andApplied Mathematics, vol. 36, no. 5, pp. 635–664, 1983.

[9] P. J. Olver, Applications of Lie Groups to Differential Equations,vol. 107, Springer, New York, NY, USA, 2nd edition, 1993.

[10] S. Yang and C. Hua, “Lie symmetry reductions and exact solu-tions of a coupled KdV-Burgers equation,”AppliedMathematicsand Computation, vol. 234, pp. 579–583, 2014.

[11] S. Shen, C. Qu, and Q. Huang, “Lie group classification ofthe N-th-order nonlinear evolution equations,” Science ChinaMathematics, vol. 54, no. 12, pp. 2553–2572, 2011.

[12] E. Casella, P. Secchi, and P. Trebeschi, “Global classical solutionsforMHD system,” Journal ofMathematical FluidMechanics, vol.5, no. 1, pp. 70–91, 2003.

[13] X. Liao, The Stability of the Theory, Method and Application,Huazhong University of Science and Technology Press, 2ndedition, 2010.

[14] W. Song, H. Li, G. Yang, and G. X. Yuan, “Nonhomoge-neous boundary value problem for (I, J) similar Solutions ofincompressible two-dimensional Euler equations,” Journal ofInequalities and Applications, vol. 2014, no. 1, article 277, 2014.

[15] M. D. Gunzburger, A. J. Meir, and J. S. Peterson, “On theexistence, uniqueness, and finite element approximation ofsolutions of the equations of stationary, incompressible mag-netohydrodynamics,”Mathematics of Computation, vol. 56, no.194, pp. 523–563, 1991.

[16] L.-J. Yang and J.-M. Wang, “Staility of a damped hyperbolicTimoshenko system coupled with a heat equation,” AsianJournal of Control, vol. 16, no. 2, pp. 546–555, 2014.

[17] G. Yang, 𝛿-Viscosity solution, blowup solution and global solutionof multidimensional Landau-Lifshitz equations [Ph.D. thesis],China Academy of Engineering Physics, Sichuan, China, 2002.

[18] V. I. Yudovitch, “Non-stationary flow of an ideal incompressibleliquid,” USSR Computational Mathematics and MathematicalPhysics, vol. 3, no. 6, pp. 1407–1456, 1963.

https://arxiv.org/abs/0901.2908

Research ArticleResolvent for Non-Self-Adjoint Differential Operator withBlock-Triangular Operator Potential

Aleksandr Mikhailovich Kholkin

Department of Higher and Applied Mathematics, Priazovskyi State Technical University, Universitetskaya Street 7,Mariupol 87500, Ukraine

Correspondence should be addressed to Aleksandr Mikhailovich Kholkin; [email protected]

Received 10 July 2016; Revised 30 September 2016; Accepted 6 October 2016

Academic Editor: Cemil Tunç

Copyright © 2016 Aleksandr Mikhailovich Kholkin. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

A resolvent for a non-self-adjoint differential operator with a block-triangular operator potential, increasing at infinity, isconstructed. Sufficient conditions under which the spectrum is real and discrete are obtained.

1. Introduction

The theory of non-self-adjoint singular differential operators,generated by scalar differential expressions, has been wellstudied. An overview on the theory of non-self-adjointsingular ordinary differential operators is provided in V. E.Lyantse’s Appendix I to the monograph of Naimark [1]. Inthis regard the papers ofNaimark [2], Lyantse [3],Marchenko[4], Rofe-Beketov [5], Schwartz [6], and Kato [7] shouldbe noted. The questions regarding equations with non-Hermitian matrix or operator coefficients have been studiedinsufficiently. For a differential operator with a triangularmatrix potential decreasing at infinity, which has a boundedfirstmoment due to the inverse scattering problem, it is statedin [8, 9] that the discrete spectrum of the operator consistsof a finite number of negative eigenvalues, and the essentialspectrum covers the positive semiaxis. The questions regard-ing an operator with a block-triangular matrix potential thatincreases at infinity are considered in [10, 11]. In the future,by the author of this paper similar questions are consideredfor equations with block-triangular operator coefficients. In[11, 12] Green’s function of a non-self-adjoint operator isconstructed.

In this article we construct a resolvent for a non-self-adjoint differential operator, using which the structure of theoperator spectrum is set.

2. Preliminary Notes

Let 𝐻𝑘, 𝑘 = 1, 2, . . . , 𝑟, be finite-dimensional or infinite-dimensional separable Hilbert space with inner product (⋅, ⋅)and norm | ⋅ |, dim𝐻𝑘 ≤ ∞. Denote H = 𝐻1 ⊕ 𝐻2 ⊕⋅ ⋅ ⋅ ⊕ 𝐻𝑟. Element ℎ ∈ H will be written in the form ℎ =col (ℎ1, ℎ2, . . . , ℎ𝑟), where ℎ𝑘 ∈ 𝐻𝑘, 𝑘 = 1, 𝑟, 𝐼𝑘, 𝐼 are identityoperators in𝐻𝑘 andH accordingly.

We denote by 𝐿2(H, (0,∞)) the Hilbert space of vector-valued functions 𝑦(𝑥) with values inH with inner product⟨𝑦, 𝑧⟩ = ∫∞

0

(𝑦 (𝑥) , 𝑧 (𝑥)) 𝑑𝑥 (1)and the corresponding norm ‖ ⋅ ‖.

Consider the equation with block-triangular operatorpotential 𝑙 [𝑦] = −𝑦 + 𝑉 (𝑥) 𝑦 = 𝜆𝑦, 0 ≤ 𝑥 < ∞, (2)where 𝑉 (𝑥) = V (𝑥) ⋅ 𝐼 + 𝑈 (𝑥) ,

𝑈 (𝑥) = (𝑈11 (𝑥) 𝑈12 (𝑥) ⋅ ⋅ ⋅ 𝑈1𝑟 (𝑥)0 𝑈22 (𝑥) ⋅ ⋅ ⋅ 𝑈2𝑟 (𝑥)⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅0 0 ⋅ ⋅ ⋅ 𝑈𝑟𝑟 (𝑥)) ,(3)


http://dx.doi.org/10.1155/2016/2964817


V(𝑥) is a real scalar function, and 0 < V(𝑥) → ∞ monotoni-cally, as 𝑥 → ∞, and it has monotone absolutely continuousderivative. Also, 𝑈(𝑥) is a relatively small perturbation; forexample, |𝑈(𝑥)| ⋅ V−1(𝑥) → 0 as 𝑥 → ∞ or |𝑈|V−1 ∈ 𝐿∞(R+).The diagonal blocks𝑈𝑘𝑘(𝑥), 𝑘 = 1, 𝑟, are assumed as boundedself-adjoint operators in𝐻𝑘, 𝑈𝑘𝑙 : 𝐻𝑙 → 𝐻𝑘.

In case where

V (𝑥) ≥ 𝐶𝑥2𝛼, 𝐶 > 0, 𝛼 > 1, (4)we suppose that coefficients of (2) satisfy relations∫∞

0

|𝑈 (𝑡)| ⋅ V−1/2 (𝑡) 𝑑𝑡 < ∞,∫∞0

V2 (𝑡) ⋅ V−5/2 (𝑡) 𝑑𝑡 < ∞,∫∞0

V (𝑡) ⋅ V−3/2 (𝑡) 𝑑𝑡 < ∞.(5)

Let us consider the functions𝛾0 (𝑥) = 14√4V (𝑥) ⋅ exp(−∫𝑥0 √V (𝑢)𝑑𝑢) ,𝛾∞ (𝑥) = 14√4V (𝑥) ⋅ exp(∫𝑥0 √V (𝑢)𝑑𝑢) . (6)

It is easy to see that 𝛾0(𝑥) → 0, 𝛾∞(𝑥) → ∞ as 𝑥 → ∞.Thesesolutions constitute a fundamental system of solutions of thescalar differential equation−𝑧 + (V (𝑥) + 𝑞 (𝑥)) 𝑧 = 0, (7)where 𝑞(𝑥) is determined by a formula (cf. with the mono-graph [13])

𝑞 (𝑥) = 516 (V (𝑥)V (𝑥) )2 − 14 V (𝑥)V (𝑥) . (8)In such a way for all 𝑥 ∈ [0,∞) one has𝑊(𝛾0, 𝛾∞) fl 𝛾0 (𝑥) ⋅ 𝛾∞ (𝑥) − 𝛾0 (𝑥) ⋅ 𝛾∞ (𝑥) = 1. (9)In case of V(𝑥) = 𝑥2𝛼, 0 < 𝛼 ≤ 1, we suppose that thecoefficients of (2) satisfy the relation∫∞

𝑎

|𝑈 (𝑡)| ⋅ 𝑡−𝛼𝑑𝑡 < ∞, 𝑎 > 0. (10)Now functions 𝛾0(𝑥, 𝜆) and 𝛾∞(𝑥, 𝜆) are defined as follows:𝛾0 (𝑥, 𝜆) = 1

4√4 (𝑥2𝛼 − 𝜆) ⋅ exp(−∫𝑥𝑎 √𝑢2𝛼 − 𝜆𝑑𝑢) ,𝛾∞ (𝑥, 𝜆) = 14√4 (𝑥2𝛼 − 𝜆) ⋅ exp(∫𝑥𝑎 √𝑢2𝛼 − 𝜆𝑑𝑢) . (11)

These functions also form a fundamental system of solutionsof the scalar differential equation, which is obtained byreplacing V(𝑥) with V(𝑥) − 𝜆 in formulas (7) and (8).

In [10] the asymptotic behavior of the functions 𝛾0(𝑥, 𝜆)and 𝛾∞(𝑥, 𝜆)was established as𝑥 → ∞. If (𝛼+1)/2𝛼 = 𝑛 ∈ N,that is, 𝛼 = 1/(2𝑛 − 1), then functions 𝛾0(𝑥, 𝜆) and 𝛾∞(𝑥, 𝜆)as 𝑥 → ∞ will have the following asymptotic behavior:𝛾0 (𝑥, 𝜆) = 𝑐 ⋅ exp(− 𝑥1+𝛼1 + 𝛼 + 𝜆2 ⋅ 𝑥1−𝛼1 − 𝛼

+ 𝑛−1∑𝑘=2

1 ⋅ 3 ⋅ . . . ⋅ (2𝑘 − 3)𝑘! ⋅ (𝜆2)𝑘 ⋅ 𝑥1−(2𝑘−1)𝛼1 − (2𝑘 − 1) 𝛼)⋅ 𝑥((1⋅3⋅...⋅(2𝑛−3))/𝑛!)⋅(𝜆/2)𝑛−𝛼/2 ⋅ (1 + 𝑜 (1)) ,𝛾∞ (𝑥, 𝜆) = 𝑐 ⋅ exp( 𝑥1+𝛼1 + 𝛼 − 𝜆2 ⋅ 𝑥1−𝛼1 − 𝛼

− 𝑛−1∑𝑘=2

1 ⋅ 3 ⋅ . . . ⋅ (2𝑘 − 3)𝑘! ⋅ (𝜆2)𝑘 ⋅ 𝑥1−(2𝑘−1)𝛼1 − (2𝑘 − 1) 𝛼)⋅ 𝑥−(((1⋅3⋅...⋅(2𝑛−3))/𝑛!)⋅(𝜆/2)𝑛+𝛼/2) ⋅ (1 + 𝑜 (1)) .

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In particular, with 𝛼 = 1 (𝑛 = 1) one has𝛾0 (𝑥, 𝜆) = 𝑐 ⋅ 𝑥(𝜆−1)/2 ⋅ exp(−𝑥22 ) (1 + 𝑜 (1)) ,𝛾∞ (𝑥, 𝜆) = 𝑐 ⋅ 𝑥−(𝜆+1)/2 ⋅ exp(𝑥22 ) (1 + 𝑜 (1)) . (13)In the case (𝛼 + 1)/2𝛼 ∉ N, set 𝑛 = [(𝛼 + 1)/2𝛼] + 1, with [𝛽]being the integral part of𝛽, to obtain the following asymptoticbehavior for 𝛾0(𝑥, 𝜆) and 𝛾∞(𝑥) at infinity:𝛾0 (𝑥, 𝜆) = 𝑐 ⋅ 𝑥−𝛼/2 exp(− 𝑥1+𝛼1 + 𝛼 + 𝜆2 ⋅ 𝑥1−𝛼1 − 𝛼

+ 𝑛−1∑𝑘=2

1 ⋅ 3 ⋅ . . . ⋅ (2𝑘 − 3)𝑘! ⋅ (𝜆2)𝑘 ⋅ 𝑥1−(2𝑘−1)𝛼1 − (2𝑘 − 1) 𝛼)⋅ exp(−1 ⋅ 3 ⋅ . . . ⋅ (2𝑛 − 3)𝑛! ⋅ (𝜆2)𝑛 ⋅ 𝑥−𝛼𝛼 ) ⋅ (1+ 𝑜 (𝑥−𝛼)) ,𝛾∞ (𝑥, 𝜆) = 𝑐 ⋅ 𝑥−𝛼/2 exp( 𝑥1+𝛼1 + 𝛼 − 𝜆2 ⋅ 𝑥1−𝛼1 − 𝛼

− 𝑛−1∑𝑘=2

1 ⋅ 3 ⋅ . . . ⋅ (2𝑘 − 3)𝑘! ⋅ (𝜆2)𝑘 ⋅ 𝑥1−(2𝑘−1)𝛼1 − (2𝑘 − 1) 𝛼)⋅ exp(1 ⋅ 3 ⋅ . . . ⋅ (2𝑛 − 3)𝑛! ⋅ (𝜆2)𝑛 ⋅ 𝑥−𝛼𝛼 ) ⋅ (1+ 𝑜 (𝑥−𝛼)) .

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In [10] for an equation with matrix coefficients, and inthe furtherance for equations with operator coefficients, thefollowing theorem is proved.


Theorem 1. If for (2) conditions (4)-(5) are satisfied for 𝛼 > 1or condition (10) for 0 < 𝛼 ≤ 1, then the equation has a uniquedecreasing at infinity operator solution Φ(𝑥, 𝜆), satisfying theconditions

lim𝑥→∞

Φ (𝑥, 𝜆)𝛾0 (𝑥, 𝜆) = 𝐼,lim𝑥→∞

Φ (𝑥, 𝜆)𝛾0 (𝑥, 𝜆) = 𝐼. (15)Also, there exists increasing at infinity operator solutionΨ(𝑥, 𝜆), satisfying the conditions

lim𝑥→∞

Ψ (𝑥, 𝜆)𝛾∞ (𝑥, 𝜆) = 𝐼,lim𝑥→∞

Ψ (𝑥, 𝜆)𝛾∞ (𝑥, 𝜆) = 𝐼. (16)Corollary 2. If 𝛼 = 1, that is, V(𝑥) = 𝑥2, then, under condition(10), the solutions Φ(𝑥, 𝜆) and Ψ(𝑥, 𝜆) have common (known)asymptotic behavior, as in the quality 𝛾0(𝑥, 𝜆) and 𝛾∞(𝑥, 𝜆) youcan take the following functions:𝛾0 (𝑥, 𝜆) = 𝑥(𝜆−1)/2 ⋅ exp(−𝑥22 ) ,𝛾∞ (𝑥, 𝜆) = 𝑥−(𝜆+1)/2 ⋅ exp(𝑥22 ) . (17)3. Resolvent of the Non-Self-Adjoint Operator

Let the following boundary condition be given at 𝑥 = 0:cos𝐴 ⋅ 𝑦 (0) − sin𝐴 ⋅ 𝑦 (0) = 0, (18)

where 𝐴 is block-triangular operator of the same structureas the potential 𝑉(𝑥) (3) of the differential equation (2), and𝐴𝑘𝑘, 𝑘 = 1, 𝑟, are the bounded self-adjoint operators in 𝐻𝑘,which satisfy the conditions−𝜋2 𝐼𝑘 ≪ 𝐴𝑘𝑘 ≤ 𝜋2 𝐼𝑘. (19)Together with problem (2) and (18) we consider the separatedsystem𝑙𝑘 [𝑦𝑘] = −𝑦𝑘 + (V (𝑥) 𝐼𝑘 + 𝑈𝑘𝑘 (𝑥)) 𝑦𝑘 = 𝜆𝑦𝑘, 𝑘 = 1, 𝑟 (20)with the boundary conditions

cos𝐴𝑘𝑘 ⋅ 𝑦𝑘 (0) − sin𝐴𝑘𝑘 ⋅ 𝑦𝑘 (0) = 0, 𝑘 = 1, 𝑟. (21)Let 𝐿 denote the minimal differential operator generated

by differential expression 𝑙[𝑦] and the boundary condition(18), and let 𝐿𝑘, 𝑘 = 1, 𝑟, denote the minimal differentialoperator on 𝐿2(H, (0,∞)) generated by differential expres-sion 𝑙𝑘[𝑦𝑘] and the boundary conditions (21). Taking intoaccount the conditions on coefficients, as well as sufficient

smallness of perturbations 𝑈𝑘𝑘(𝑥), and conditions (19), weconclude that, for every symmetric operator 𝐿𝑘, 𝑘 = 1, 𝑟,there is a case of limit point at infinity.Hence their self-adjointextensions 𝐿𝑘 are the closures of operators 𝐿𝑘, respectively.The operators 𝐿𝑘 are semibounded below, and their spectraare discrete.

Let 𝐿 denote the operator extensions 𝐿, by requiring that𝐿2(H, (0,∞)) be the domain of operator 𝐿.The following theorem is proved in [10].

Theorem3. Suppose that for (2) conditions (4)-(5) are satisfiedfor 𝛼 > 1 or condition (10) for 0 < 𝛼 ≤ 1. Then the discretespectrum of the operator 𝐿 is real and coincides with the unionof spectra of the self-adjoint operators 𝐿𝑘, 𝑘 = 1, 𝑟; that is,𝜎𝑑 (𝐿) = 𝑟⋃

𝑘=1

𝜎 (𝐿𝑘) . (22)Comment 4. Note that this theorem contains a statement ofthe discrete spectrum of the non-self-adjoint operator 𝐿 onlyand no allegations of its continuous and residual spectrum.

Along with (2) we consider the equation𝑙1 [𝑦] = −𝑦 + 𝑉∗ (𝑥) 𝑦 = 𝜆𝑦 (23)(𝑉∗(𝑥) is adjoint to the operator𝑉(𝑥)). If the spaceH is finite-dimensional, then (23) can be rewritten as�̃� [�̃�] = −�̃� + �̃�𝑉 (𝑥) = 𝜆�̃�, (24)where �̃� = (�̃�1 �̃�2 . . . �̃�𝑟) and the equation is called the left.

For operator functions 𝑌(𝑥, 𝜆), 𝑍(𝑥, 𝜆) ∈ 𝐵(H) let𝑊{𝑍∗, 𝑌} = 𝑍∗ (𝑥, 𝜆) 𝑌 (𝑥, 𝜆)− 𝑍∗ (𝑥, 𝜆) 𝑌 (𝑥, 𝜆) . (25)If 𝑌(𝑥, 𝜆) is operator solution of (2) and 𝑍(𝑥, 𝜆) is operatorsolution of (23), the Wronskian does not depend on 𝑥.

Nowwe denote𝑌(𝑥, 𝜆) and𝑌1(𝑥, 𝜆) as the solutions of (2)and (23), respectively, satisfying the initial conditions𝑌 (0, 𝜆) = cos𝐴,𝑌 (0, 𝜆) = sin𝐴,𝑌1 (0, 𝜆) = (cos𝐴)∗ ,𝑌1 (0, 𝜆) = (sin𝐴)∗ , 𝜆 ∈ C.

(26)

Because the operator function 𝑌∗1 (𝑥, 𝜆) satisfies equation−𝑌∗1 (𝑥, 𝜆) + 𝑌∗1 (𝑥, 𝜆) ⋅ 𝑉 (𝑥) = 𝜆𝑌∗1 (𝑥, 𝜆) , (27)the operator function �̃�(𝑥, 𝜆) š 𝑌∗1 (𝑥, 𝜆) is a solution to theleft of the equation−�̃� (𝑥, 𝜆) + �̃� (𝑥, 𝜆) ⋅ 𝑉 (𝑥) = 𝜆�̃� (𝑥, 𝜆) (28)and satisfies the initial conditions �̃�(0, 𝜆) = cos𝐴, �̃�(0, 𝜆) =sin𝐴, 𝜆 ∈ C.


Operator solutions of (23) decreasing and increasingat infinity will be denoted by Φ1(𝑥, 𝜆), Ψ1(𝑥, 𝜆), and thecorresponding solutions of (28) are denoted by Φ̃(𝑥, 𝜆) andΨ̃(𝑥, 𝜆). The system operator solutions 𝑌(𝑥, 𝜆), Φ̃(𝑥, 𝜆) ∈𝐵(H) of (2) and (28), respectively, will take the form ofWronskian𝑊{Φ̃, 𝑌} = Φ̃(𝑥, 𝜆)𝑌(𝑥, 𝜆) − Φ̃(𝑥, 𝜆)𝑌(𝑥, 𝜆).

Let us designate𝐺 (𝑥, 𝑡, 𝜆)= {{{𝑌 (𝑥, 𝜆) (𝑊{Φ̃, 𝑌})

−1 Φ̃ (𝑡, 𝜆) 0 ≤ 𝑥 ≤ 𝑡−Φ (𝑥, 𝜆) (𝑊{�̃�, Φ})−1 �̃� (𝑡, 𝜆) 𝑥 ≥ 𝑡. (29)It is proved in [12] that the operator function 𝐺(𝑥, 𝑡, 𝜆)

is Green’s function of the differential operator 𝐿; that is, itpossesses all the classical properties of Green’s function. Inparticular, for a fixed 𝑡 the function 𝐺(𝑥, 𝑡, 𝜆) of the variable𝑥 is an operator solution of (2) on each of the intervals [0, 𝑡),(𝑡,∞), and it satisfies the boundary condition (18), and at afixed 𝑥, the function 𝐺(𝑥, 𝑡, 𝜆) satisfies (28) in the variable𝑡 on each of the intervals [0, 𝑥), (𝑥,∞), and it satisfies theboundary condition (cos𝐴)∗ ⋅ 𝑦(0) − (sin𝐴)∗ ⋅ 𝑦(0) = 0.

By definition (28), function 𝐺(𝑥, 𝑡, 𝜆) is meromorphic byparameter 𝜆with the poles coinciding with the eigenvalues ofthe operator 𝐿.

We consider the operator 𝑅𝜆 defined in 𝐿2(H, (0,∞)) bythe relation(𝑅𝜆𝑓) (𝑥) = ∫∞

0

𝐺 (𝑥, 𝑡, 𝜆) 𝑓 (𝑡) 𝑑𝑡= −∫𝑥0

Φ (𝑥, 𝜆) (𝑊{�̃�, Φ})−1 �̃� (𝑡, 𝜆) 𝑓 (𝑡) 𝑑𝑡+ ∫∞𝑥

𝑌 (𝑥, 𝜆) (𝑊{Φ̃, 𝑌})−1 Φ̃ (𝑡, 𝜆) 𝑓 (𝑡) 𝑑𝑡.(30)

Theorem 5. The operator 𝑅𝜆 is the resolvent of the operator 𝐿.4. Proof of Theorem 5

One can directly verify that, for any function 𝑓(𝑥) ∈𝐿2(H, (0,∞)), the vector-function 𝑦(𝑥, 𝜆) = (𝑅𝜆𝑓)(𝑥) is asolution of the equation 𝑙[𝑦] − 𝜆𝑦 = 𝑓 whenever 𝜆 ∉ 𝜎(𝐿).We will prove that 𝑦(𝑥, 𝜆) ∈ 𝐿2(H, (0,∞)).

Since operator solutions Φ(𝑥, 𝜆) and Ψ(𝑥, 𝜆) form afundamental system of solutions of (2), the operator solution𝑌(𝑥, 𝜆) of (2) satisfying the initial conditions (26) can bewritten as𝑌(𝑥, 𝜆) = Φ(𝑥, 𝜆)𝐴(𝜆)+Ψ(𝑥, 𝜆)𝐵(𝜆), where𝐴(𝜆) =𝑊{Ψ̃, 𝑌}, 𝐵(𝜆) = −𝑊{Φ̃, 𝑌}; that is,𝑌 (𝑥, 𝜆) = Φ (𝑥, 𝜆)𝑊{Ψ̃, 𝑌} − Ψ (𝑥, 𝜆)𝑊{Φ̃, 𝑌} . (31)

Similarly, the operator solution �̃�(𝑥, 𝜆) of (28) can berepresented in the following form:�̃� (𝑥, 𝜆) = 𝑊{�̃�, Φ} Ψ̃ (𝑥, 𝜆) − 𝑊{�̃�, Ψ} Φ̃ (𝑥, 𝜆) . (32)

By using formulas (31) and (32), we can rewrite relation (30)as follows:(𝑅𝜆𝑓) (𝑥)= −∫𝑎

0

Φ (𝑥, 𝜆) (𝑊{�̃�, Φ})−1 �̃� (𝑡, 𝜆) 𝑓 (𝑡) 𝑑𝑡+ 𝜒1 (𝑥, 𝜆) − 𝜒2 (𝑥, 𝜆) + 𝜒3 (𝑥, 𝜆) − 𝜒4 (𝑥, 𝜆) , (33)where 𝑎 > 0 and𝜒1 (𝑥, 𝜆) = Φ (𝑥, 𝜆) (𝑊{�̃�, Φ})−1𝑊{�̃�, Ψ}⋅ ∫𝑥

𝑎

Φ̃ (𝑡, 𝜆) 𝑓 (𝑡) 𝑑𝑡,𝜒2 (𝑥, 𝜆) = Φ (𝑥, 𝜆) ∫𝑥

𝑎

Ψ̃ (𝑡, 𝜆) 𝑓 (𝑡) 𝑑𝑡,𝜒3 (𝑥, 𝜆) = Φ (𝑥, 𝜆)𝑊{Ψ̃, 𝑌} (𝑊{Φ̃, 𝑌})−1⋅ ∫∞𝑥

Φ̃ (𝑡, 𝜆) 𝑓 (𝑡) 𝑑𝑡,𝜒4 (𝑥, 𝜆) = Ψ (𝑥, 𝜆) ∫∞

𝑥

Φ̃ (𝑡, 𝜆) 𝑓 (𝑡) 𝑑𝑡.(34)

Let us show that each of these vector-functions 𝜒1(𝑥, 𝜆),𝜒2(𝑥, 𝜆), 𝜒3(𝑥, 𝜆), and 𝜒4(𝑥, 𝜆) belongs to 𝐿2(H, (0,∞)).Since the operator solution Φ(𝑥, 𝜆) decays fairly quickly as𝑥 → ∞, then |Φ(𝑥, 𝜆)| ∈ 𝐿2(0,∞). It follows that𝜒1 (𝑥, 𝜆) ≤ 𝑐 (𝜆) ⋅ |Φ (𝑥, 𝜆)| ⋅ ∫𝑥

𝑎

Φ̃ (𝑡, 𝜆) ⋅ 𝑓 (𝑡) 𝑑𝑡≤ 𝑐 (𝜆) ⋅ |Φ (𝑥, 𝜆)| ⋅ (∫𝑥

𝑎

Φ̃ (𝑡, 𝜆) 𝑑𝑡)1/2⋅ (∫𝑥𝑎

𝑓 (𝑡) 𝑑𝑡)1/2< 𝑐 (𝜆) ⋅ |Φ (𝑥, 𝜆)| ⋅ (∫∞

𝑎

Φ̃ (𝑡, 𝜆) 𝑑𝑡)1/2⋅ (∫∞𝑎

𝑓 (𝑡) 𝑑𝑡)1/2 ≤ 𝑐1 (𝜆) ⋅ |Φ (𝑥, 𝜆)| ,(35)

and therefore 𝜒1(𝑥, 𝜆) ∈ 𝐿2(H, (0,∞)). Similarly we get that𝜒3(𝑥, 𝜆) ∈ 𝐿2(H, (0,∞)). First we prove the assertion forthe function 𝜒2(𝑥, 𝜆), when 𝛼 > 1 and the coefficients of (2)satisfy the conditions (4)-(5). In this case, we have𝜒2 (𝑥, 𝜆) ≤ |Φ (𝑥, 𝜆)| ∫𝑥

𝑎

Ψ̃ (𝑡, 𝜆) 𝑓 (𝑡) 𝑑𝑡. (36)By virtue of the asymptotic formulas for the operator solu-tionsΦ(𝑥, 𝜆) and Ψ(𝑥, 𝜆) we obtain that𝜒2 (𝑥, 𝜆) ≤ 𝑐1 (𝜆) 𝛾0 (𝑥, 𝜆) ∫𝑥

𝑎

𝛾∞ (𝑡, 𝜆) 𝑓 (𝑡) 𝑑𝑡. (37)


Let us rewrite this relation in the following form:𝜒2 (𝑥, 𝜆)≤ 𝑐1 (𝜆) 𝛾0 (𝑥, 𝜆) 𝛾∞ (𝑥, 𝜆) ∫𝑥𝑎

𝛾∞ (𝑡, 𝜆)𝛾∞ (𝑥, 𝜆) 𝑓 (𝑡) 𝑑𝑡. (38)By using the definition of the functions 𝛾0(𝑥, 𝜆) and 𝛾∞(𝑥, 𝜆)(see (6)) and by applying the Cauchy- Bunyakovsky inequal-ity we obtain

𝜒2 (𝑥, 𝜆) ≤ 12𝑐1 (𝜆) 1√V (𝑥) (∫𝑥𝑎 √ V (𝑥)V (𝑡)⋅ exp(−2∫𝑥

𝑡

√V (𝑢)𝑑𝑢)𝑑𝑡)1/2⋅ (∫∞0

𝑓 (𝑡)2 𝑑𝑡)1/2 .(39)

Since 𝑡 ≤ 𝑥, we get exp(−2 ∫𝑥𝑡√V(𝑢)𝑑𝑢) ≤ 1, and then the

latter estimate for 𝜒2(𝑥, 𝜆) can be rewritten as follows:𝜒2 (𝑥, 𝜆) ≤ 𝑐2 (𝜆) 14√V (𝑥) (∫𝑥𝑎 1√V (𝑡)𝑑𝑡)1/2≤ 𝑐2 (𝜆) 1

4√V (𝑥) (∫∞𝑎 1√V (𝑡)𝑑𝑡)1/2 .(40)

By formula (4), we get𝜒2 (𝑥, 𝜆) ≤ 𝑐3 (𝜆)4√V (𝑥) , (41)and hence if 𝛼 > 1 and the coefficients of (2) satisfy theconditions (4) and (5), we have 𝜒2(𝑥, 𝜆) ∈ 𝐿2(H, (0,∞)). Inthe case of V(𝑥) = 𝑥2𝛼, 0 < 𝛼 ≤ 1, the assertion can be provedsimilarly.

For the function𝜒4(𝑥, 𝜆)wewill conduct the proof for thecase when V(𝑥) = 𝑥2𝛼, 0 < 𝛼 ≤ 1, and the coefficients of (2)satisfy condition (10). As in (37) we have𝜒4 (𝑥, 𝜆) ≤ 𝑐1 (𝜆) 𝛾∞ (𝑥, 𝜆) ∫∞

𝑥

𝛾0 (𝑡, 𝜆) 𝑓 (𝑡) 𝑑𝑡, (42)which can be rewritten as follows:𝜒4 (𝑥, 𝜆)≤ 𝑐1 (𝜆) 𝛾0 (𝑥, 𝜆) 𝛾∞ (𝑥, 𝜆) ∫∞

𝑥

𝛾0 (𝑡, 𝜆)𝛾0 (𝑥, 𝜆) 𝑓 (𝑡) 𝑑𝑡. (43)Let us use the asymptotic behavior of the functions 𝛾0(𝑥, 𝜆)and 𝛾∞(𝑥, 𝜆), for example, in the case (𝛼 + 1)/2𝛼 = 𝑛 ∈ 𝑁,

that is, 𝛼 = 1/(2𝑛 − 1) (see (12)). Setting 𝑎(𝛼, 𝜆) = ((1 ⋅ 3 ⋅ . . . ⋅(2𝑛 − 3))/𝑛!) ⋅ (𝜆/2)𝑛, we obtain𝜒4 (𝑥, 𝜆) ≤ 𝑐2 (𝜆) 𝑥−𝛼 ∫∞𝑥

𝛾0 (𝑡, 𝜆)𝛾0 (𝑥, 𝜆) 𝑓 (𝑡) 𝑑𝑡 ≤ 𝑐2 (𝜆)⋅ 𝑥−𝛼 (∫𝑥

𝑎

( 𝛾0 (𝑡, 𝜆)𝛾0 (𝑥, 𝜆))2 𝑑𝑡)1/2 (∫∞0 𝑓 (𝑡)2 𝑑𝑡)1/2 ,𝜒4 (𝑥, 𝜆) ≤ 𝑐3 (𝜆) 𝑥−𝛼(∫∞𝑥

( 𝑡𝑥)2𝑎(𝛼,𝜆)−𝛼⋅ exp −2𝑥𝛼+1 ((𝑡/𝑥)𝛼+1 − 1)1 + 𝛼 𝑑𝑡)1/2 .

(44)

Replacing variables 𝑡 = 𝑥𝑢, we get𝜒4 (𝑥, 𝜆) ≤ 𝑐3 (𝜆) 𝑥−𝛼+1/2(∫∞1

𝑢2𝑎(𝛼,𝜆)−𝛼⋅ exp −2𝑥𝛼+1 (𝑢𝛼+1 − 1)1 + 𝛼 𝑑𝑢)1/2 .

(45)

Since the inequality exp(−𝑥𝛼+1(𝑢𝛼+1−1)/(1+𝛼)) ≤ 𝑥−2 holdsfor all 𝛼 ∈ (0, 1] and 𝑢 ∈ [1,∞) with sufficiently large 𝑥, wehave 𝜒4 (𝑥, 𝜆) ≤ 𝑐3 (𝜆) 𝑥−𝛼−1/2(∫∞

1

𝑢2𝑎(𝛼,𝜆)−𝛼⋅ exp −𝑥𝛼+1 (𝑢𝛼+1 − 1)1 + 𝛼 𝑑𝑢)1/2 .

(46)

Hence it follows that |𝜒4(𝑥, 𝜆)| ≤ 𝑐4(𝛼, 𝜆)𝑥−𝛼−1/2, andtherefore 𝜒4(𝑥, 𝜆) ∈ 𝐿2(H, (0,∞)). In case, where 0 < 𝛼 ≤ 1and (𝛼 + 1)/2𝛼 ∉ 𝑁, and where 𝛼 > 1, the proof is similar.

Thus, 𝑅𝜆𝑓 ∈ 𝐿2(H, (0,∞)) for any function 𝑓 ∈𝐿2(H, (0,∞)). This completes the proof.Since the resolvent𝑅𝜆 is ameromorphic function of𝜆, the

poles of which coincide with the eigenvalues of the operator𝐿, the statement of Theorem 3 can be refined.Theorem6. If conditions (4)-(5) where 𝛼 > 1 or condition (10)where 0 < 𝛼 ≤ 1 is satisfied for (2), then the spectrum of theoperator 𝐿 is real and discrete and coincides with the union ofspectra of self-adjoint operators 𝐿𝑘, 𝑘 = 1,𝑚; that is,𝜎 (𝐿) = 𝑟⋃

𝑘=1

𝜎 (𝐿𝑘) . (47)5. Application

Here we consider (2) with matrix coefficients and use thesame notation as in Section 3 (note that could be consideredsecond-order equation with block-triangular coefficients of


a more general form [14]). Suppose that every symmetricoperator 𝐿𝑘 is lower semibounded. Let 𝐿 be an arbitraryextension of the operator 𝐿, defined boundary condition atinfinity, and𝐿𝑘 an arbitrary self-adjoint extension of the oper-ator 𝐿𝑘. If the conditions at infinity determine the Friedrichsextension 𝐿0𝑘 of the semibounded symmetric operator 𝐿𝑘, thecorresponding extension of 𝐿 will be denoted 𝐿0. Besides, letus assume that coefficients of (2) for the problem of semiaxisare such that discrete spectrum of 𝐿 operator coincides withthe union of discrete spectra of 𝐿𝑘 operators, 𝑘 = 1, 𝑟,(sufficient conditions are specified above inTheorem 6).

Denote by nul𝑎𝑇 the algebraic multiplicity of 0 as aneigenvalue of 𝑇.

Denote by 𝑁0𝑎(𝜆) the number of eigenvalues 𝜆0𝑛 < 𝜆

Research ArticleGlobal Existence of Weak Solutions to a Fractional Model inMagnetoelastic Interactions

Idriss Ellahiani,1 EL-Hassan Essoufi,1 andMouhcine Tilioua2

1Laboratoire MISI, FST Settat, Université Hassan I, 26000 Settat, Morocco2Laboratoire M2I, FST Errachidia, Equipe MAMCS, Université Moulay Ismaı̈l, BP 509, Boutalamine, 52000 Errachidia, Morocco

Correspondence should be addressed to Mouhcine Tilioua; [email protected]

Received 6 July 2016; Accepted 8 September 2016

Academic Editor: Cemil Tunç

Copyright © 2016 Idriss Ellahiani et al.This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The paper deals with global existence of weak solutions to a one-dimensional mathematical model describing magnetoelasticinteractions. The model is described by a fractional Landau-Lifshitz-Gilbert equation for the magnetization field coupled to anevolution equation for the displacement. We prove global existence by using Faedo-Galerkin/penalty method. Some commutatorestimates are used to prove the convergence of nonlinear terms.

1. Introduction

The nonlinear parabolic hyperbolic coupled system describ-ing magnetoelastic dynamics in 𝑄 = (0, 𝑇) × Ω (𝑇 > 0 and Ωis a bounded open set of R𝑑, 𝑑 ⩾ 1) is given by (see [1])𝛾−1m𝑡 = −m × (Heff + m𝑡) . (1)

𝜌u𝑡𝑡 − div (S (u) + 12L (m)) = 0. (2)Equation (1), well known in the literature, is the Landau-

Lifshitz-Gilbert (LLG) equation. The unknown m, the mag-netization vector, is a map from Ω to 𝑆2 (the unit sphereof R3) and m𝑡 is its derivative with respect to time. Thesymbol × denotes the vector cross product in R3. Moreoverwe denote by 𝑚𝑖, 𝑖 = 1, 2, 3, the components of m. Theconstant 𝛾 represents the damping parameter.Heff representsthe effective field which is given by

Heff = 𝑎Δm + ℓ (m, u) , (3)where 𝑎 is a positive constant and the components of thevector ℓ(m, u) are given byℓ𝑖 = 𝜆𝑖𝑗𝑘𝑙𝑚𝑗𝜖𝑘𝑙 (u) . (4)

Here 𝜖𝑖𝑗(u) = (1/2)(𝜕𝑖𝑢𝑗 +𝜕𝑗𝑢𝑖) stand for the componentsof the linearized strain tensor 𝜖, 𝜆𝑖𝑗𝑘𝑙 = 𝜆1𝛿𝑖𝑗𝑘𝑙 + 𝜆2𝛿𝑖𝑗𝛿𝑘𝑙 +𝜆3(𝛿𝑖𝑘𝛿𝑗𝑙 + 𝛿𝑖𝑙𝛿𝑗𝑘) with 𝛿𝑖𝑗𝑘𝑙 = 1 if 𝑖 = 𝑗 = 𝑘 = 𝑙 and 𝛿𝑖𝑗𝑘𝑙 = 0otherwise.

Equation (2) describes the evolution of the displacementu, 𝜌 is a positive constant, and the tensors S(u), L(m) aregiven by

S𝑘𝑙 = 𝜎𝑖𝑗𝑘𝑙𝜖𝑖𝑗 (u) ,L𝑘𝑙 = 𝜆𝑖𝑗𝑘𝑙𝑚𝑖𝑚𝑗. (5)𝜎 = (𝜎𝑖𝑗𝑘𝑙) is the elasticity tensor satisfying the following

symmetry property: 𝜎𝑖𝑗𝑘𝑙 = 𝜎𝑘𝑙𝑖𝑗 = 𝜎𝑗𝑖𝑘𝑙. (6)Many studies have been done on the fractional Landau-

Lifshitz equation; we quote here, for example, [2], wherethe existence of weak solutions under periodical boundarycondition has been proven for equation of a reduced modelfor thin-film micromagnetics. In [3], the main purpose is toconsider the well-posedness of the fractional Landau-Lifshitzequation without Gilbert damping. The global existence ofweak solutions is proved by vanishing viscosity method.


http://dx.doi.org/10.1155/2016/9238948


Note that the existence and asymptotic behaviors of globalweak solutions to the one-dimensional periodical fractionalLandau-Lifshitz equation modeling the soft micromagneticmaterials are studied in [4]. For the magnetoelasticity cou-pling, in [1], the authors study the three-dimensional case andestablish the existence of weak solutions taking into accountthree terms of the total free energy. Existence and uniquenessof solutions have been proven in [5] for a simplified modeland in [6] a one-dimensional penalty problem is discussedand the gradient flowof the associated typeGinzburg-Landaufunctional is studied. More precisely the authors prove theexistence and uniqueness of a classical solution which tendsasymptotically for subsequences to a stationary point of theenergy functional. Our aim here is to study the coupledsystem of magnetoelastic interactions with fractional LLGequation.

The rest of the paper is divided as follows. In the nextsection we present the model equation we will be interestedin. Section 3 recalls some useful lemmas. Finally in Section 4we prove a global existence result of weak solutions to theconsidered model.

2. The Model and Main Result

Weassume thatΩ is a subset ofR and the displacement is onlyin one direction. Specifically, we consider a simple variablespace 𝑥 and assume that Ω = (0, 2𝜋). We take the followingsystem:

𝛾−1m𝑡 = −m × (Heff + m𝑡)𝜌u𝑡𝑡 − div (S (u) + 12L (m)) = 0, (7)

with associated initial and boundary conditions

u (⋅, 0) = u0,u𝑡 (⋅, 0) = u1,m (⋅, 0) = m0,m0 = 1

in Ω,(8)

u = 0,m (𝑥, 𝑡) = m (𝑥 + 2𝜋, 𝑡)

on Σ fl 𝜕Ω × (0, 𝑇) . (9)The effective field is given by

Heff = 𝑎Λ2𝛼m + ℓ (m, u) , (10)where Λ = (−Δ)1/2 denotes the square root of the Laplacianwhich can be defined via Fourier transformation [7]. In thispaper we are interested in the case 𝛼 ∈ (1/2, 1). For thevector u, we assume that u = (0, 0, 𝜔) and we keep the threecomponents of the vectorm = (𝑚1, 𝑚2, 𝑚3).

It is a common practice (see [5]) to replace the firstequation of system (7) by the quasilinear parabolic equation(Ginzburg-Landau type equation):

m𝜀𝑡 + 𝛾−1m𝜀𝑡 × m𝜀 = −H𝜀eff − m𝜀2 − 1𝜀 m𝜀. (11)Here 𝜀 is a positive parameter and m𝜀 : Ω → R3. 𝜀-

penalization in (11) replaces the magnitude constraint |m| =1.Throughout, we make use of the following notation. ForΩ, an open bounded domain of R3, we denote by L𝑝(Ω) =(𝐿𝑝(Ω))3 and H1(Ω) = (𝐻1(Ω))3 the classical Hilbert spaces

equipped with the usual norm denoted by ‖ ⋅ ‖L𝑝(Ω) and ‖ ⋅‖H1(Ω) (in general, the product functional spaces (𝑋)3 are allsimplified toX). For all 𝑠 > 0,𝑊𝑠,𝑝 denotes the usual Sobolevspace consisting of all 𝑓 such that𝑓𝑊𝑠,𝑝 fl F−1 (1 + |⋅|2)𝑠/2 (F𝑓) (⋅)𝐿𝑝 < ∞, (12)whereF denotes the Fourier transform andF−1 its inverse.Let �̇�𝑠,𝑝 denote the corresponding homogeneous Sobolevspace. When 𝑝 = 2, 𝑊𝑠,𝑝 corresponds to the usual Sobolevspace 𝐻𝑠.

Nowwe give a definition of the solution in the weak senseof problem (7)-(8)-(9).

Definition 1. Letm0 ∈ H𝛼(Ω), |m0| = 1 a.e., 𝜔0 ∈ 𝐻10 (Ω), and𝜔1 ∈ 𝐿2(Ω). One says that the pair (m, 𝜔) is a weak solutionof problem (7)-(8)-(9) if

(i) for all 𝑇 > 0, m ∈ 𝐿∞(0, 𝑇;H𝛼(Ω)), m𝑡 ∈𝐿2(0, 𝑇; L2(Ω)), |m| = 1 a.e., 𝜔 ∈ 𝐿2(0, 𝑇;𝐻10 (Ω)), and𝜔𝑡 ∈ 𝐿2(0, 𝑇; 𝐿2(Ω));(ii) for all 𝜑 ∈ 𝐿2(0, 𝑇;H𝛼(Ω)) and 𝜓 ∈ 𝐻10 (𝑄) one has𝛾−1 ∫

𝑄m𝑡 ⋅ 𝜑 𝑑𝑥 𝑑𝑡 + ∫

𝑄(m × m𝑡) ⋅ 𝜑 𝑑𝑥 𝑑𝑡

= 𝑎∫𝑄

Λ𝛼m ⋅ Λ𝛼 (m × 𝜑) 𝑑𝑥 𝑑𝑡+ ∫𝑄

(ℓ (m, 𝜔) × m) ⋅ 𝜑 𝑑𝑥 𝑑𝑡− 𝜌∫𝑄

𝜔𝑡𝜓𝑡𝑑𝑥 𝑑𝑡 + ∫𝑄

𝜔𝑥𝜓𝑥𝑑𝑥 𝑑𝑡+ 𝜆∫𝑄

𝑚1𝑚3𝜓𝑥𝑑𝑥 𝑑𝑡 = 0;

(13)

(iii) m(0, 𝑥) = m0(𝑥) and 𝜔(0, 𝑥) = 𝜔0(𝑥) in the tracesense;

(iv) for all 𝑇 > 0 we have𝑎2 ∫Ω Λ𝛼m (𝑇)2 𝑑𝑥 + ∫𝑄 m𝑡2 𝑑𝑥 𝑑𝑡+ 𝜌2 ∫Ω 𝜔𝑡 (𝑇)2 𝑑𝑥 + 14 ∫Ω 𝜔𝑥 (𝑇)2 𝑑𝑥


≤ 𝑎2 ∫Ω Λ𝛼m02 𝑑𝑥 + 𝜌2 ∫Ω 𝜔12 𝑑𝑥+ 14 ∫Ω 𝜔𝑥 (0)2 𝑑𝑥 + 𝐶 (Ω, 𝜆) ,(14)

where 𝐶(Ω, 𝜆) is a positive constant which dependsonly on Ω and 𝜆.

The main result of this paper is the following.

Theorem 2. Let 𝛼 ∈ (1/2, 1),m0 ∈ H𝛼(Ω) such that |m0| = 1a.e., 𝜔0 ∈ 𝐻10 (Ω), and 𝜔1 ∈ 𝐿2(Ω). Then there exists atleast a weak solution for problem (7)-(8)-(9) in the sense ofDefinition 1.

The proof of Theorem 2 will be given in Section 4.

3. Some Technical Lemmas

In this section we present some lemmas which will be usedin the rest of the paper. We start by recalling the followinglemma due to Simon (see [8]).

Lemma 3. Assume 𝐴, 𝐵, and 𝐶 are three Banach spaces andsatisfy 𝐴 ⊂ 𝐵 ⊂ 𝐶 with compact embedding 𝐴 → 𝐵. Let Θ bebounded in 𝐿∞(0, 𝑇; 𝐴) and Θ𝑡 fl {𝑓𝑡; 𝑓 ∈ Θ} be bounded in𝐿𝑝(0, 𝑇; 𝐶), 𝑝 > 1. ThenΘ is relatively compact in𝐶([0, 𝑇]; 𝐵).

There is another lemmawhose proof can be found in [[9],page 12].

Lemma 4. Let Θ be a bounded open set of R𝑑𝑥 × R𝑡, ℎ𝑛 and ℎin 𝐿𝑞(Θ), 1 < 𝑞 < ∞ such that ‖ℎ𝑛‖𝐿𝑞(Θ) ≤ 𝐶, ℎ𝑛 → ℎ a.e. inΘ; then ℎ𝑛 ⇀ ℎ weakly in 𝐿𝑞(Θ).

The following lemma will ensure a compact embeddingfor the space 𝑊𝑠,𝑝.Lemma5. LetΘ be a bounded open set ofR𝑑, which is uniformLipschitz. Let 𝑠 ∈ [0, 1[, 𝑝 > 1, 𝑑 ≥ 1. If 𝑠𝑝 < 𝑑 then theinjection of 𝑊𝑠,𝑝(Θ) in 𝐿𝑘(Θ) is compact, for any 𝑘 < 𝑑𝑝/(𝑑 −𝑠𝑝).

The proof can be found in [[10], Theorem 4.54., p 216].We give now a lemma that will play a very important role inthe convergence of approximate solutions (see [11–13] for aproof).

Lemma 6 (commutator estimates). Suppose that 𝑠 > 0 and𝑝 ∈ (1, +∞). If 𝑓, 𝑔 ∈ S (the Schwartz class) thenΛ𝑠 (𝑓𝑔) − 𝑓Λ𝑠𝑔𝐿𝑝≤ 𝐶 (∇𝑓𝐿𝑝1 𝑔�̇�𝑠−1,𝑝2 + 𝑓�̇�𝑠,𝑝3 𝑔𝐿𝑝4 ) , (15)Λ𝑠 (𝑓𝑔)𝐿𝑝 ≤ 𝐶 (𝑓𝐿𝑝1 𝑔�̇�𝑠,𝑝2 + 𝑓�̇�𝑠,𝑝3 𝑔𝐿𝑝4 ) (16)with 𝑝2, 𝑝3 ∈ (1, +∞) such that 1/𝑝 = 1/𝑝1 + 1/𝑝2 = 1/𝑝3 +1/𝑝4.

Here is another lemma which can be viewed as a result ofthe Hardy-Littlewood-Sobolev theorem of fractional integra-tion; see [7] for a detailed proof.

Lemma 7. Suppose that 𝑝 > 𝑞 > 1 and 1/𝑝+𝑠 = 1/𝑞. Assumethat 𝑓 ∈ 𝐿𝑞; then Λ−𝑠𝑓 ∈ 𝐿𝑝 and there is a constant 𝐶 > 0such that

𝑓�̇�−𝑠,𝑝 fl Λ−𝑠𝑓𝐿𝑝 ≤ 𝐶 𝑓𝐿𝑞 . (17)We finish this section with the following result (the proof

can be found in [2]).

Lemma 8. If 𝑓 and 𝑔 belong to 𝐻2𝛼𝑝𝑒𝑟(Ω) fl {𝑓 ∈𝐿2(Ω)/Λ2𝛼𝑓 ∈ 𝐿2(Ω)}, then∫Ω

Λ2𝛼𝑓 ⋅ 𝑔 𝑑𝑥 = ∫Ω

Λ𝛼𝑓 ⋅ Λ𝛼𝑔𝑑𝑥. (18)4. Proof of Theorem 2

Our goal is to show global existence of weak solutions for thefractional problem (7)-(8)-(9).

4.1. The Penalty Problem. Let 𝜀 > 0 be a fixed parameter. Weconstruct approximated solutions m𝜀 converging, as 𝜀 → 0,to a solution m of the problem. System (7) is reduced to thefollowing problem:

𝛾−1m𝜀𝑡 × m𝜀 + m𝜀𝑡 + 𝑎Λ2𝛼m𝜀 + ℓ (m𝜀, 𝜔𝜀)+ m𝜀2 − 1𝜀 m𝜀 = 0𝜌𝜔𝜀𝑡𝑡 − 𝜔𝜀𝑥𝑥 − 𝜆 (𝑚𝜀1𝑚𝜀3)𝑥 = 0

(19)

in 𝑄 = Ω × (0, 𝑇), where the vector ℓ(m, 𝜔) is given byℓ(m, 𝜔) = (𝜆𝑚3𝜔𝑥, 0, 𝜆𝑚1𝜔𝑥), 𝜆1 = 𝜆2 = 0, 𝜆3 = 𝜆, and𝜎1313 = 1.

System (19) is supplemented with initial and boundaryconditions

𝜔𝜀 (⋅, 0) = 𝜔0,𝜔𝜀𝑡 (⋅, 0) = 𝜔1,m𝜀 (⋅, 0) = m0,m0 = 1

a.e. in Ω,𝜔𝜀 = 0,m𝜀 (𝑥, 𝑡) = m𝜀 (𝑥 + 2𝜋, 𝑡)

on Σ.

(20)

qualitative theory of functional differential and integral … · 2019. 8. 7. · differential and...

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