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Stationary Navier-Stokes Problem in aTwo-Dimensional Exterior Domain

Giovanni P. Galdi

Department of Mechanical Engineering, University of Pittsburgh, 15261 Pittsburgh, USA.

Dedicated to Professor Salvatore Rionero on the occasion ofhis 70th birthday

Contents

Introduction 1Outline of the Article . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 4Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 6

Part I. Analysis of Some Linearized Problems 71.1 The Stokes Approximation . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 81.2 Some Applications of Theorem 1.1 . . . . . . . . . . . . . . . . . . . .. . . . . . . 13

1.2.1 Symmetric Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 131.2.2 Self-Propelled Motions . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 14

1.3 The Oseen Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 171.4 The Oseen Approximation in the Limit of Vanishing Reynolds Number . . . . . . . . 231.5 A Variant to the Oseen Approximation . . . . . . . . . . . . . . . . .. . . . . . . . 25

Part II. The Nonlinear Problem: Unique Solvability for Small Reynolds Number and RelatedResults 272.1 Unique Solvability at Small Reynolds Number . . . . . . . . . .. . . . . . . . . . . 292.2 Limit of Vanishing Reynolds Number . . . . . . . . . . . . . . . . . .. . . . . . . . 322.3 Perturbation Theory at Finite Reynolds Number . . . . . . . .. . . . . . . . . . . . . 33

Part III. The Nonlinear Problem: On the Solvability for Arbitrary Reynolds Number 353.1 Existence: Leray Method . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 363.2 Existence: Fujita Method . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 383.3 Some Existence Results when� � � . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.4 On the Pointwise Asymptotic Behavior of�-Solutions . . . . . . . . . . . . . . . . . 423.5 Asymptotic Structure of�-Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Part IV. The Nonlinear Problem: On the Existence of Symmetric Solutions for ArbitraryLarge Reynolds Number 554.1 A Remark About Symmetric Solutions . . . . . . . . . . . . . . . . . .. . . . . . . 574.2 A Key Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 584.3 Existence of Symmetric Solutions for Arbitrary Large Reynolds Number . . . . . . . 64

References 68

Introduction

As is well known, one of the most representative and fascinating issues in mathematical fluid dynamicsis the steady-state, plane, exterior boundary-value problem associated to the Navier-Stokes equations.

The problem, in its classical formulation, consists in finding a vector function� � �� and ascalar function�, depending only on� � �� and satisfying the following system of equations

��

div � � �

�in �

��

��

(1)

In (1),� is the exterior of a two-dimensional compact, connected set�, � � �� is a fixed constantvector, �� is a prescribed vector function at the boundary� of �, and� is a givennon-negative real number.

From the physical point of view, problem (1) is related to thestationary motion of a viscous,incompressible fluid around a long, straight cylinder�, assuming that the fluid is at rest at largedistances from�. The vector�� is the (possibly zero) translational velocity of�, supposed to beorthogonal to its axisa. In a region of flow sufficiently far from the two ends of� and including�,one can expect that the velocity field of the fluid is independent of the coordinate parallel toa and,moreover, that there is no motion in the direction ofa. Under these hypotheses, the correspondingmathematical problem becomes two-dimensional and is described by (1), where�, scaled by a charac-teristic velocity , is the dimensionless Eulerian velocity of the particle of the fluid,� is the associatedpressure field, and� (the complement of�) is the relevant region of flow. Moreover,�� representsthe (dimensionless) velocity of the fluid at� �� ). Finally, � is a dimensionless parameter, theReynolds number, that can be written as �� , where� is a length scale (typically, the diameter of�), and �� is the coefficient of kinematical viscosity of the fluid.

The case�� and � �� in (1) deserves special attention. In fact, it describes thesignificantphysical situation of when the cylinder has impermeable, immobile walls and translates into thefluid with constant velocity��. Actually, it was just this problem that, in 1851, received the firstmathematical treatment ever, in the pioneering work of Sir George Gabriel Stokes on the motion of apendulum in a viscous liquid [42]. In particular, in the wakeof his successful study of the motion ofa spherein a viscous fluid, Stokes looked for solutions to (1) with�� in the limiting case whenthe viscosity of the fluid is much larger that the quantity �, where is taken as the magnitude ofthe velocity of the cylinder. This amounts to take� � � in (1)� and get a correspondinglinearizedproblem that is nowadays calledStokes approximation. However, to his surprise, Stokes found thatthis linearized problem, even in the simplest case when� is a circle, hasno solution, and he concludedwith the following statement [42], p. 63,

“It appears that the supposition of steady motion is inadmissible.”

Such an observation constitutes what we currently refer to as Stokes Paradox.

This is definitely a very intriguing starting point for the mathematician who is interested in theresolution of the boundary-value problem (1). In fact, it appears that, if the problem admits a solution,the nonlinear term�� has to play a key role in its determination.

1

This latter fact was recognized, and made quantitative, by C. W. Oseen, more than half a centurylater, in 1910, in his fundamental paper [37]; see also [38],15 and Chapter III. Specifically, Oseenproposed another approximation that takes into account, somehow, the nonlinear term by replacingit with �� . This procedure leads to a corresponding linearization of problem (1), called theOseen approximation. The well-posedness of the boundary-value problem corresponding to the Oseenapproximation was proved, in its full generality, almost one decade later by H. Faxen [13]

The first mathematical study of the full nonlinear problem (1) in its complete generality is dueto J. Leray in 1933 [33]. Actually, in [33] Leray investigated also the three-dimensional counterpartof (1). By using topological degree theory (Leray-Schauder theorem) in conjunction with ana prioriestimate for all possible solutions (in a given functional class) to (1) (see (2) below), Leray was ableto show that forany � there exists a smooth pair�� that satisfies (1)��, provided�� and� areregular enough and the total flux of�� through� is zero, namely,

�

��

where� is the unit outer normal to�. The important question that Leray could not answer waswhether or not the velocity� satisfies the prescription at infinity (1)�. Notice that this lack ofinformation only appeared in the two-dimensional problem,while in the three-dimensional case hewas able to show the validity of (1)� uniformly pointwise, if � � � and in a generalized sense, if� �� . The discrepancy between the two- and three-dimensional cases is due to the following reason.The solution constructed by Leray verifies the condition

�

�� (2)

Now, if � � ��, Leray proved the following inequality [33], p.47,�

�

��

�

��

so that the method of construction he used, and (2) imply�

�

�� (3)

which, in turn, furnishes (1)� in a generalized sense.1 If � � ��, we only have theweakerinequality[33] p. 54-55 �

�

��

�

�� (4)

Now, it is not hard to bring examples of plane solenoidal fields satisfying (4) (or (2)), vanishing at� and growing as a power of�� , for sufficiently large��. As a consequence, in order to showthe (possible) validity of (1)�, in the two-dimensional case, the equations of motion (1)�� must playa fundamental role. It is worth emphasizing that Leray’s method applies to both situations� �� and� � � as well, furnishing the same partial answer in either case.

Analogous conclusions were reached almost thirty years later, in 1961, by H. Fujita [17], whofound essentially the same results as Leray’s, by a different method of constructing solutions, theso-calledGalerkin method.

1If � � �, Leray further showed that (3) eventually implies (1)� uniformly pointwise.

2

The drawback of both Leray’s and Fujita’s solutions can be summarized by saying that theonlyinformation available on the asymptotic behavior of the solution, is that the velocity field� has afinite Dirichlet integral which, by what we noticed, does not even ensure the boundedness of�. Thesesolutions are called�-solutions.

The above interlocutory results left open the worrisome possibility that a Stokes paradox couldalso hold for thefully nonlinearproblem (1). If this chance turns out to be indeed true, it might castserious doubts on the reliability of the Navier-Stokes fluid model, in that it would not be able to catchthe physics of such an elementary phenomenon.

The possibility of a nonlinear Stokes paradox, was ruled outby R. Finn and D. Smith in a deeppaper appeared in 1967 [16], where they show thatif � �� , and� and�� are sufficiently regular, then(1) has a solution, at least for “small”�. Moreover, these solutions arePhysically Reasonable, in thesense that they meet all the basic physical requirements, such as energy equation, and they show thepresence of awakein the direction�, opposite to the direction of the velocity of the cylinder. Finally,the solutions are unique in a ball (of “small” radius) of a suitable Banach space. The method usedby Finn and Smith is completely different than Leray’s and Fujita’s, and is based on very accurateand detailed estimates of the Green’s tensor and of the fundamental solution associated to the Oseenapproximation.

Another approach to the problem addressed by Finn and Smith was given, more recently, by Galdi[18]. The approach, based on a suitable��-theory of the Oseen approximation, is very flexible and infact allowed the resolution of other, more complicated problems, such as the analogous boundary-valueproblem (1) for the case of acompressible(density-varying) viscous fluid [25]

Though significant, these results leave open several fundamental questions. The most importantis, of course, that of whether problem (1) has a solution forall � � �, as it happens, in fact, inthe analogous three-dimensional case. Another important question regards the solvability of (1) when� � �. In such a circumstance, no result whatsoever is available,other than the incomplete oneobtained by Leray and Fujita that we described before.

As we already noticed, the solvability of problem (1) for arbitrary � � � would be established, themoment we could show that the velocity field of solutions constructed by Leray or Fujita (�-solutions)doestend to the prescribed value�. The question of the asymptotic behavior of the velocity field of a�-solution was taken up in a series of remarkable papers by Gilbarg and Weinberger first [29], [30],and then by Amick [2], [3]. Specifically, in the case when�� , the above authors showed that (i)Every solution to (1)�� that satisfies (2) is uniformly pointwise bounded; and (ii)For every solutionto (1)�� that satisfies (2), there existssomevector �� such that the average over the angle of�� tends to� as�� . If, in particular,� is symmetricaround the direction of� (�� ), one can showthat � tends to�� uniformly pointwise [2], [22]. Notice that, in general, no information is availableabout��. Actually, �� can be in principle be zero, even though� �� .

Therefore, another spontaneous question arises:Does the vector�� coincide with the prescribedvector �? Even though it is very probable that the answer is in the positive, at the present time, noanswer is known. Of course, had this question have an affermative answer for all�, problem (1)would admit a solution for all� as well.

Very recently, when� is symmetric around the direction of� �� , Galdi has proved the followingresult concerning the solvability of (1) for arbitrary large �, in the class of symmetric solutions [22].Assuming� directed along the��-axis, such solutions�� , have�� and� even in�� and�� odd in ��. Let us denote by (1)� problem (1) with� � �� , and by� the class of symmetricsolutions�� to (1)� with � having a finite Dirichlet integral. (Notice that, in the class� the velocityfield � tends uniformly pointwise to zero at infinity.) It is clear that � contains the trivial solution

3

� � �, � �const. Nowif the trivial solution is theonly solution in �, then the set of� for which(1) is solvable in the class of symmetric solutions corresponding to a prescribed� (�� ) and�� contains anunboundedset,��, of the positive real line. This result leaves open two interesting linesof investigation. The first, and more important, is the proof of the validity of its assumption, and, thesecond, is the study of the property of the set��. Unfortunately, to date, no result is available ineither direction.

Objective of the present article is to furnish a complete, consistent and, as far as possible, self-contained, presentation of the state of the art of the uniquesolvability of problem (1). We shall alsogive some new results and point out several open questions, and, whenever is the case, we suggestpossible ways of resolution.

Outline of the Article

The article is divided into four parts. Part I is dedicated tothe mathematical analysis of somelinearized versions of problem (1), including the Stokes and the Oseen approximations, while Parts IIto IV concern the full nonlinear problem.

Specifically, in Part I, Section 1.1, we study the well-posedness of the boundary-value problem,(1)� , say, to which (1) reduces by taking formally� � �. Even though, as we noticed before, (1)�

does not have a solution for an arbitrary choice of� and��, nevertheless it is also known that thereare physically interesting situation where (1)� furnishes results in a reasonable agreement with theexperience. Our main objective is to investigate under which conditions on�� and �, problem (1)�admits one and only one regular solution. This objective is accomplished by showing that (1)� isuniquely solvableif and only if � and�� satisfy a “compatibility condition” (see (1.12)). In the casewhen� is a circle of radius one, this condition takes the followingsimple form

� ��

��

�

��

Applications of this result are furnished in Section 1.2. A noteworthy application is that given to theself-propelled motion of micro-organisms likeCiliata. In a commonly accepted model of Ciliata, thelayer model, the motion of minuscule hair-like organelles (cilia) placed on the surface of the mainbody of the animal produces a distribution of velocity, which serves as a propeller [8], [10], [9], [24].Due to the small velocity and to the microscopic size of the micro-organism, the typical Reynoldsnumber involved is� � ��, and, therefore, the Stokes approximation is applicable. By using theresults of Section 1.1, we give necessary and sufficient conditions for self-propulsion, that contain, as aparticular case, those furnished by other authors by different methods [8], [9]. Section 1.3 is dedicatedto the study of the basic mathematical properties of the Oseen approximation. As mentioned before,the associated boundary-value problem, (1)�, say, is obtained from (1) by replacing the nonlinear term�� with �� . In particular, we present results of existence, uniqueness and correspondingestimates of solutions to (1)�. We also furnish results on the asymptotic behavior that show, amongother things, the existence of a “wake” in the direction of�. In Section 1.4, we study the behavior ofsolutions to the Oseen problem as� � and show that they tend to solutions to the Stokes problemcorresponding to the same data, if and only if the data satisfy the compatibility condition determinedin Section 1.1. In the last section of Part I, Section 1.5, we study the functional-analytic propertiesof a problem obtained by perturbing the Oseen problem by a suitable linear operator, furnishing, inparticular, sufficient conditions for the existence and uniqueness of solutions to the perturbed problem.

4

In Part II we begin the study of the unique solvability of the full nonlinear problem (1) at “small”Reynolds number�. Specifically, in Section 2.1, by using the results proved in Section 1.3, we showthat, there is�� such that, for any� � � � ��, (1) has at least one solution in a suitable Banachspace,� say. Moreover, the solution islocally unique, in the sense that it is the only one within a ballof � of appropriately “small” radius. Whether or not these solutions are unique “in the large” remainsan open question. This circumstance has an undesired consequence. In fact, even though solutionsdetermined here and the solutions constructed by Finn and Smith [16] belong to the same functionalclass, we can not conclude that (for small�) they coincide. In Section 2.2, we analyze the behaviorof solutions previously found in the limit of� �, and show that they tend to the solutions of thecorresponding Stokes problem if and only if the data satisfythe compatibility conditions establishedin Section 1.1. An interesting issue obtained as a byproductof this result is that at small, nonzeroReynolds number the force exerted by the fluid on� is independent of the shape of�, a fact firstdiscovered by Finn and Smith [16], and, more recently, reconsidered, with a completely differentapproach, in [44], [45], [43]. In Section 2.3 we are interested in the construction of a solution to (1)�� , corresponding to�, in a neighborhood of another solution�� , corresponding to��. Usingthe results of Section 1.5, we give sufficient conditions for the existence of�� and show that, underthese conditions,�� is analytic in�� .

The remaining Parts III and IV are dedicated to the existenceof solutions to (1) forarbitraryReynolds numbers. In the first two Sections 3.1 and 3.2 we briefly review the methods of constructionof Leray [33] and that of Fujita [17] which provide existenceof solutions to (1)�� with � havinga finite Dirichlet integral (D-solutions), for any value of�. The drawback with these solutions istwo-fold. On the one hand, the lack of information about the behavior of� at large distance and, asa consequence, the impossibility of checking the validity of condition (1)�. On the other hand, in thecase when�� , it is not excluded that such solutions are identically zero. Before investigatingthese two questions, in Section 3.3 we prove some existence results for problem (1) when� � �.In particular, we show that if� has two orthogonal directions of symmetry and if the data satisfysuitable parity conditions, then, for any� � �, (1) has at least one�-solution. Notice that, unlike thecase� �� , in such a case we show that� satisfies also (1)�. The remaining two sections of Part IIIare devoted to the study of the asymptotic behavior of�-solutions. Specifically, in Section 3.4, wedescribe the results of Gilbarg and Weinberger [29], [30], and of Amick [2], [3], and show that, if�� , for any�-solution there exists a vector�� such that

��

� ��

��

Moreover, we also prove that, in fact,� tends to�� uniformly pointwise, a property known, so far inthe literature, only for symmetric solutions [3]. Our proofis very simple and based on the vorticityequation in conjunction with a suitable pointwise estimateof the Sobolev type. However, since we donot know whether or not�� , the big question that remains open is whether or not (1) is solvable forarbitrary large�. We shall return on this problem in Part IV. The asymptotic structure of�-solutionswhose velocity field� tends, uniformly pointwise, to anonzerovector,��, is the object of Section 3.5.Following the approach of Galdi and Sohr [28], finalized by the very recent results of Sazonov [39],we shall show that every such a solution is “physically reasonable” in the sense of Finn and Smith,and so, in particular, the velocity field and the pressure field behave, roughly speaking, as velocityand pressure fields of the corresponding Oseen problem. In this respect, we emphasize that, if�� ,the rate of decay of the velocity field� of a �-solution is, in general,not predictable. Actually, as

5

seen by means of counter-examples (see (2.5)), in general� is not representable, at large��, in termsof negative powers of��.

The last Part IV is based on the work of Galdi [22], and is aimedat furnishing sufficient conditionsfor the existence of symmetric, “physically reasonable” solutions to (1) in the case when� is symmetricaround one direction. As explained previously in the Introduction, the basic assumption, (�) say, isthat problem (1) with�� hasonly the trivial solution in the class of symmetric�-solutions.This result is based on a key lemma derived in Section 4.2, Theorem 4.1, which furnishes apositivelower bound, in terms of��, for the Dirichlet integral of the velocity field of a solution constructed byLeray method. This bound furnishes, in particular, that symmetric Leray solutions, corresponding to�� and� �� , are not trivial, a fact first discovered by Amick [2]. If this latter conclusion holdsalso for solutions that are not necessarily symmetric or forgeneric symmetric�-solutions is an openquestion. Using Theorem 4.1 and other preparatory results derived in Section 4.1, in Section 4.3 wethen show that if the basic assumption (�) is satisfied, the set of� for which (1) with �� and� �� has a symmetric solution contains an unbounded set��. Proving or disproving (�) remainsan open question. If�� is proven to be true, the next step is to investigate the properties of��.A possible way is to use an analytic continuation argument, along the lines of the results proved inSection 2.3.

Notation

In this paper we shall use the notation of [20]. However, for the reader’s convenience, we collect herethe most frequently used symbols.

�� is the set of positive integers.��, � � �, is the Euclidean�-dimensional space and�� is the associated

canonical basis. Throughout this paper we shall essentially deal with the case� � �. The coordinates[respectively, components] of a point� � �� [respectively, vector�] will be denoted by�� and�� [respectively,�� and ��]. We shall also use polar coordinates�� where �� and�� . The corresponding components of the vector� will be denoted by�� and��, respectively.

Given a second-order tensor� and a vector�, of components�� and ��, respectively,in the basis��, by � � � [respectively,� � �] we mean the vector whose components are givenby �� [respectively,��]. Moreover, if � � �� is another second-order tensor, by thesymbol� � � we mean the second-order tensor whose components are given by � � � . We also set� � � � �� , where the superscript “�” denotes transpose, and��

�� .

For � � �, we set�� , and�� . If� � �, we shall simply write�� and��, respectively.

If � is a domain of��, we denote by�� its diameter and by�� its complement.If � is an exterior domain (the complement of the closure of a bounded domain) weshall take

the origin of coordinates in the interior of��. Moreover, for� � ��, we set�� ,�� and�� , � � � � ��.

With the greek letter� we shall indicate the relevant “region of flow” of the fluid,that is, anexterior domain of��. If � �� , we shall assume, without loss, that�� (the “cross-section ofthe cylinder”) is contained in��.

Unless explicitly stated, all domains involved in this paper are contained in��.��, � � �, ��, ��, ! � �, � � " � �, denote the usual space of functions of

class�� on �, and Lebesgue and Sobolev spaces, respectively. Norms in�� and �� are

6

denoted by� � ��, � � ��. Unless confusion arises, we shall usually drop the subscript “�” inthese norms. The trace space on� for functions from �� will be denoted by ��and its norm by� � ��.

By ��, � � �� " � �, we indicate the homogeneous Sobolev space of order�!� "�on �, [40] [20], that is, the class of functions# that are (Lebesgue) locally integrable in� and with��# � ��, $ � �$�� $��, �$� � �, where

��

��

� �$� � $� � $��

For # � ��, we set2

�#��

��

��

�

��#��

��

�

where, again, the subscript “�” will be generally omitted.By��

� �� we indicate the completion of�� in the norm�#��, and denote by��

� ��,��"� � � � ��", the dual space of��

� ��. The��

� -�� duality pairing will be indicated by� �� .

Finally,�� denotes the subset of�� constituted by solenoidal vector functions, and�� is the completion of�� in the��-norm.

Part I. Analysis of Some Linearized Problems

In this Part I we shall describe the most significant mathematical properties related to several lineariza-tions of problem (1). Specifically, in Section 1.1, we shallinvestigate the oldest linearization, namely,the so calledStokes approximation. This approximation, especially for plane flow, is very well-knownbecause it leads to the famousStokes paradox(see Theorem 1.2) according to which, roughly, acylinder can not move by steady, translational motion in a fluid of “very large” viscosity, since thecorresponding boundary-value problem has no solution (in any “reasonable” class). We shall show,however, that such an approximation is valid in several other physically interesting problems and, inparticular, we will furnish acharacterizationon the data in order that the boundary-value problemhas one and only one solution; see Theorem 1.1. Applicationsof this result includeself-propelledmotions of a body in a viscous liquid, and are presented in Section 1.2. In Section 1.3, we shall surveythe relevant properties of another and, in a sense, more appropriate linearization, that is, theOseenapproximation. We shall collect the significant results of the corresponding boundary-value problemwhich will be the cornerstone of the nonlinear theory developed in Parts II–IV. We shall also study inwhich sense the solutions of the Oseen boundary-value problem converge to those of the correspondingStokes boundary-value problem; see Section 1.4. Finally, in Section 1.5, we shall analyze a variantto the Oseen linearization, obtained by adding to the Oseen operator a suitable linear operator that,in Part II, will play an important role in the nonlinear treatment of existence of solutions at finiteReynolds numbers.

2Typically, we shall omit in the integrals the infinitesimalvolume or surface of integration.

7

1.1 The Stokes Approximation

If we formally take� � � in (1) we obtain the so-calledStokes approximation:

�� div ��

�in �

��

��

(1.1)

In general, this boundary-value problem does not have a solution. In fact, as we mentioned in theIntroduction, it certainly does not have a solution in the physically relevant circumstance when�� and� �� , leading to the so-calledStokes paradox. However, there are also other well-known specialcases where problem (1.1) has one and only one solution. For example, an elementary solution ina closed form can be constructed if� is the exterior of a circle,� � � and �� % � �, for someconstant vector% orthogonal to the plane of flow; see,e.g. [7], p. 18.

The investigation of the solvability of problem (1.1) has been the object of several researches; see,e.g. [11], [4], [12], [1], [5]. One of the main goals of this section is to establish anecessary andsufficient conditionon the data�� and � for the (unique) solvability of (1.1) in a suitable functionalclass; see eq. (1.12). To this end, we recall some preliminary facts.

The first result concerns the asymptotic behavior of solutions to (1.1) possessing a very mild degreeof regularity locally in� and at infinity. To this end, we recall that theStokes fundamental solutionis a pair constituted by a tensor field& � �&�� and a vector field" � �"�� defined by

&��

�

��

�

��

��

"��

��

��

�

(1.2)

By direct inspection, one checks that& and" satisfy the following equations

�&��

�"��

�&��

for � �� .

Denote by' � �'� � '��#� �� the stress tensorassociated to the velocity field# and associatedscalar field�, namely,

'� � �� #�

where

��#� ��

�#��

�#��

�

is thestretching tensor.

We have the following result, whose proof can be found in [20], Theorems V.1.1 and V.3.2.

8

Lemma 1.1 Let ��

��, � � " � �, be a pair of vector and scalar fields,

respectively, solving(1.1)�� in the sense of distributions. Then,�� . Moreover, if at leastone of the following conditions is satisfied

(i) �� (��, all ��

(ii)�

��

�� (�� ,

for some� � � and some) � ��, there exist vector and scalar constants��, �� such that, as�� ,

�� !&�� *��

�� !"�� +��(1.3)

where

! � ��

��'�� (1.4)

and, for all �,� � �,��*�� -��+�� -��

(1.5)

Our next result (see Lemma 1.3) concerns the structure of thenullspace of the problem (1.1)��,in the homogeneous Sobolev spaces��. For this reason, and also because these spaces willplay an important role also in our approach to the mathematical theory of the nonlinear Navier-Stokesproblem, we wish to collect here their most significant properties. Specifically, we have the followingresult, whose proof is given in [20], Theorem II.6.1.

Lemma 1.2 Let � be an exterior domain and let# � ��. Then

(i) If � " � �, there exists a unique#� � �� such that, for all sufficiently large� we have

� ��

��#�� #�� .��#��

where.� � ��" � �� "�� if " � � and .� � � if " � �;

(ii) If " � � we have

��

�

��

� ��

��#��

This estimate is sharp, in the sense that there are functions# such that

� ��

�

��

� ��

��#��

and# �� , for all " � �� ; 3

3Take for example,� � �� , � � �.

9

(iii) If � � " ��, we have

��

�

��

� ��

��#��

Assume, moreover, that� is locally Lipschitzian, and let# � ��, � � " � �. Then,# � �� , � � " � �, if and only if #�� and the constant vector#� in part (i) is zero.Finally, # � �� , " � �, if and only if#�� .

Remark 1.1 Even though�� is the completion of�� in the norm� � ��, functions from�� may grow at large spatial distances, if" � �. The fields in (1.5)� below furnish an explicitexample (see also Footnote 3).

In order to prove the mentioned characterization, we need tointroduce some suitable “auxiliaryfields”. These are particular solutions to the Stokes system (1.1)��, and can be introduced in severaldifferent ways. Following [27], we introduce them as a basisof the null space of solutions to theStokes system (1.1)�� in the space�� . Specifically, we have [27]

Lemma 1.3 Let � be an exterior domain of class��. Let �� be the linear subspace of�� , � � " ��, constituted by the distributional solutions# to the following problem:

�# � ��

div# � �

�in �

�#� �� (1.6)

Then, if� � " �, �� , while if � � " ��, �� . In this latter case, there exists abasis�/� � �� in �� satisfying the following properties.

(i) For all � � " �� and 0 � �� , we have

�/� � ��

��

(ii) There exist/� � � �� and �� , 0 � �� , such that, as�� , the following representa-tion holds

/� �� /� ��&�� -��

�� "�� -��

(1.7)

(iii) For 0 � �� , we have �

��' �/� � �� (1.8)

10

Remark 1.2 In some special cases, the fields�/� � �� are known in a closed form. For example,if � is the exterior of the unit circle, we have

/��

/��

��

��

��

/��

��

��

/��

(1.9)

Finally, we recall the following lemma on the solvability ofa non-homogeneous version of problem(1.6) [27].

Lemma 1.4 Let � be an exterior domain of class��. Then, for every1 � �� , � � " � �,satisfying�1� /� � �, 0 � �� , the problem

�# � �� 1

div# � �

�in � (1.10)

has one and only one solution�#� �� , in the sense of distributions.

We are now in a position to furnish the desired characterization.

Theorem 1.1 Let � be an exterior domain of class��. Let �� , � � " � �, andlet ��

� �� satisfy(1.1)�� in the sense of distribution,(1.1)� in the trace sense,

and (1.1)� in the following averaged sense

��

� ��

�� (1.11)

Then,�� and � must satisfy the following condition4

� �

�

�� ' �/� � �� 0 � �� (1.12)

Conversely, let�� , � � " ��, and � � �� satisfy(1.12). Then, there is a uniquesolution�� to (1.1)�� that assumes the boundary data(1.1)� in the sense of trace andthat satisfies(1.1)� uniformly pointwise. Moreover, as�� ,

�� 2�� (1.13)

where��2�� -�� all �,� � ��

4Notice that, since��

��

�� for all � �� (Lemma 1.3(i)), it follows that� ��

�� is well-defined as an element of� ��, so that (1.12) makes sense.

11

Proof. Multiplying (1.1)� by /� and integrating by parts over��, we obtain�

��/� � ' ��

�

��

' �� /� �

Equation (1.11) along with Lemma 1.1 implies that�� have the asymptotic behavior given in (1.3)with �� . Moreover, the vector! in (1.3) must vanish. Employing this information, recalling(1.5) and passing to the limit3 � in the previous relation, we thus find

�

�' �� /� � �� (1.14)

We next multiply (1.6) with# � /� , � � �� , by �� , and integrate by parts over�� to obtain�

�� ' �/� � ��

�

�� ' �/� � ��

�

��

' �/� � �� (1.15)

Observing that' �/� � �� ' �� /� �

we may let3 � in (1.15) and use (1.14), (1.3) and (1.7) to deduce (1.12). Conversely, assumethat (1.12) holds. We look for a solution�� to (1.1), with�� of the form

�� 4 � � * � �� (1.16)

In this relation

*��

��

�

�

��

and � ��, is a solenoidal extension in� of compact support of the field

�� *�� (1.17)

The existence of such is well-known; see [20], Exercise II.3.4. Moreover

� �� 5 �� (1.18)

Furthermore,4 solves the following problem

�4 � �� 1

div4 � �

�in �

4��

��

4��

(1.19)

where (in the sense of distributions)1 � ��

12

From (1.18) it follows that1 � �� , for some� � " � �. Also, recalling that �� is the field(1.17), we find

�1� /� � � �� /� � � �� *�� /� � ��

�� ' �/� � ��

In view of the assumption (1.12) on the data, we obtain�1� /� � � �, 0 � �� . So, from Lemma1.4, we deduce that problem (1.19) admits a unique solution�4� �� . By Lemma1.2(i) we thus get, in particular,

��

� ��

��4��

Combining this information with (1.16) we obtain that�� is a solution to (1.1), where the condition(1.1)� is attained in the following way

��

� ��

��

Then, by Lemma 1.1(ii) we deduce that�� has the asymptotic behavior given in (1.13). The theoremis completely proved. �

An important, immediate consequence of Theorem 1.1 is the following.

Theorem 1.2 (Stokes Paradox)Let � be as inTheorem 1.1. Then, there is no solution to(1.1) with�� and � �� .

Proof. If �� , the condition (1.12) is equivalent to� � �, giving a contradiction. �

1.2 Some Applications of Theorem 1.1

In this section we shall consider some application to physically relevant situation of the characterizationproved in Theorem 1.1.

1.2.1 Symmetric Domains

In the case when� is the exterior of the unit circle, the auxiliary fields�/� � �� are given in (1.9).By a simple calculation, one shows that

'��/� � ��

�

�� 6� 0 � �� (1.20)

Therefore, the necessary and sufficient condition (1.12) becomes

� ��

��

�

�� (1.21)

This relation can be satisfied under several physically relevant assumptions on� and��. For example,consider the case when� is a circular disk� uniformly rotating around an axis orthogonal to itsplane,5 with angular velocity%, in a fluid that is rest at infinity. We then have� � � and�� %��,

5As explained in the introduction, we recall that the disk is the cross-section of a “long” cylinder in the plane of theflow.

13

and it is at once verified that (1.21) is satisfied. Another example is furnished by the case when�is in a fluid subject to a simple shear in the��-direction. In this circumstance we have� � � and�� , where� is a given constant. Again, it is readily seen that condition(1.21) is satisfied.Similar conclusions can be drawn in the more general case when � (� ��) possesses two orthogonalstraight lines of geometric symmetry. Specifically, assuming that these lines coincide with the�� and�� axes, respectively, we suppose

��

��

In such a case, the fields�/� � �� possess the following symmetry properties

/�� /

�� /�� /

�� /�� /��

��

/�� /�� /�� /�� /

�� /��

��

Therefore, taking into account that the unit normal� satisfies

�� by direct inspection we see that the right-hand side of (1.12) is zero, provided�� is chosen as above.

1.2.2 Self-Propelled Motions

A very interesting application of Theorem 1.1 when� �� is related toself-propulsionof a rigid body� [23], [24], Part II. In such a situation, the fluid is at rest at infinity and� moves by constant motion.The motion of� is not due to external forces but, rather, to a suitable distribution of velocity�� at� (� �), that furnishes the needed “thrust”. This happens, for example, in modeling the motionof certain micro-organisms, such asCiliata; see [8] and [24], Part II. One of the basic questions forthis type of problems is the following one: in which ways can we choose the field�� in order that� moves with a (constant) rigid motion velocity& � �� % � �, where� �� (so that� doesmove)?6 Within the Stokes approximation, this amounts to find#�� and& satisfying the followingproblem [23]

�#� � ��div#� � �

�in �

#�� % � �

��

#��

�

��' �#��

�

�� ' �#��

(1.22)

6Of course,� is orthogonal to the plane of motion.

14

The last two equations in (1.22) are consequences of Newton’s laws of conservation of linear andangular momentum, respectively, for the body�. They express the fact that totalexternalforce andtorque acting on� are identically zero, that is, that� is self-propelled. In view of Theorem 1.1 weknow that, given�� such that

��

�� ' �/� � �� 0 � �� (1.23)

there is one and only one solution��7 to the following problem

�� 7

div� � �

�in �

��

��

��

(1.24)

The following properties are easily established.�

��' ��7 � � � � �

8 � ��

�� ' ��7 � � � ��

(1.25)

Actually (1.25)� is a direct consequence of (1.24)� and of Lemma 1.1. To establish (1.25)�, wemultiply both sides of (1.24)� by � and integrate by parts over��. Using the asymptotic properties(1.3)� for � we then let3 � and obtain the following relation

�

�� 8�

which shows (1.25)�. We are now in position to analyze the solvability of (1.22).We begin to observethat, as a consequence of Theorem 1.1, problem (1.22)�� has a solution (for sufficiently smooth�and��) if and only if

� � %�� 9� 0 � �� (1.26)

where�� is defined in (1.23), and

9 �

�

�� ' �/� � �� 0 � �� (1.27)

Again by Theorem 1.1, and by Lemma 1.1, condition (1.26) implies the vanishing of the total force;see (1.22)�. We next multiply (1.24)� by #�, and integrate by parts over��. Using the asymptoticproperties (1.3)� for #� we then let3 � and obtain the following relation

�

�� % � �� ' ��7 � � � �

�

�� #�� (1.28)

Likewise, multiplying (1.22)� by�, integrating by parts over��, taking into account the asymptoticproperties of� and#�, and then letting3 �, we find

�

�� #��

�

�� ' �#�� (1.29)

15

From (1.28), (1.29) and (1.25)�, we deduce that the vanishing of the torque, condition (1.22)�, isequivalent to the following one

8% � :� (1.30)

where

: �

�

�� ' ��7 � � �� (1.31)

Let; � ' �/

� � �� 0 � ��

;� � ' ��7 � � ��and define the following 3-dimensional subspaces of��

<�� # � �� # � ,;� for some, � ��

�� (1.32)

Notice that< dependsonly on the geometric properties of�, like shape and symmetry. Denote by�!the projection of�� onto � ��. The next theorem shows, among other things, that a sufficientcondition to self-propel� is that the boundary velocity (the “thrust”) has a non-zero projection on��. Moreover, the velocity of� is uniquely determined by this projection. Precisely, we have thefollowing result.

Theorem 1.3 Let � be as inTheorem 1.1Then, for any�� , � � " �� satisfying�!�� , there exists a solution�#�� & � �� % � �� to problem(1.22)with & � �� .Moreover, the translational velocity� and the angular velocity% are given by

� � 9 �:��8 � 0 � ��

% � :�8(1.33)

where��, 8, 9 and: are given in(1.23), (1.25), (1.27) and (1.31), respectively. So, in particular,� �� if and only if9 �� :��8.

Moreover, let�� be another boundary velocity with�!�� !�� anddenote by��#�� & � �� % � �� the corresponding solution. Then�& � & .

Proof. Let �� be given as stated. We then choose� and% as in (1.33) and solve problem (1.22)��.This is certainly possible by Theorem 1.1, because the data satisfy the compatibility condition expressedby (1.33)�. By what we have seen before, this implies that also (1.22)� is satisfied. Moreover, thechoice of% in (1.33)� ensures, again as shown earlier, that also condition (1.22)� is satisfied. Finally,the last part of the theorem follows from (1.33), from the fact that if �!�� , then9 � : � �, andfrom the linearity of the problem. The result is completely proved. �

It might be of some interest to evaluate the “self-propelled” conditions (1.33) in the case when�is a disk (of radius 1, say), since this situation is of a certain relevance in the study of the motionof several micro-organisms [8]. We then take�� 1��, and would like to find the conditions on1�� for which �!�� , so that�� is an appropriate “thrust”. From (1.27), (1.20) we find

9� � � �

��

� ��

�1�� 9� �

�

��

� ��

�1��

16

Moreover, we have� � �� [7], p.18, so that�� and, by a simple calculation, we also find

8 � � � � : � ��

�1��

Therefore, with the specified choice of�� will move with the following translational and angularvelocities

��

��

� ��

�1��

�

��

� ��

�1��

% ��

��

� ��

�1��

1.3 The Oseen Approximation

Even though the Stokes linearization may provide some insights and useful information in certainphysically interesting problems (as illustrated in the previous section), it is not able to give any kindof information on one of the most important problems in fluiddynamics, namely, the motion of abody of simple symmetric shape, such as a cylinder, steadilytranslating through a viscous fluid witha velocity orthogonal to its major axis of symmetry. Actually, there are several other situations, alsofor three-dimensional flows, where the Stokes linearization furnishes results that are at odds with theobservation or, even, with the assumptions that are at the basis of the linearization itself. For instance,in the case of the motion of aspherein a Navier-Stokes fluid, the Stokes approximation does notshowany “wake” behind the sphere. Moreover, the ratio of the inertial term (� � ��) to the viscous term(��) goes to infinity as soon as we move far away from the boundary, contradicting the basic logicof the linearization, that assumes this ratio to be “small” everywhere in the region of flow.

Motivated by these considerations, C.W. Oseen proposed another type of linearization of (1.1) when� �� ; see [38]. Choosing, without loss,� � ��, the linearization is obtained by setting� � # � ��in (1.1) and by disregarding the nonlinear term# � �#. Such an assumption leads to the followingproblem

�#� � #��

� �� 9

div# � =

��

in �

#�� #��

#��

(1.34)

where, for future purposes, we have allowed for a non-zero “body force”�9 acting on the fluid, anda prescribed value= (not necessarily zero) for the divergence of#.

Associated with problem (1.34), Oseen introduced the correspondingfundamental solution>� "defined as follows

>��

��

� ��

"��

��

��

��

� ��

(1.35)

17

where

��

�"�?� �� ? �

�

�

� ��

�� ?�@��? ��?�

"��

��

�� @� ��

� (1.36)

and@� is the modified Bessel function of the second kind of order zero. By a direct calculation weshow �

��

��

�>��

��

��>��

for � �� ;

and that

"��

��

�� 6 � ��

Moreover, from the property of@��A� for small and large values of�A�, we can show the followingrelations (see [20], Section VII.3 for details)

>��

�

��

�

��

��

� (��

� &��

��

�

�� (�� as��

(1.37)

with � � �� , & Stokes fundamental tensor (1.2)�, and

>�� B

��

��

�� B� � � # ��B

��

�

>�� >�� B

��

�� B

��

��

#

��

�

>�� B ��

��

��

�;� �� # ��B

$��

��

as��

(1.38)

whereB is the angle made by a ray that starts from� and is directed toward�, with the direction ofthe positive��-axis, and

; � �� B��

Finally, the “remainder”��)� satisfies

��)�

� -�)�� as) �, � � �.

Using (1.38), one can show the following asymptotic estimates. If � is interior to the parabola

�� B� � � (1.39)

18

we have�>��

5

�� as �� (1.40)

while if�� B� � �� for some* � �� (1.41)

we have�>��

5

�� as �� (1.42)

Moreover,�>��

5

�� 0 � �� as �� (1.43)

Remark 1.3 The fact that the asymptotic decay of>�� for large �� is faster outside than insidethe parabolic region defined in (1.39), is representative of the existence of a “parabolic wake” in thepositive��-direction.

So far as the behavior of the first derivatives of> is concerned, differentiating, we derive the followinguniform bounds as��

��>��

��

�� 5

�� >��

��

�� 5

��

��>��

�

�� 5

��

��>��

�

�� 5

�� 0 � ��

(1.44)

It should be also observed that, as it can be easily proved,>�� and>��, 0 � �� have afaster decay rate outside the wake than that given in (1.44).

The next result is the Oseen counterpart of Lemma 1.1 given for the Stokes linearization. We referto [20], Theorem VII.6.2, for a proof.

Lemma 1.5 Let� be an exterior domain of class�� and let�#� ��

��, � � " ��,

be a pair of vector and scalar fields, respectively, satisfying (1.34)�� with 9 � = � �, in the senseof distributions. Then,#� � � ��. Moreover, if# satisfies at least one of the conditions (i), (ii)of Lemma 1.1, then there exist vector and scalar constants#�, �� such that as�� ,

#�� #�� >�� *��

�� "�� +��

(1.45)

where

� � ��

��'��#� ��#��

��

�

��'��#� �� # � ��#��

(1.46)

and, for all �,� � �,��*�� -�� +�� -��

(1.47)

19

Remark 1.4 It can be shown that, in fact, the “remnant”* of the preceding lemma has the sameasymptotic behavior as the corresponding first derivatives of the tensor>; see (1.44). Therefore, inparticular,* decays faster outside than inside the wake.

Remark 1.5 The previous lemma shows, in particular, that every solution to the Oseen problemsatisfying the stated assumptions, behaves, asymptotically, as the Oseen fundamental tensor; see alsothe previous remark. In view of Remark 1.3, this implies, that these solutions show a wake structurein the positive��-direction, as expected on a physical ground.

We shall next present a number of theorems that will play a fundamental role in further developments.In fact, on the one hand, they insure existence and uniqueness for the Oseen problem (1.34), and, onthe other hand, they furnish key functional-analytic properties that will allow us to show, among otherthings, existence and uniqueness for thenonlinear problem (1.1), at least for “small” data. To thisend, for# � �#�� #��, we introduce the following notation:

�#�� #�� #��#�� #

��

��

� �#�� " � #�� (1.48)

As usual, if we need to specify the domain� on which we take the norm� � ��, we will write � � ��.

Remark 1.6 If � is an exterior domain, every# with �#�� #�� in �, satisfies the condition

��

#�� uniformly

In fact, since# � �� , from [20] Theorem II.5.1, we deduce# � �� .Since�"�� "� � � and, also,# � �� , the stated property follows from [20], RemarkII.7.2.

The following result holds.

Theorem 1.4 Let � be an exterior domain of class��. Given

9 � �� = � �� #� � �� " � #��

there exists one and only one corresponding solution#� � to the Oseen problem(1.34) such that

# � ��

#� � ��

with ;� �� , ;� �

�� . Moreover,#� � verify the following estimate

�#�� #�� 5 �9��#�� =��

�� (1.49)

where the positive constant5 depends on"�� and�.

Proof. A full proof of the theorem is given in [20], Theorem VII.7.1 and Exercise VII.7.1. Here, forreader’s convenience, we shall sketch a proof when� � �� and= � �. In such a case the proof is

20

obtained by using Fourier transform in conjuction with elementary multipliers theory. For simplicity,we shall also set� � �. We look for a solution to (1.34) corresponding to9 � �� of the form

#��

��

�

��&��

�

��

�

��7 �� (1.50)

Replacing (1.50) into (1.34) furnishes the following algebraic system for& and7 :

�� 0��&�� 0��7 �� 9�� ! � ��

0�&�� (1.51)

where�9 ��

��

�

��9 ��

is the Fourier transform of9 . Solving (1.51) for& and7 delivers

&�� 9�� 7 �� 0�� 9��

� (1.52)

where

��

�� 0��

We recall the following theorem of P.I. Lizorkin [35]. Giventhe integral transformation

'1 ��

��

�

��"�� 1��

with "�� continuous together with the derivatives

"

��"

��"

��

for �� , then if, for some$ � �� and� � �,

�� "

��

�� (1.53)

'1 is bounded from�� into ��, � � " ��, �� " � $, and we have

�'1�� 1��

where� � 5�"� �� , 5�"� �� . It is at once checked that the functions��, ;� � � �� , satisfythe assumption (1.53) with$ � �. Therefore, from (1.50) and (1.52), we find

�� 5 �9�� (1.54)

Likewise, the functions�� ;� ��!� � � ��

21

satisfy (1.53) with$ � �, and so, again from (1.50) and (1.52), we have

�#�� #

��

��

5�9�� (1.55)

Moreover, by a simple calculation, we verify that, for anyC�!� � � �� verifies (1.53) with$ � ��#, �� with $ � ��#, �� with $ � �� and �� with $ � �. Thus, from (1.50) and(1.52), by Lizorkin’s theorem we find

�#�� #�� #�� #�� 5�9�� " � #�� (1.56)

The summability properties stated in the theorem along withthe estimate (1.49) are then a consequenceof (1.54)-(1.56). Notice that, as observed in Remark 1.5, the solution# satisfies the condition at infinity(1.34)�, uniformly pointwise. �

Remark 1.7 If 9 and = are of compact support, the solutions determined in Theorem1.4 have theasymptotic behavior described in Lemma 1.5.

The theorem just proved has some simple but important consequences. To show them, for� � " � #��,we define the following Sobolev-like spacesD�� andD�� by

D�� # � div# � � � �#�� D�� # � D�� #��

�� (1.57)

We observe the validity of the embeddings

D�� E �� ! � �� for all 3 � �� (1.58)

So, if � is locally Lipschitzian, a function# from D�� leaves a trace#�� on � and the map

# � D�� #��

is continuous. In particular, if� is locally Lipschitzian, the following space is well-defined

D�� # � D�� #��

�� (1.59)

TheD-spaces are suitable spaces for velocity. We next introducethe appropriate space for the pressureF �� defined by

F �� (1.60)

It is readily seen that theD�F -spaces become Banach spaces when endowed with their “natural”norms

�#�� #��#�� #�� #��

�� It also shown, by standard methods, that they are reflexive and separable. We next introduce theOseenoperator��#� �� formally defined as

��#� �� #� � #��

�� (1.61)

where� � �. 7 As an immediate corollary to Theorem 1.4 we obtain the following result.7More generally, we could take� �� , and all the properties stated below for the Oseen operator would continue to hold.

22

Theorem 1.5 Let � be as inTheorem 1.4. The Oseen operator(1.61) is a linear isomorphism from�D�� D�� F �� into ��.

Proof. Clearly, the map

�� #� �� D�� D�� F �� #� ��

is well-defined. It remains to show that the problem

�#� � #��

� �� 9

div# � �

��

in �

#��

��

#��

has a unique solution�#� �� D�� D�� F �� for every9 � ��. But this isexactly the statement of Theorem 1.4 with= � #� � �. �

The validity of an inequality of the type (1.49) with anexplicit dependence of the constant5 on� � �� , for some�� , may be of fundamental importance for treating the nonlinear problem(1) when� �� . Because of the Stokes paradox (see Theorem 1.4), one also expects that the constant5 becomes unbounded as� approaches zero. Now, if we restrict" in the interval� � " � %�&, onecan prove the validity of an inequality of the type (1.49), with a constant5 which can be renderedindependent of�, for � ranging in�� , but where the norm of# involves� in an known way. Tothis end, for# � �#�� #��, set

�#�� #�� #�� #�� #�� (1.62)

If we need to specify the domain� on which we take the norm� � ��, we will write � � ��. We thenhave the following result, for whose rather technically complicated proof we refer to [20], TheoremVII.5.1, and [21], Lemmas X.4.1 and X.4.2.

Theorem 1.6 Let � be an exterior domain of class��. Then, given

9 � �� #� � �� " � %�&�

there is a unique solution#� � to (1.34)with = � �, such that�#� �� D��F ��. Moreover,there is�� such that for all� � � ��, this solution satisfies the following estimate

�#�� #�� 5�� #�� 9��

��

1.4 The Oseen Approximation in the Limit of Vanishing Reynolds Number

A question that may spontaneously arise is the relation between the solutions to the Oseen problem(1.34) and those to the Stokes problem (1.1), in the limit of vanishing�. In this section we present

23

an interesting result relating the solutions to the Oseen problem (1.34) with9 � � and#� � �� to the corresponding solutions to the Stokes problem (1.1),namely,

�#� � ��div#� � �

�in �

#��

#��

(1.63)

Assume� and�� are prescribed as in Theorem 1.4. We begin to observe that, bythe usual methodbased on a suitable solenoidal extension of the boundary data along with the Riesz representationtheorem [20], Remark V.2.2, one can easily show the existence of a solution�#�� satisfying (1.63)��. Of course, nothing in principle can be said aboutthe attainability of the condition at infinity (1.63)�, unless#�� satisfies the compatibility condition(1.12). However, despite this lack of information, this solution is unique in the class of solutions withvelocity field in��. This is an immediate consequence of Lemma 1.2 and Lemma 1.3.Actually,denoting by�4�G� the difference between two such solutions, we have that4�G satisfy (1.63)�� with�� . Thus, since4 � �� and4�� , from Lemma 1.2 it follows that4 � �� .Therefore, by Lemma 1.3 we get4 � �. The solution#� admits the following representation [20],Theorem V.3.2,

#�� #��

�

�� '��&� � "�� &�� '��#� ��*�� (1.64)

for someconstant vector#� � ��, and where&� � & � �� , 6 � �� .The following theorem gives the answer to the question raised above.

Theorem 1.7 Let � and �� be as inTheorem 1.4, and let #� � be the corresponding solution to(1.34) given in that theorem with9 � = � �. Moreover, let�#�� , #� � ��, be the uniquelydetermined solution to(1.63)��. Then, as� �, �#� �� tends to�#�� , uniformly on compactsubsets of�, together with first and second derivatives. Furthermore,

��

�!�#�� #�� (1.65)

where#� is given in(1.64) and

!�#� � ��

��' �#� ��

Finally, the limit process preserves the prescription at infinity, that is, #� � �, if and only if ��satisfies condition(1.12), namely,

�

�� ' �/� � �� (1.66)

24

Proof. We will sketch here only the proof of the second part, referring to [20], Section VII.8, for acomplete proof of the theorem. The solution to the Oseen problem (1.34) with9 � = � �, given inTheorem 1.4 admits the following representation [20], Theorem VII.6.2,

#��

�

�� '��>� � �� >�� '��#� ��

�� >�� *��(1.67)

where>� � > � �� , 6 � �� . From (1.37) and (1.66) we formally find

#��

�!��#� ��

�

��

�

�� '��&�� "��

�&�� '��#� ��*� � (�� as��

We now pass to the limit� � in this latter relation. Invoking the first part of the theorem and using(1.64), we thus obtain (1.65). Finally, the validity of the characterization (1.66) is a consequence ofTheorem 1.1. �

1.5 A Variant to the Oseen Approximation

The objective of this section is to study some functional property of the following variant to the Oseenproblem

�#� � #��

� ��#� � �#� # � �#�� 9

div# � �

��

in �

#��

��

#��

(1.68)

where#� is a prescribed function fromD�� (see (1.57)), and9 � ��.We begin with a very simple but useful result.

Lemma 1.6 Let � be an arbitrary domain in�� and let�, 4 be two divergence-free vectors in�for which the norm(1.62) with � � " %�&, is finite. Then the following inequality holds for all� � �

�� 4�� 4��

Proof. Taking into account that� and4 are both divergence-free, we obtain

� � �4 ��4��

� ��4��

��

��4��

� ��4��

��

and so, by the Holder inequality and (1.62),

�� 4�� 4�� 4��

�

��4�� 4��

��

(1.69)

25

From elementary��-interpolation inequalities we find (with"� � "��" � ��) that

�4�� 4��

�� 4��

�� 4��

��

��

��

and the lemma becomes a consequence of this relation and (1.69). �

For a given4 � D��, consider the operator

@!�� # � D�� @!��#� � ��#� � �#� # � �#�� " %�&� (1.70)

In view of Lemma 1.6, the operator@!�� is well-defined. Therefore, using Theorem 1.5 and recallingRemark 1.5, problem (1.68) can be re-written in the following functional form

��#� ��@!��#� � 9� �#� �� D�� D�� F �� (1.71)

where��#� �� is the Oseen operator (1.61), andD�� , F �� are defined in (1.59) and (1.60),respectively.

The operator@!�� enjoys the following important property.

Lemma 1.7 @!�� is compact.

Proof. Let �#�� D��, with �#�� SinceD�� is reflexive, we may select asubsequence, which we continue to denote by�#�� that converges weakly to some# � D��.Set&� � #� � #. In view of the embedding (1.58) we get, in particular,

�&�� for all 3 � �.

By Rellich theorem and by (1.58) we then deduce

��

�&�� for all 3 � �. (1.72)

From Lemma 1.6, and from the fact that�&�� , for all 3 � �, we also find

�@!��&�� 5�� 4��&�� 4��&��

�

5�� 4��&�� 4��

��

This inequality together with (1.72) implies

� �'(��

�@!��&�� 5��4��

and the lemma follows from the fact that��

�4�� . �

From Theorem 1.5, Lemma 1.6 and well-known results on compact perturbations of isomorphisms,e.g. [32], Theorem IV.5.26, we then obtain the following theorem

26

Theorem 1.8 Let� be an exterior domain of class��. LetH��! be the linear subspace of�D��D�� F ��, � � " %�&, constituted by the solutions of the problem

��#� ��@!��#� � ��

Then,�H��! � �. If, in particular, �H��! � �, then, for any9 � ��, � � " � %�&,problem (1.68) has a unique solution�#� �� D�� D�� F ��, and this solutionsatisfies the following estimate

�#�� #�� 5�9��

Part II. The Nonlinear Problem: Unique Solvability for Small ReynoldsNumber and Related Results

The subject of Part II is to develop a perturbation theory forthe boundary-value problem (1), when� �� . Under this latter assumption, we can take, without loss,� � ��. Thus, setting� � # � ��,#� � �� , we at once obtain that (1) goes into the following equivalent boundary-value problem

�#� � #��

� �# � �#��

div# � �

��

in �

#�� #��

#��

(2.1)

One of the main goals of this section is to show that the nonlinear Navier-Stokes problem (2.1)possesses a solution if the Reynolds number� is sufficiently small. However, the validity of thisresult is not so evidenta priori, as we will know explain. A way of showing existence is to prove theexistence of a fixed point (in a subset< of an appropriate Banach space) of the mapping

� � 4 � < � ��4� � # � <

where# solves the problem

�#� � #��

� �4 � �4 ��

div# � �

��

in �

#�� #��

#��

(2.2)

In the limit of � � there will be a competition between the linear term

�#

��(2.3)

27

and the nonlinear one�4 � �4� (2.4)

If in the range of vanishing�, the contribution of the former is negligible with respect to that ofthe latter, it would be very unlikely to prove existence, because the linear part in (2.2) would thenapproach the Stokes system for which, as we know from Section1.1, solvability is established onlyunder suitable compatibility conditions on the data. Fortunately, what happens is that (2.3) “prevails”on (2.4) and the machinery produces nonlinear existence. Infact, we shall show a stronger result,namely, that, provided� is sufficiently small, a solution to (1) with� �� can be constructed in theform of a series, that is converging in a suitable Banach spaceD. Moreover, each coefficient of theseries can be evaluated as the solution to a suitable Oseen problem. Concerning uniqueness, we shallshow that this solution is the only one that lies in a suitableball of an appropriate Banach space.

This type of solutions are called by R. Finn and D. Smith, who first discovered their existence[16], Physically Reasonable(PR). The reason for such a name is because they satisfy all requirementsexpected on a physical ground such as uniqueness, validity of the energy equality (see Remark 2.2)and moreover, as we shall see in Section 3.5, they show the presence of a wake behind the body�(that is, in the� � ��-direction).

Another goal of this section is the construction of a perturbation theory at arbitrary Reynoldsnumbers. Specifically, we shall show that if�� is such that�H� � � (see Theorem 1.8), then wecan construct a solution�#� �� to (2.1) that is (real) analytic in� in a neighborhood of��. We arethus able to obtain the solution to (2.1) by analytic continuation with respect to�. This process willstop if, for some��, either�#�� as� �� , or �H� �� In this latter case, one can givesufficient conditions for the existence of a bifurcating solution [26].

Let us now consider the solvability of problem (1) when� � �. In this regard, we wish toemphasize that, to date, the existence of solutions to the nonlinear problem (1) when� � � forarbitrarily prescribed (sufficiently smooth) data�� is open, no matter what the magnitude of theReynolds number�. The major difficulty here is the choice of the function space where solutionsshould exist. Such a difficulty is mainly due to the fact thatit is not clear what is the asymptoticspatial behavior that solutionsa priori might have. Actually, this behavior can not be, in general, ofthe type��, (� � ��) for somefixed positive�, as the following example shows

��

�� %

��

�

��

�

� � ��

��

��

� ��

�

��

(2.5)

In the solution (2.5), due to G. Hamel [31],% is an arbitrary constant and we assume� �� and� � �� . Therefore, taking� sufficiently close to 1, (2.5) provides an example of a solution thatdecays more slowly than any negative power of�. The fields in (2.5) show another undesired featureof problem (1) when� � �. In fact, taking� � �� we see at once that the velocity field� assumesthe boundary data�� , �� , for all values of the constant%. Consequently, solutions (2.5)also furnish anexample of non-uniquenessto problem (1) with� � �. In Part IV, we will considerthe problem of uniqueness in relation to the solvability of problem (1) with� �� for arbitrary large�. Coming back to the question of existence, it is very probable that a solution to (1) with� � � doesnot exist unless the data�� satisfy certain compatibility conditions. This guess is strongly suggested

28

by the results presented in Section 1.1 for the Stokes approximation; see, in particular, Theorem 1.1.In fact, in Section 3.3 of the next Part III, we shall show thatproblem (1) with� � � has at leastone solution, provided� and �� satisfy certain symmetry conditions. Such a solution exists for allReynolds number.

2.1 Unique Solvability at Small Reynolds Number

In this section we shall prove that problem (2.1) has one and only one solution in a ball of a suitableBanach space, provided� is positive and “sufficiently small”. This solution can be expressed in theform of a series.

To reach this goal, we propose a very simple result on the convergence of certain power series.

Lemma 2.1 Let ��, �� , be a sequence of positive real numbers satisfying the condition:

��

��

�� (2.6)

where� is a positive constant independent of�. Then the power series

=��

��

��

is convergent provided �� , and we have

=�� (2.7)

Proof. Consider the sequence of positive numbers�� defined as follows

��

�

��

�� (2.8)

From (2.6) and (2.8) we find that

� � � for all � � �� (2.9)

If we multiply (2.8) by�, sum from� to � and use Cauchy’s product formula for series, from (2.8)we formally obtain

�� (2.10)

where

��

��

��

��

The solution to (2.10) that reduces to�� at � � � is given by

��

��

��

��

�

29

which has an analytic branch provided �� . The lemma then follows from this fact, from(2.9), and from the inequality

��

�

We are in a position to show the main result of this section.

Theorem 2.1 Let � be an exterior domain of class��, and let

#� � �� " � %�&�

There exists a positive constant�� such that, if for some� � �� ,

� �� #�� %5� � (2.11)

with 5 given inTheorem 1.6, then problem(2.1) has at least one solution�#� �� D��F ��This solution can be written in the form of a series

#��

��

� #��

��

� �� (2.12)

where�#�� is the solution to the Oseen problem(1.34)with 9 � = � �, and, for� � �,

�#�� #��

�

�

��

#� � �#��

div#��

��

in �

#��

��

#��

(2.13)

The two series in(2.14) are converging inD�� and F ��, respectively. Furthermore, thesolution satisfies the estimate

�#�� #�� 5 �� #�� (2.14)

Finally, if �#�� D��F �� is another solution corresponding to the same data and suchthat

�� #�� $5 � (2.15)

then# � #� and � � ��.

Proof. Let us temporarily setI � � on the right-hand side of (2.1) and considerI as a positive smallparameter. We then look for a solution to (2.1) of the form

#��

��

��

I #��

��

��

I �� (2.16)

30

where the coefficients#�� and#�� , � � �, satisfy the conditions stated in the theorem. Weshall show that (2.16) are converging inD�� F �� also forI � �. Applying the results ofTheorem 1.6 to problem (2.13), and taking into account Lemma1.6 we find that, for some�� and all� � �� ,

&�� 5 ��

��

&�&�� (2.17)

where& � �#�� #��

Moreover, again by Theorem 1.6, we have

&� 5 �� #�� (2.18)

We thus obtain that the sequence�&�� verifies the assumptions of Lemma 2.1 with� � 5 �� . From (2.18) it then follows that the condition &��I � � is satisfied if

�%5��I� �� #��

Thus, the series (2.16) will converge forI � � if condition (2.11) holds. Moreover, in view of (2.7)and (2.18), we also recover the estimate (2.14). The existence proof is thus completed. It remainsto show uniqueness. Denote by�#�� D�� F �� another solution corresponding to thesame data, and set4 � #� #�, � � �� . We then have that4� � satisfy the following problem

�4 � � 4��

� �� 9

div4 � �

��

in �

4��

��

4��

where9 � ��4 � �#� � # � �4� �

From Theorem 1.6 and Lemma 1.6 it follows that

�4�� 5�� 4��#�� #��

�� (2.19)

By a direct computation that uses (2.11) and (2.14), we find

�� #�� $5 �

and so, from this inequality and from (2.15) we obtain

5�� #�� #��

��

so that (2.19) implies4 � �, thus completing the proof of the theorem. �

31

Remark 2.1 An important question that the previous theorem leaves openis that of whether or not thesolution there constructed is unique in the class of solutions �#�� merelybelonging toD��F ��, but not necessarily satisfying the smallness condition (2.15).

Remark 2.2 It is verified at once that the solutions of Theorem 2.1 satisfy the energy equality:

�

�

��

�

�� ' ��

This is immediately established by multiplying (2.2)� by # � �� , integrating by parts and using theasymptotic properties following from the fact that�#� �� D�� F ��.

2.2 Limit of Vanishing Reynolds Number

In this section we collect some results related to the behavior of solutions determined in Theorem 2.1in the limit � �. In fact, these results are quite similar to those obtained for the Oseen linearizationin Section 1.3. Specifically, we have the following theorem, for whose proof we refer to [21], TheoremX.7.1.

Theorem 2.2 Let the assumptions ofTheorem 2.1hold and let#, � be the solution constructed inthat theorem. Moreover, let#�, ��, #� � ��, be the uniquely determined solution to the Stokesproblem(1.63)��; seeSection 1.3. Then, as� �, �#� �� tends to�#�� , uniformly on compactsets, together with their first and second derivatives. Furthermore, there is a#� � �� such that

��

#�� #� (2.20)

and we have��!�#�� #� (2.21)

where

!�#� �

�

��' �#� ��

Finally, the limit process preserves the prescription at infinity, i.e., #� � � if and only if the datasatisfy condition(1.66)

An interesting consequence of this theorem is the derivation of an asymptotic formula (in the limit ofvanishing Reynolds number) for the force9 � �!�#� exerted by the fluid on a body moving in itwith constant velocity��. Specifically, taking#� � ��, from the results of the first part of Section1.3, we have that the limit solution#� is identically equal to��, and so from (2.21) it follows, inthe limit � �, that

9 � ( �� )� �� (2.22)

where(�� denotes a vector quantity tending to zero with�. This formula shows that in the limit ofvanishingly small Reynolds number, the total force exertedfrom the fluid on the body is determinedentirely by the velocity at infinity�� and that it is directed along the line of this vector (only “drag”and no “lift”). Surprisingly enough, it does not depend on the shape of the body. This type of problemhas been addressed also in [44], [45], [43].

32

2.3 Perturbation Theory at Finite Reynolds Number

Let us suppose that we know the existence of a solution�#�� to (2.1) in the classD��F ��corresponding to a certain�� (� �). Our objective in this section is to investigate the existence ofa solution to (2.1) corresponding to a Reynolds number in a neighborhood of��. Therefore, writing� � �� I, �I� I�, I� � �, we look for a solution to (2.1) of the form# � #� � 4, � � �� Gwhere4 andG satisfy the following boundary-value problem

�4� ��4

�� #� � �4 � 4 � �#��

� I

�#��

�4

�� #� � �4 � 4 � �#� � #� � �#�

�

�� I�4 � �4 ��G

div4 � �

��

in �

4��

��

4��

(2.23)

We have the following theorem.

Theorem 2.3 Let � be an exterior domain of class��, and let�#�� be a solution to(2.1) with#� � D��, � � " � %�&. Then, if �H��! � � (seeTheorem 1.8), there existsI� � �

such that problem(2.23) has at least one solution�4�G� � �D�� D�� F ��, for all�I� I I�. Moreover, this solution can be expressed as power series inI, converging in thespace�D�� D�� F ��.

Proof. We look for a solution in the form

4��

��

��

I�4�� G��

��

��

I�G�� (2.24)

Formally replacing these expressions in (2.23) and equating to zero the terms of equal power inI, wefind, for all � � �,

�4�� 4

�� #� � �4� � 4� � �#�� 9� ��G�

div4� � �

��

in �

4��

��

4��

(2.25)

where

9� �4��

� ��

��

��

4 � �4��

��

4 � �4�� (2.26)

33

From the assumptions, from Theorem 1.8 and from (2.25)-(2.26), we obtain, for� � �,

�4�� 4�� G�� (2.27)

where the positive constant� depends only on#� and��. We want to show that there is a (sufficientlylarge) constant� such that

�4�� 4�� G�� for all � � �� (2.28)

We will use an induction argument that we have learned from [6]. Clearly, in view of (2.27), condition(2.28) is true for� � �. Thus, assuming that,

� �4�� 4�� G�� 0�� (2.29)

for all � 0 ��, � � �, we have to prove that (2.29) holds also for0 � �. From (2.29), Theorem1.8 and Lemma 1.6, we obtain

� ��

��

��

��

��0�� 0��

�

��

��

��0�� 0�� !�

(2.30)

where�� depends only on#� and��. Observing that (� � �)

��

��

��

0�� 0��

��

0�� 0��

��

�� 5��

with a constant5� independent of�, from (2.30) we find

� ��

�5� �

��

�5� �

��

��5�

�� (2.31)

Recalling that� , �� depend only on#� and ��, we can choose� so large that the quantity inbrackets in (2.31) is less than 1. In this way we obtain (2.29)also for 0 � �, and the inductionproof is completed. From the estimate (2.29) for the coefficients of the series (2.24), we deduce thatthese series in�D�� D�� andF ��, respectively, are both bounded from above by thenumerical series

�

�

��

��

��I��

Thus, they both converge if��I� �. The proof of the theorem is completed. �

Remark 2.3 According to the theorem just proved, we are able to obtain the solution�#� �� to theboundary-value problem (2.1) by analytic continuation with respect to the Reynolds number�. Theprocess will break if, for a certain��, either �#�� as� �� , or the problem (1.68) has anonzero solution. In this latter case, bifurcation may occur. Sufficient conditions for the occurrenceof bifurcation are given in [26], Section 7.

34

Part III. The Nonlinear Problem: On the Solvability for Arbitrary Rey-nolds Number

Since the appearance of the seminal paper of J. Leray in 1933 [33], it is known that the systemof equations (1)�� possesses at least one solution, provided the boundary value �� satisfies thezero-outflow condition

��

�� (3.1)

This solution presents two important properties: (i) it is smooth in� (�� ), and (ii) itexists forall values of the Reynolds number�; see Section 3.1. The main, basic question that Lerayleft open (see [33], p. 54-55) was the proof of whether or not this solution satisfies the condition atinfinity (1)�. In fact, concerning the asymptotic behavior of the solution, he was only able to provethat the velocity field is in��, that is,

�

�� (3.2)

where� is a constant depending only on�, ��, and�. As we know from Lemma 1.2, this propertyalone is not enough to control the behavior at infinity of the velocity field.

Apparently, Leray’s problem did not catch the attention of mathematicians for more than fortyyears, till when, in a series of remarkable papers, Gilbarg and Weinberger first [29], [30], and thenAmick [2], [3] investigated in great detail if and when a Leray’s solution (and, more generally, asolution with velocity fields in��) satisfies the prescription at infinity (1)�. Specifically, in thecase when�� , the above authors showed the validity of the following assertions; see Section 3.2.

(i) Every solution to (1)�� that satisfies (3.2) (and so, in particular, every Leray’s solution) isuniformly pointwise bounded;

(ii) For every solution to (1)�� that satisfies (3.2), there exists�� such that

��

� ��

��

More information about�� can be obtained, if� (� ��) is symmetricaround the direction of� (��,say). This means that�� implies �� . In such a case, one can show theexistence ofsymmetricsolutions� � �� , that is,

�� (3.3)

provided�� verifies the same parity properties as� does. For symmetric solutions one then provesthat �� ,�, for some, � �� [22], and that

��

�� uniformly� (3.4)

see [2]. Actually, in Section 3.2, we shall furnish a new (andsimpler) proof of (3.4) that extends toflows that are not necessarily symmetric.

Even though the above results represent a significant contribution to the original achievementof Leray, the fundamental, outstanding question remains still open: Does� satisfy the condition at

35

infinity (1)�? Or, in other words,can we show that�� ? Notice that the possibility that�� isnot excluded.

A positive answer to this question would imply that (1) has a solution for arbitrary large Reynoldsnumbers.

In this connection, we observe that, recently, Galdi has given another contribution to the problem,in the case of symmetric solutions [22]. Specifically, he has shown that if the problem (1) with�� , hasonly the zero solution in the class of solutions satisfying (3.2)and (3.3), then problem(1) has at least one symmetric solution in a range of Reynoldsnumbers belonging to anunboundedset� of the positive real axis. This result will be described in detail in Part IV.

Interestingly enough, we are able to give some results of existence forall Reynolds numbers, if� � �. These results require that� is symmetric with respect two orthogonal directions, and that theboundary data�� satisfy suitable parity conditions. We shall give a simple,self-contained proof of thisfact in Section 3.3. The proof would also follow from much more elaborated arguments presented inSection 3.4. These results are interesting in that, as we emphasized in the introduction to Part III, thecase� � � is a completely unexplored territory, even for “small” (non-vanishing) Reynolds number.

3.1 Existence: Leray Method

As mentioned in the previous section, Leray was the first to show existence of regular solutions tothe Navier-Stokes system (1)�� for arbitrary values of the Reynolds number� [33]. In this sectionwe shall briefly describe Leray’s method of constructing solutions, and recall some of their propertiesthat we shall use later on.

In the rest of this article, with the exception of Section 3.3, we will be concerned with the physicallyrelevant case when� �� . With this in mind, we find it convenient to rewrite (1) in a different form,that is obtained by introducing the new velocity field# � � �. If we do this, and continue to denoteby � the rescaled velocity field, and moreover we take, without loss,� � ��, we then obtain that (1)can be re-written as follows

��

� � � � �

�in �

�� (3.5)

along with the condition at infinity��

�� (3.6)

A solution to (3.5)-(3.6) was sought by Leray [33] by means ofthe following procedure of “invadingdomains”. Let�3�� be an unbounded, increasing sequence of positive numbers, with 3� � �.For each�, consider the sequence of problems:

��

�in ��

��

�� at �� 3��

(3.7)

36

Leray’s proof is based on the observation that every solution to (3.7) formally obeys the followingapriori estimate �

��

�� (3.8)

where� depends only on�, �� and �, but not on�. Such a uniform bound, along with Odqvistestimates for the Green’s tensor of the Stokes problem in bounded domains [36], and what we callnowadays the Leray-Schauder theorem [34] allowed Leray to prove existence of a regular solution to(3.7) for all � � ��, provided� and�� have a suitable degree of smoothness. Letting� � andusing the uniform bound (3.8), one can then show the existence of a regular solution to (3.5), whosevelocity field has a finite Dirichlet integral. If� is symmetric around the��-axis, this method deliverssymmetric solutions, in the sense of (3.3).

If we use this procedure along with well-known regularity theory for the classical Stokes problemin a bounded domain (see,e.g. [21]), we can reformulate the original result of Leray in thefollowingconvenient form.

Theorem 3.1 Assume that� is of class�� and that�� , ; � #. Then, there exist asubsequence of�� –that we still denote by�� – and two fields� � �� and�such that

(i)�

��

�� , for some� depending only on�, �� and�;

(ii) � �� , � �� , for all bounded subdomains��;

(iii) �� "�� "�� as � �;

(iv) �� satisfy(3.5),

(v) � � �� and the following energy inequality holds

�

�

��

�

�� ' �� (3.9)

Finally, if � (� ��) is symmetric around the��-axis, in addition to the above properties we havealso that�� satisfy(3.3).

Remark 3.1 One fundamental issue that comes with the method of Leray of invading domains (andwith any other method we are aware of, like Fujita’s, see nextsection) is related to the physicallyremarkable case when�� , and is the following one (cf. [14], p. 88): Is the solution�� nontrivial? Actually, we are not assured,a priori that � is nonidentically zero. As a matter of fact,Leray’s construction in thelinear case would lead to an identically vanishing solution, as a consequenceof the Stokes paradox. To see this, let us disregard in (3.7) the nonlinear term�� for each� � ��,and take�� as well. Applying Leray’s procedure, we then obtain that thelimit field, �� say,solves the Stokes problem with zero boundary data and that, in view of property (i),�� has a finiteDirichlet integral. Therefore, by Lemma 1.3 we infer�� . In the general nonlinear case, theanswer to the question is still unknown. However, in Part IV,we shall prove that� is nontrivial atleast for symmetric flow, a fact first discovered by Amick [2], 4.2.

37

Remark 3.2 The (approximating) solutions�� to (3.7), satisfy the following energyequality

�

�

��

��

�� ' ��

This is at once established by multiplying both sides of (3.7)� by �� , and integrating by partsover �� . Notice that, in the limit� � the energy equality is lost and we only obtain anenergyinequality; see (3.9). This loss is essentially due to the little (uniform) information that theapproximating solutions bring about their behavior at infinity. Should one be able to show that thelimit solution satisfies the energy equality, the problem of existence for arbitrary Reynolds numberswould be “almost” solved, at least in the class of symmetric solutions; see Remark 4.1

3.2 Existence: Fujita Method

An alternative method of constructing solutions to (3.5), based on the so called “Galerkin approxima-tion” was introduced by H.Fujita [17] in 1961. We will sketchit in the following, referring to [21],Chapter X, for details. Throughout this section, we shall denote by� �� the duality pairing in��.

Assuming� locally Lipschitzian, given an arbitrary�� satisfying (3.1), we canfind � �� , all 3 � �, which equals�� at � and�� at �� 3�, for some3� � �; see [21], Lemma IX.4.1 and Remark IX.4.2. Moreover,cf. [21] loc. cit., for a given. � �,the field can be chosen in such a way as to verify the following inequality

��# � �� #�� .�#�� for all # � �� (3.10)

A sequence of approximating solutions to (3.5)�� #� � is then searched in the form

#� ��

��

5��J�

��#��J�� #� � �#�� J�� #� � �� J�� #�� J��

� ��J�� J�� ! �

(3.11)

where�J�� is a basis of�� that is orthonormal in��. For each! � �� wemay establish existence to the nonlinear system (3.10), provided we show a uniform bound for�#��in terms of ; see [21], Lemma VIII.3.2. Multiplying (3.10)� by 5��, summing over� from 1 to!and observing that

� � �#�� #�� #� � �#�� #�� for all ! � ��

we obtain�#�� #� � �� #�� #�� #�� (3.12)

From (3.12), using Holder’s inequality, (3.10), and the fact that the support of� is bounded, onecan show the following estimate

�#�� (3.13)

where�� is a positive constant depending only on . This latter inequality, on the one hand,proves that the nonlinear system (3.11) has at least one solution ([21], Lemma VIII.3.2) and, on the

38

other hand it implies that there exists# � �� and a subsequence, that we continue to denote by�#�� such that

#� # weakly in�� #� # strongly in�� (3.14)

for any compact�� . Passing to the limit! � in (3.11)� and employing (3.14), we easilyobtain that� � #� satisfies the following relation

��J�� J�� for all � � �� (3.15)

However, given anyB � ��, bothB and�B can be approximated in the uniform norm by a finitelinear combination ofJ�; see [20], Lemma VII.2.1. Thus, from (3.15), we infer

��B� � �� B� � �� for all B � �� (3.16)

Since� � �� for all 3 � ��, from (3.16) and well-known regularity results for the Navier-Stokes equations (seee.g. [21], Corollary VIII.5.1) we have that� � �� and that there exists� � �� such that�� satisfies (3.5)��. Moreover,� assumes the boundary data�� in the senseof trace. Also, in view of (3.13), we find that� � ��, which, as in Leray’s method, is theonlyinformation that Fujita’s method provides about the asymptotic behavior of the solution. Finally, using(3.11), we can show that� satisfies the energy inequality (3.9).

The same type of argument would lead to a symmetric solution,in case when� is symmetricaround the��-axis. This is achieved by using, instead of�� , its subspace constituted by vectorfields satisfying the parity condition (3.3)

Remark 3.3 As in the case of Leray’s construction, the solution� just constructed with the Galerkinapproximation may reduce, when�� to the trivial one� � �. In fact, we recall that� is of theform # � , with an extension of� �� and# � �� . In dimension 2 the field belongs to�� , since�� contains also the functions that are constant in a neighborhood of infinity;see Lemma 1.2. Thus, the possibility# � � can not be ruled out, which would give� � �. While,as mentioned in Remark 3.1, one can show that symmetric solutions constructed with the method ofLeray are nontrivial (Part IV), it is not known if the same conclusion can be drawn for the same typeof solutions constructed via the Galerkin approximation.

3.3 Some Existence Results when � � �

As we mentioned in the introduction to Part II, it is not knownif (1) possesses a solution if� � �,in the case when�� is arbitrarily (sufficiently smooth) prescribed. However, it is not difficult toshow that if� is symmetric with respect to two orthogonal directions and�� satisfy suitable parityconditions, (1) with� � � has at least one solution for every value of�. Specifically, assuming thatthese directions coincide with the�� and�� axes, respectively, we suppose

��

�� (3.17)

and that

�� (3.18)

In order to prove the existence result, we need two simple lemmas.

39

Lemma 3.1 Let � be a locally Lipschitzian, exterior domain, and let# � �� satisfying either(i) #�� #��, or (ii) #�� #�� . Then, there exists5 � 5�� suchthat �

�

#�

�� 5�

��#�� (3.19)

Proof. We first assume� �� and condition (i). The proof under condition (ii) is exactlythe same,with the change�� . Let J be a non-decreasing function that is zero in a neighborhood of �and is one for sufficiently large��, and set4 � J#. Since, by hypothesis,#�� for all�� , we find that4�� for all �� . Thus, by theHardy inequality, it follows that

�

��#�

4�

��

��#�

4�

�� 5

�

��#��4��

By the properties ofJ, we find that

�

��#��4�� 5

��

��#��

�

$�#��

where@ is a bounded subset containing the support of�J. Thus, from the Hardy inequality, weobtain �

��#�

4�

�� 5��

��#��

�

$�#��

Since an analogous inequality holds for the half-plane�� , and since#�� 4��, for allsufficiently large��, �� 3, say, we conclude

�

��#�

#�

�� 5��

��#��

�

$�#�� (3.20)

Recalling that#�� , we obtain that# obeys the followingPoincare inequality, for all K � �

�

�

#� 5��

�

��#��

where 5 � 5�K� � �; see,e.g. [20], Exercise II.4.10. If� �� , the lemma then follows fromthis latter inequality and (3.20). If� � ��, the proof goes exactly as before, without the use of thefunctionJ. �

Remark 3.4 As we shall show in the following lemma, functions satisfying (3.19) tend to zero atinfinity in a suitable sense. If# merelybelongs to�� and doesnot necessarily satisfy the parityconditions of Lemma 3.1, the following weaker inequality holds

�

�

#�

�� 5

�

��#��

if �� , which does not prevent# from growing logarithmically fast at large distances; see Lemma1.2.

40

Lemma 3.2 Let the assumptions ofLemma 3.1be satisfied. Then

��

� ��

��#�� (3.21)

Proof. We shall again assume condition (i) of Lemma 3.1, since the proof goes exactly the same wayif (ii) is assumed instead. From the fact that# � ��, we have

� ��

��

� ��

�

�

�

��#��

��

��

�� as� �.

Moreover, by the mean value theorem, there is�� such that

� ��

�

��#��

��

��

��

��

��

� ��

��

� ��

�

�

�

��#��

��

��

��

Therefore � ��

�

��#��

��

��

��

�� as� �. (3.22)

However, by condition (i) in Lemma 3.1, for all sufficientlylarge� we have#�� , and so

�)��

�#�� 5� ��

�

��#��

��

��

��

��

which, in turn, by (3.22) furnishes

�)��

�#�� as� �. (3.23)

Set

L��

� ��

�

��#��

�

��

��

For any� � �� , we have

L�� L��

��

��L

��

�� L�� as� �. (3.24)

Using Cauchy inequality, we get

��L

��

��

� ��

�

�#��

��#

�

��

� ��

�

#�

��

� ��

�

��#

�

��

��

and so, from Lemma 3.1, we deduceL � ��. The lemma then follows from this property andfrom (3.24). �

With Lemma 3.1 in hands, we can then prove the following existence result.

41

Theorem 3.2 Let � and �� satisfy the assumptions(3.17) and (3.18). Assume, moreover that� islocally Lipschitzian and that�� . Then, for any� �� 8 problem(1) has at least onesolution �� , that satisfies(1)� in the trace sense, and(1)� in the followingsense

��

� ��

�� (3.25)

Proof. Using, for instance, Fujita method restricted to the subspace of�� constituted by vectorfield satisfying parity properties similar to (3.18), we can find a pair�� that solves (1)�� in thesense specified in the theorem. Moreover, since�� satisfies condition (i) of Lemma 3.1 and�� satisfiescondition (ii) of the same lemma, from Lemma 3.2 we deduce thevalidity of (3.25), and the resultfollows. �

Remark 3.5 A simple example where the symmetry assumptions of Theorem 3.2 are satisfied, is givenby the case when� is the unit disk and�� 1�� =�� where1 and= are even functions of�. Notice that the solutions of Hamel given in (2.5) satisfy all these requirements.

3.4 On the Pointwise Asymptotic Behavior of �-Solutions

We now draw attention to the behavior at infinity of a solution �� to (3.5). The only assumptionwe shall makea priori on �� , is that� � ��. Usually, these solutions are referred to as�-solutions. Solutions constructed by Leray in Theorem 3.1 and by Fujitain Section 3.2 are�-solutions.Notice that�-solutions are infinitely differentiable in�.

We shall mainly focus on the behavior of the velocity field itself, referring to Remark 3.5 and toLemma 3.3 for information regarding the behavior of the derivatives of�, and of� and its derivatives.

Our study will be done through a number of intermediate stepsdue, mostly, to Gilbarg andWeinberger [30], and to Amick [2]. We shall give here only themain ideas, referring the reader tothose papers and to [21], Section X.3, for full details.

The first result concerns the pointwise convergence of the pressure field� at large distances.

Lemma 3.3 Let �� be a�-solution. Then, there exits�� such that

��

��

Proof. See [30],4, and [21], Theorem X.3.3. �

In order to investigate the behavior at infinity of the velocity field �, we begin to prove that� isuniformly bounded. To this end, we notice that, defining thetotal head pressureas

� ��

and thevorticity as

% ��

� �� (3.26)

8In fact, the result continues to hold also for� � �; see Section 1.2.1.

42

by a simple calculation based on (3.5)��, we show that

� � � � � � %� � (3.27)

Consider now (3.27) in�%��%� , for arbitraryK�, K�, with K� � K� � K�, K� sufficiently large. We maythen apply Hopf’s maximum principle and obtain that can not attain a maximum in�%��%� , unlessit is a constant. It also follows that

�)��

��

has no maximum. We thus deduce that

��

�)��

"��

� �� #��

implying, by Lemma 3.3 that9

��

�)��

�� (3.28)

However, we don’t know if� is finite or infinite. As a consequence, the maximum principle is notenough to obtain the boundedness of�, and we need more information about the function . In theparticular case�� this is achieved through the following profound result, dueto C. J. Amick [2],Theorem 11, that we shall state without proof.

Lemma 3.4 Let �� be a D-solution to(3.5) corresponding to�� . Then there exists a Jordanarc

. � ) � �� .�)� � �%

such that

(i) .�� %;

(ii) �.�)�� as ) �.

In addition, the function is monotonically decreasing along., namely,

�.�)�� .�;�� for all ;� ) � �� , ; �t. (3.29)

With this result in hand, we can show the following one.

Lemma 3.5 Let � and �� be as inLemma 3.4. Then

� � ��%��

and there is an� � �� such that

��

�)��

�� uniformly. (3.30)

9We assume, without loss, that� � �.

43

Proof. Since�� tends to zero for large��, by (3.29) we deduce that

��.�)�� 5� for all ) � �� , (3.31)

with 5 independent of). Using the assumption that� � ��, we have that

� ��

��

� ��

�

�

�

��

�

��

�� as� ��

implying � ��

�

��

�

��

�� as� �, (3.32)

for some sequence�� with �� . Since. is connected and extends to infinity, for any� � �� we can find at least one)� � �� such that

.�)��

for some�� . Thus, in view of (3.31), it follows that

�� 5� for all � � ��. (3.33)

From the identity

��

�

�� ?�

?�? �

and from (3.32) and (3.33), we find

�)��&��

�� 5�� for all � � ��, (3.34)

with 5� independent of�. We next apply the maximum principle to (3.27) in the annulus�� tofind

�)��

�� )��

$��

�

��

% �)

��&��&��

�� (3.35)

However, by (3.34) and Lemma 3.3 we deduce

�)��&��&��

�� 5�� for all � � ��

so that, again Lemma 3.3 and (3.35) deliver

�)��

�� 5�� for all � � ��

with a constant5� independent of�. Therefore,� � ��%�. Since the second part of the lemma isan immediate consequence of the first and (3.28), the proof is complete. �

A conclusion similar to that of Lemma 3.5 can be reached by a more elementary proof that does notuse Lemma 3.4, in the case when�� is a solution constructed with Leray’s method. Actually, wehave the following result whose proof can be found in [29],2.

44

Lemma 3.6 Let �� be a solution to(3.5) in the sense ofTheorem 3.1. Then, there exists a constant�� independent of� � ��, such that

�� for all � � ��

Thus, fromTheorem 3.1(iii)we find, in particular,� � ��.

We shall next investigate if� approaches some vector�� at infinity. We have the following twopossibilities:

(i) the number� in (3.28) is zero;

(ii) the number� in (3.28) belongs to��. 10

In case (i) we have��

�� uniformly�

and we deduce at once that�� . On the other hand, if� � �, using the ideas of Gilbarg andWeinberger [30],5, we proceed as follows. We set

1 ��

��

� ��

�1�� (3.36)

and recall theWirtinger inequality

� ��

��1�� 1��

� ��

�

��1��

�

��

�� (3.37)

We need two preliminary lemmas

Lemma 3.7 Let � and �� be as inLemma 3.4. Then

(i) ��

� ��

�� ,

(ii) ��

�� ,

where� is defined in(3.28).

Proof. By the Wirtinger inequality (3.37) and the Cauchy inequality we have��

��

� ��

��

��

��

��

��

��

� ��

�

��

��

��

5

� ��

��

10Of course, by Lemma 3.5, if�� it follows that �.

45

therefore,

��

� ��

�� C � ��

However, again by (3.37), we have� �

%

�

�

��

��

��

which impliesC � �, and (i) is proved. To show (ii), we observe that (3.30) implies that, given anysequence�� with �� , there is a corresponding sequence�� such that

��

�� (3.38)

However, as in the proof of Lemma 3.5, we show the existence ofa sequence�� suchthat (3.32) holds. Since

��

� ��

�

�� ?�

?

��

�?�

by (3.38) and the triangle inequality we find that

��

�� uniformly in �� (3.39)

Moreover, since � ��

�� for all � � ��

it follows that

��

� ��

�

�� ?�

?

��

�?

which together with (3.32) and (3.39) allows us to conclude that

��

�� (3.40)

Now, for � � �� we have

��

��

� �

��

� ��

�

�

��

��

�

��

� ��

��

� ��

�

�

��

�

��

��

and so, in view of (3.40), the property (ii) follows by letting � � in this inequality. �

Lemma 3.8 Let � be as inLemma 3.4and assume that the number� in (3.28) is finite. Then,

��% � ��%��

46

Proof. From (3.5)� we find�% � � � �% � �� (3.41)

Let J� � J�� be a “cut-off” function that is one for3 � �� and is zero for�� 3, and satisfies��J�� 3��, �,� � �� , for a constant� independent of3. Setting+� � �J�, multiplying(3.41) by+� % and integrating by parts over�% we deduce that

�

�+��%��

�

�

�

�%��+� � � � �+��

�

�

�

�&

�%�

�� %�

�� (3.42)

By the properties ofJ�, it follows that

��+�� +�� +�� 5 (3.43)

for some constant5 independent of3, so that by (3.43), identity (3.42) gives�

�+��%��

for a constant� independent of3. Letting3 � and using the monotone convergence theoremcompletes the proof. �

We are now in a position to prove a first result on the behaviorof the velocity field of a�-solutionat infinity.

Theorem 3.3 Let �� be a�-solution to(3.5) and let� � �� be the number defined in(3.28).Then, if� �� (this certainly happens whenever�� ), there is�� such that

��

� ��

�� (3.44)

If � ��,

��

� ��

�� (3.45)

Proof. Let J � J�� be the argument of��, that is,

�� J�� J��

J � �� (3.46)

Clearly, we have

J��

�� (3.47)

where the prime means differentiation. Multiplying the first component of (3.5)� by �� , the secondcomponent by�� , and adding up, for sufficiently large�� we find that

%

��

��

� ��

��

�

�

�� (3.48)

47

We take the average over� of both sides of (3.48) to deduce that

%

��

��

�

��

� ��

�

$��

��

�

��

�

%�� (3.49)

From Lemma 3.7 we know that�� converges to� � �. Assume, for a while, that� � �. Thenwe may findK � �� such that

�� for all � � K� (3.50)

We then divide both sides of (3.49) by�� and integrate over� � �� and over� � �� ,�� K, to obtain

J�� J��

��

� ��

��

� ��

�

�

��%

��

��

��

��

Using (3.50) and the Schwarz and the Wirtinger inequalitieswe see that

�J�� J��

��%��

� ��

��

� ��

�

��

��

��

and, therefore, letting�� and recalling Lemma 3.8, we obtain

��

J�� J� (3.51)

for someJ� � �� . For� � � we define the vector

�� J�� J��

If � � ��, from Lemma 3.7(ii), (3.46), and (3.51) we conclude that

��

��

which along with Lemma 3.7(i) implies (3.44). If� � �, we have�� and (3.44) follows from(3.28) even in a stronger, pointwise sense. Finally, if� � �, (3.45) follows directly from Lemma3.7. Notice that, in view of Lemma 3.5, this latter circumstance can not occur if�� . The theoremis proved. �

Our next task is to show that, in fact,� tends uniformly pointwise to��. To this end we need twopreliminary results.

Lemma 3.9 Let �� be a�-solution and letK � �. Then� � ��%� � ��%�, for any� " ��.

48

Proof. From the identity

�� %

��

we obtain that�4 � 9 in �� (3.52)

where

4 � G%�� 9 � G%%

�� G% � �� G%

andG% � � � J%, with J% the “cut-off” function introduced in the proof of Lemma 3.8.By Lemma3.8 we deduce, in particular, that% � ��%�, and since�� , it follows that9 � ��.By well-known results of existence and uniqueness for the Poisson equation in the plane, we deduce4 � �� which, by the properties ofG% and the regularity of� in turn implies�� %�.Since�� , we obtain�� %�. By the Sobolev embedding theorem wethus find�� %�, � " ��. Since

�� %��

by a similar reasoning we show�� %�, � " � �. The proof of the lemma is, therefore,accomplished. �

Lemma 3.10 Let K � �, and assume# � ��%�, for some" � �. Then, there exists a constant5depending only on" such that

�#�� 5��

��#�� #��&��

�� for all � � �%.

Proof. The proof follows standard arguments. Let� � �� and let�� be a polar coordinatesystem with the origin at�. We have

#�� #��

� ��

�

#�K� ��

K�K� (3.53)

Thus��

��#��

�� #��

�

� �

�K��

�� K ��

��#��&��

��#��" � �" � �

��#��&��

(3.54)

Multiplying both sides of this inequality by�� and integrating over�� and� � �� we get

�#��&��

� ��

��#�� 5��#��&�� (3.55)

where5� � 5��"� � �. We now go back to (3.53) and, by the same arguments leading to(3.54), weobtain

��#��

��#�� 5��#��&��

49

with 5� � 5��"� � �. Multiplying by �� and integrating over�� we find

�#��

��#��&�� 5��#��&��

with 5� � 5��"� � �. The lemma then follows from this latter inequality and (3.55). �

Coupling the results of Theorem 3.3, Lemma 3.9 and Lemma 3.10we obtain the following.

Theorem 3.4 Let �� be a�-solution to(3.5). Then, there exists�� such that

��

�� uniformly� (3.56)

Proof. In view of Lemma 3.9 and Theorem 3.3, givenI � � there exists3� � � such that

�� I��5� ��

�� I��5� � for all � � 3�

where5 � 5�"� is the constant entering the inequality in Lemma 3.10. Applying Lemma 3.10 with# � � � �� andK � 3� � � we thus conclude

�� I � for all � such that�� 3� � �.

The theorem is proved. �

Remark 3.6 The preceding theorem asserts that every solution to (3.5) with �� , with correspond-ing velocity field having a finite Dirichlet integral tendsuniformly pointwise tosomevector ��. Thefundamental question that remains open is whether or not�� , so that also condition (3.6) maybe satisfied. Actually, the vector�� can, in principle, even be zero. So, the question of solvability of(3.5)-(3.6) for “large” values of� is still open. However, using the result of Theorem 3.4, in Part IVwe shall show that if� is symmetric around the��-axis and if a certain homogeneous problem relatedto (3.5)-(3.6) hasonly the zero solution, then problem (3.5)-(3.6) is solvable forarbitrarily large� inthe class of symmetric solutions. A crucial step in getting this result is the knowledge of a detailedasymptotic behavior of�-solutions that satisfy (3.56), which will be the object of Section 3.5.

Remark 3.7 We would like to collect here the main results concerning thebehavior at infinity ofthe derivatives of� and�, when�� is a�-solution. We begin to observe that, by Lemma 3.9, itfollows that�� converges uniformly pointwise to zero. By using arguments similar to those employedin Lemma 3.8 and Lemma 3.9, it is possible to show that

��

�� uniformly, for any�,� � �;

see [21], Theorem X.3.2. Using this property along with (1)�, one can also prove that

��

�� uniformly, for any�,� � �;

50

see [21], Theorem X.3.2. These results are silent about therate of decay of� and � and theirderivatives at large distances. If�� , then, as we shall see in the next section, the fields� and�present the same asymptotic structure of the Oseen fundamental tensor. In the general case, very littlecan be said, and the available results concern only the firstderivatives of the velocity field. Precisely,using Lemma 3.8 and the fact that%, by (3.41), satisfies the maximum principle in�K�� K�, one canshow that if� is bounded (as it happens when�� ) then

��

��%�� uniformly; (3.57)

(cf. Gilbarg and Weinberger [30] Theorem 5), and that, moreover,

��

�� uniformly�

see Gilbarg and Weinbergerloc. cit., Theorem 7. The proof of (3.57) goes as follows. From Lemma3.8, and from the assumption that� � �� we have that

� ��

��

�

�

� ��

�

��%� � ��

��%%

�

�� for all � � ��

which implies the existence of�� such that

� ��

�

��%

��

��%�� %��

�

�� as� �� (3.58)

However,

%��

��

� ��

�%�� B��B�

�

�

� ��

��%�� B��

��%�� B�

B

�� B

which, by (3.58), implies

�� %�� as� �, uniformly in ��

The result then follows by applying the maximum principle to(3.41) in the annuli�� .

3.5 Asymptotic Structure of �-Solutions

The objective of this section is to show that every�-solution satisfying (3.56) forsome�� admitsan asymptotic expansion, for large��, whose dominating term is the Oseen fundamental solution�>� "�. In particular� � �� and� satisfy all the summability properties at large distance possessed bythe fundamental solution. This result is by no means obviousand, for symmetric solution, is due toAmick [3], while in the general case is obtained from the workof Galdi and Sohr [28] and Sazonov[39].

We wish to emphasize that, if�� , the structure of a�-solution at large distances is anopen question. In this regard, it should be noted that, when�� , �-solutions that are regular in aneighborhood of infinity need not be represented there by anexpansion in negative powers of� (� ��)with coefficients independent of�. Actually, the fields given in (2.5) for� � �� provide examplesof �-solutions that decay to zero at infinity more slowly than any negative power of�.

51

Thus, assuming�� , by means of an orthogonal transformation, we can always bring �� intothe vectorM ��, for someM �� . As a matter of fact, the specific value ofM plays no role at all inour proof, so that, for simplicity, we shall putM � �. Furthermore, even though some of the resultscontinue to hold also when�� , for simplicity we take�� . We are then lead to the investigationof the asymptotic structure of�-solutions to the following problem

��

� � � � �

�in �

��

��

��

(3.59)

To reach our objective, we begin to recall, without proof, the following result due to Smith [41](part(i)) and to Galdi [19], Lemma X.5.1 (part (ii)).

Lemma 3.11 Let �� be a regular solution to(3.59) satisfying either of the following conditionsfor all large ��, and for some3 � �

(i) � � �� -��'� � for someI � ��

(ii)�

�� for some� " � %.

Then, the following asymptotic representations hold as��

�� ! � >�� !��

�

�� ! � >��

�� "��

�� ! � "�� #��

(3.60)

where�� is a constant,! is defined in(1.4), and!, "�, � � �� , and# are “remnants” satisfyingthe following asymptotic estimates

!�� -�� "�� -�� #�� -�� (3.61)

where,� � #�� and,� � �. In particular, the following estimate holds, uniformly in�,

�� 5 �� 5 �� 5 �� (3.62)

with 5 independent of�.

Following Finn and Smith [16], we shall call solutions satisfying (3.60)-(3.62),Physically ReasonableSolutions.

Remark 3.8 The results of Lemma 3.11 imply that the solutions constructed in Theorem 2.1 arephysically reasonable.

The following result, due to Galdi and Sohr [28] shows that ifthe component�� of a�-solution�� to (3.59) is in�� for some� � ; ��, then�� is physically reasonable.

52

Lemma 3.12 Let �� be a solution to(3.59)�� such that

� � ��

��

Assume, further, that there existsK � � such that

�� %�� for some; � �� (3.63)

Then�� is physically reasonable.

Proof. In view of the preceding lemma, it is enough to show that� enjoys suitable summabilityproperties in a neighborhood of infinity. To reach this goal, we begin to notice that, by Lemma 3.9,

�� %�� (3.64)

Thus, from (3.63) and Sobolev-like inequalities one shows that

��

�� (3.65)

From the assumptions, (3.63), (3.64) and (3.59)� we also have

�� %�� (3.66)

For3 � K � ��, let J� be a smooth “cut-off” function defined by

J��

�� if �� 3��

� if �� 3�

Setting# � J�� J�� J��

from (3.59) we deduce that#� � satisfy the following system in��

�#� #

��

#

��#� �� 9

� � # � =�(3.67)

where

9 � ��J� � �� J�� J��

J��

� ��J�

= � � � �J�� J��

� � J��

��

Clearly, we have9 � �� = � �� for all " � ��

53

Moreover, we observe that in view of (3.64) and (3.65), by taking 3 sufficiently large, the quantities

�� can be made less than any prescribed constant. For" � �� #��, we set

��#�� #�� #��

where� �� is defined in (1.48). Since for all" � �� , by the Holder inequality, we have that��#

��#�

��

��#

��

��

� ��#�� (3.68)

from Theorem 1.4 we deduce that

��#�� 5�� #�� 9�� =��

�� (3.69)

for some5� � 5��"� � �. Thus, assuming, for instance

��

� 5��

we may use estimate (3.69) to show theexistenceof a solution, �#� ��, say, to (3.67), such that��#�� . Thus, to show that# (and hence� � ��) possesses the same summability properties as#it would be enough to show the following uniqueness result:

Let #� � be a solution to the following problem in��

�#� #

��

#

��#� �� 9

� � # � =�(3.70)

with# � ��

��#�� (3.71)

Assume that�� can be made less than a fixed, prescribed constant. Then# � �.

We don’t know if this result holds under thesolecondition (3.71). However, Galdi and Sohr [28]prove that if (3.63) holds, then

#

�� #� � �� for all " � �� .

Thus, applying to (3.70) the estimates of Theorem 1.4, we obtain

�#�� 5��#

��

��

� �� 5��#��

with 5 � 5�"�. This relation implies# � �, provided we take3 so large as to make�� 5.The theorem is proved. �

The next step is to show that a solution to (3.59) with� � �� satisfies (3.63). This fact hasbeen recently proved by L.I. Sazonov [39], and we shall outline his proof here. The idea is to showthat the total head pressure

� ��

� (3.72)

54

belongs to an��-weighted space, with a weight of the type��', for someI � �:

��' � �� (3.73)

It is well-known that, by using a simple argument based on elliptic regularity, the function decays(pointwise) exponentially fast outside a sector containing the positive��-axis (see [2], p. 106-107).Thus, taking into account that within the sector it is�� , for some� � �, in order to show(3.73), it is enough to prove, for sufficiently large� � �, that

�

�

�

��'�

�� (3.74)

where*� � �� . The property (3.74) is established in [39], p. 205, by combiningthe fact that obeys a maximum principle (see (3.27)) with a weight-function technique. Once (3.73)has been established, by means of an integral formula, basedon the complex variable, relating to�, in [39], Lemma 4, it is shown that �

�

��

��' �� (3.75)

We next recall the well-known representation formula for� in terms of%

��

��K

�

&��

�

��

�

&��

%��

��

where�� . Integrating the previous inequality overK from 3�� to 3and using (3.75) and Schwarz inequality we readily find that

�� 5�

&�

� ��3�

��%��

�� 5 ��' ��3�� )

�&��%��3� �

Using the decay property (3.57), from the preceding inequality we get

�� 5 ��' ��3�� 3��

Taking �� 3�� and minimizing this latter inequality with respect to3, we finally obtain�� 5��' ��, which furnishes� � ��%�, ; � �%. Coupling this information with Lemma 3.12 weconclude with the following result.

Theorem 3.5 Let �� be a solution to(3.59), with � � ��. Then�� is physically reason-able.

Part IV. The Nonlinear Problem: On the Existence of Symmetric Solu-tions for Arbitrary Large Reynolds Number

In Part III we have shown that the velocity field of a�-solution to problem (3.5) with�� ,necessarily tend to some�� uniformly pointwise, and, in fact, such solutions are even physically

55

reasonable if�� . However, we are not able to relate�� with the prescribed value��; see (3.6). Asa result, we still do not know if the problem

��

� � � � �

�in �

��

��

��

(4.1)

has a solution for “large” Reynolds number�.The objective of this Part IV is to give a contribution along this direction. Specifically, let us

denote by (NS)� the problem (4.1) with� � �. Clearly, the zero solution� � �, �=const. is asolution to (NS)�. Furthermore, assume� symmetric around the��-axis (say) and denote by� theclass of symmetric�-solutions�� , that is,�-solutions satisfying (3.3). Then, we shall prove thatif the zero solution is theonly solution to (NS)� in the class�, then problem (4.1), is solvable in�,for arbitrary large Reynolds numbers and the corresponding solutions are physically reasonable. Inparticular, denoting by� the set of� for which (4.1) has at least one symmetric solution associatedto a given��, we show that� contains anunboundedset,��, of the positive real axis.11

The crucial point in the proof of this result is to show a boundfrom below for the��-norm of�in terms of� (Section 4.2). To our knowledge, this is the first contribution relating the solution�� to its prescribed valueat infinity.

The important question with this theorem is, of course, to verify the validity of its assumption.Stated in a different way, for our result to be true it is sufficient that every symmetric solution to thehomogeneous problem (NS)� with � having a finite Dirichlet integral is identically zero. Actually,even a weaker version of this statement would be enough; see Remark 4.7. We wish to emphasizethat the problem here is not related to local regularity of solutions to (NS)� (they are of class��,and even real-analytic in�) but, rather, to their behavior at large distances. Actually, we know thatany�-solution �� to (NS)� tends to zero uniformly pointwise together with all its derivatives ofarbitrary order (see Remark 3.5), but this is not enough to make the “classical” energy method foruniqueness to work (see Remark 4.1). It should be said that, if � � �� (unfortunately, a case ofno interest in the present situation), then our assumption is easily shown to be satisfied; see [30] andRemark 4.2. However, we wish also to mention that, if� � �, � �� , and� at � is not zero,example of nonuniqueness are well-known [31].

Proving or disproving the assumption of the theorem will certainly shed new light on this long-standing problem. I regret I was not able to get any result in this direction, and I leave it to theinterested mathematician as a challenging open question.

Another interesting problem that we leave open is the study of the properties of the set��, liketopological or measure-theoretical ones. The properties of the set�� can be studied, for instance,by means of the results of Theorem 2.3. In particular, one maytry a continuation argument to showthat�� coincides with the whole positive real axis. This requires,on the one hand, that solutions�#�� with Reynolds number�� must have the velocity field in the spaceD��, for some� � " � %�&, and, on the other hand, that they do not belong to the nullspaceH!�� of the operator�� @!�� ; see Theorem 1.8. While the first issue finds a positive answer as a result of Theorem

11Without loss of generality, we may take� � �.

56

3.5, we can not exclude,a priori, that solutions with�� belong toH!��

As a final remark, we believe that our result, as is stands, can be in principle extended to the caseof non-symmetric solutions. However, this would require a substantial technical effort in generalizingthe result of Amick used in the proof of Lemma 4.5 to non-symmetric flow.

4.1 A Remark About Symmetric Solutions

In this section we shall show how the results proved in Theorem 3.4 specialize to the case of symmetricsolutions. In fact, by a very simple observation, we can relate �� to ��.

Specifically, have the following result.

Lemma 4.1 Let �� be as inTheorem 3.4. Then,�� M��, whereM � ,�, for some, � �� .

Proof. We claim that�� M�� for someM � ��. In fact, since the component4 of � along the��-axissatisfies

4�� 4��we find that4�� , for all �� . Therefore, our claim is a consequence of Theorem 3.4.We next multiply (4.1)� by � � M�� and integrate by parts over�� to get

�

��

�� M��

��' ��

�

�� M�� ' ��

If M �� , we use the asymptotic properties of Theorem 3.5 and let3 � in the previous relation.We then find that�� obey the following energy equality

�

�� M ��

�

��' �� (4.2)

From (3.9) and (4.2) we deduce

�M� ��

��' �� (4.3)

In view of the symmetry properties of�� , and of (3.9) it follows that�

��' �� +�� (4.4)

for some+ � �. Therefore, from (4.2) and (4.3) we findM � ,� for some non-zero, � �� ,and, finally, from (4.2), (4.4) we find�,�+ � �, which implies, � �. The lemma is proved. �

Remark 4.1 From the proof of the previous lemma it follows thatif we could construct�-solutionssatisfying the energyequalityand if , �� , then �� and we would show existence for arbitrary�. The possibility of being, � � would be ruled out if we could show that the problem (NS)� hasonly the zero solution in the class�; see the introduction to Part IV.

57

4.2 A Key Result

In this section we prove a fundamental inequality for asymmetric Leray solution; see Theorem 4.1.By this nomenclature, we mean a symmetric�-solution to (4.1)�� that has been constructed by themethod of Leray described in Section 3.1, for a given� � �, provided� is of class��, which willbe assumed throughout. We recall that these solutions satisfy the properties stated in Theorem 3.1 andLemma 3.6.

Everywhere in this section, we denote by�� #�� 4�� a symmetric solution to (3.7) andby %� � #�� 4�� the corresponding vorticity. We also set

��

��

��

and

� ��

��

��

�

where�� is a symmetric Leray solution. Furthermore, we put

��1 �� 1�"�� )��

�)��

��1��

Furthermore, we indicate by5 a constant depending at most on�, and whose numerical value is notessential to our aims. In particular,5 may have several different values in a single computation.

Finally, in order to avoid cumbersome notation, we shall denote the components�� and�� of thevelocity field �, by # and4, respectively. We shall also set�� and �� . We recall that� � �� is contained in��.

The main objective of this section is to prove the following key result.

Theorem 4.1 Let �� be a symmetric Leray solution corresponding to a given�. Then, there existsa polynomial7 � 7 �� with coefficients depending only on�, such that7 �� and

�� 7 ��

Remark 4.2 The proof of this theorem will be achieved through several intermediate steps. Beforedoing this, however, we wish to point out a particular, immediate consequence of our result, namely,that a symmetric Leray solution corresponding to� � � can never be trivial,0��, � � �, �=const. Thiswas proved for the first time by Amick [2], Theorem 29. It is not known if the same result is true fornon-symmetric solutions, or for symmetric solutions constructed by a method different than Leray’s(like the Fujita method).

Lemma 4.2 The following inequality holds, for allK� � �3�� #3��

�� 5��

� ��

��K��

��

58

Proof. From the identity

�� 3�� K��

%�

��

we obtain

� ��K��

%�

��

�

��

%�

��

�

The lemma then follows after squaring both sides of this inequality and integrating over� � �� .�

Lemma 4.3 The following inequality holds�

��

��%�� 5�� 3��

��

where� and�� are the constants introduced inTheorem 3.1(i)and Lemma 3.6, respectively.

Proof. We recall that the vorticity%� satisfies the following equation

�%� � � � �%� � �� (4.5)

Let J�� be a smooth, non-increasing function which is one for�� #3�� and is zero outside�� , and let2�� be a smooth, non-increasing function which is zero for�� and is one for�� . We may take��J�� 53�� . Setting+ � J��2, multiplying (4.5) by+�%� andintegrating over�� , we find

�

��

+��%��

��

+%��%� � �+ ��

��

+�%�� +� (4.6)

From Theorem 3.1(i), Lemma 3.6, and the properties of+ we get�

��

+�%�� + 5��3��

��

�%��

��

��%�� 5��3��

By a similar argument and by Cauchy inequality,

��

��

+%��%� � �+ ��

�

��

+��%��

��

��+��%��

��

�

��

+��%�� 5��3��

Replacing these two last displayed inequalities into (4.6), and recalling that+�� , �� #3�� ,we obtain �

�(��(��%�� 5

�� 3��

��

which proves the lemma. �

Lemma 4.4 There exists�3� � �3�� #3�� such that

59

(i)� ��

�� 3�� 5�� , for all �� with �� 3�;

(ii)� ��

�

�� 3��

��

�

�� 5�� )�

�� 3� �� ;

(iii) �)�

�� 3�� 5�� ,

where the quantity�� is defined inLemma 4.3

Proof. For simplicity of notation, we shall omit the subscript�. We also recall the average over�� of a function1 ; see (3.36). Recalling that�# � %��, �4 � �%��, from (3.7) we find

�

��%

�� ##�

� 4#��

�

�� %

�� #4�

�44��

(4.7)

Thus, settingM�� )

��

we have �

��

��M

�

��

��%��

��

��

and so, in particular, �

��

� ��M

� ��

��

From this relation and the integral mean-value theorem we deduce

� ��

�

�� 3� ��

��

�

5M� �3��

� ��

�

�� 3� ��

��

�

5��

� for some �3 � �3�� #3� �. (4.8)

Let �� be any point such that�� 3. Since

�� 3� ��

��

� ��

�

�� 3� ��

��

��

part (i) follows from this inequality and (4.8)�. Moreover, again from (4.8)�,

�� 3� �� 3��

�

�� 3� ��

�� 5�� (4.9)

60

Also,

�� 3��

��

��

��

��

��

� �

��

� ��

�

��4

�

�� 5�� (4.10)

Consequently, combining (4.9) and (4.10), we conclude

�� 3� �� 5��

which proves part (iii). Finally, part (ii) follows from part (iii) and (4.8)�. The lemma is proved.�

Lemma 4.5 Let �3� be the number defined inLemma 4.4. The following inequality holds

�)�

�� 3�� 5��

��

�

Proof. We shall again omit the subscript� and use the notation (3.36). Multiplying (4.7)� by �� and (4.7)� by �� , adding up, and using (3.7)�, we find

�

��

�

�%

�� #4

�� 4#

�

�� (4.11)

Integrating this equation from 0 to��, dividing by ��, and observing that� ��

�#4

��

� ��

�4#

��

we deduce that�

��

�

��

� ��

�

�#� #�4

�� 4 � 4�#

�

��

and so, integrating from� to � � 3,

��

�

� ��

�

��

��

��

�

� ��

�

�

�

��

�

��

��

��

From the Wirtinger inequality (3.37), we find� ��

��

� ��

�

��

�

��

��

and we may thus conclude, in particular,

�� 5�� for all � � �3�� #3� �� (4.12)

Now,

�� 3� �� 3��

�

�� 3� ��

��

��

�

and so, by (4.12),

�� 3� �� 5

�&��

��

�

�� 3� ��

��

��'� �

The result then follows from this inequality and Lemma 4.4, parts (ii) and (iii). �

61

Lemma 4.6 The following inequality holds

�� 5��

�� (4.13)

Proof. From Lemma 4.2 and Lemma 4.4(i) we obtain

�� 5�� for all �� with �� 3�� (4.14)

Let us next consider the total head pressure

��

From Amick [2], 4.2, we know that there is a continuous, injective map

.� � ; � �� ;� � ��

with the following properties

(a)(

��)�

.��;� � �� $ �� 3��;

(b) dist�� , �� .��;

(c) .�� 3��;

(d) The function ��;�� is monotone decreasing on�� .

Denote by�� the point where the curve.��meets the circumference�� 3��. By the monotonicityproperty of � we find

��

� ��

�)�

�� 3��

��

and so, by Lemma 4.5,

�� 5��

��

Replacing this latter relation into the inequality in (4.14) with �� , we prove the lemma. �

Lemma 4.7 Let�� be a symmetric Leray solution corresponding to�. Then, the following inequalityholds

�� 5�<� � <� � <�� <

��

where< � ��

62

Proof. Multiplying both sides of (3.7)� by �� and integrating over�� we have

��

��

��

��

��

��

� ��

Therefore,�� 5��

Once we replace this inequality in (4.13), we find

�� 5��<

��

�� <��

��<��

�� <� � <

��

�� (4.15)

where<� � ��

Recalling the definition of�� given in Lemma 4.3 and using the property of the approximating solutiongiven in Theorem 3.1(iii), it follows that

��

<� � <� ��

�� (4.16)

Thus, the lemma follows , by letting� � in (4.15), and by using (4.16) and the elementary Young’sinequality. �

Lemma 4.8 Let �� be as inLemma 4.7. Then,

�� 5 �L� � L�� L� � L

��

whereL � ��

Proof. Let � � � � � #. From the Stokes estimates applied to (3.5)� in the domain��, (see [20],Theorems V.4.1 and V.5.1), we find

�� 5 �� (4.17)

for all ! � � and " � ��. Without loss, we add a constant to the pressure field in sucha waythat �

��

� � ��

Thus, by known results (see [20], Lemma IV.1.1)

�� 5�� (4.18)

By the Sobolev embedding theorem, we have also

�� 5��and so, using the Poincare inequality:

�� 5��

63

we have�� 5�� (4.19)

In view of (4.18) and (4.19), (4.17) gives, in particular,

�� 5 �� L� � for all " � �� . (4.20)

Choosing in this relation! � �, " � "� � �, � � � � ��, � � # and using (4.19) it follows that

�� 5 �� L

� 5L� (4.21)

We next employ (4.20) with! � �, " � "�, � � � � �� and� � �� to get

�� 5��

�� L�� (4.22)

However, again by the Sobolev embedding theorem,

�� 5��

Consequently, from (4.21) and (4.22) we derive

�� 5�L� � L�� (4.23)

By Sobolev embedding theorem, this last relation implies

�� 5�L� � L�� (4.24)

Iterating one more time the preceding procedure, we finallydeduce

�� 5 �L� � L�� L� � L

�

and, again by the Sobolev embedding theorem, we conclude

�� 5 �L� � L�� L� � L

�� (4.25)

The result is then a consequence of (4.24), (4.25). �

We are now in a position to give aproof of Theorem 4.1. In fact, it is an immediate consequenceof Lemma 4.7 and Lemma 4.8.

4.3 Existence of Symmetric Solutions for Arbitrary Large Reynolds Number

We begin to introduce a suitable regularity class. Specifically we denote by� the class of pairsconstituted by a vector field# � �#�� #�� and scalar fieldG such that:

(i) Symmetry:#�� #�� #�� #��

G�� G��

(ii) Finite Dirichlet Integral: �

��#��

64

The objective of this section is to prove the following result.

Theorem 4.2 Let � be symmetric around the��-axis. Assume that the following problem

�# � # � �#��G

� � # � �

�in �

# � � at �

��

#�� uniformly

(4.26)

has only the zero solution in the class� . Then, there is a set� with the following properties:

(i) � � ��;

(ii) � % �� 5� for some5 � 5�� ;

(iii) � is unbounded;

(iv) For anyM �� , the problem

��

� � � � �

�in �

��

�� M�

(4.27)

has at least one solution in the class�.

Before we give the proof of Theorem 4.2, we wish to make the following remarks.

Remark 4.3 A sufficient condition for the validity of the hypothesis ofTheorem 4.2 is the existenceof an unbounded sequence of numbers�3��, such that

��

� ��

�

��#��3�� G�3��

�#�3�� (4.28)

where�� 3�� . Actually, multiplying (4.26)� by # and integrating by parts over��, we find�

��

��#��

��

�##

�� #� � G�# � �

�� (4.29)

One can show that all solutions to (4.26) in the class� satisfy the following property (see [21], LemmaX.3.2)

�# � ��

Consequently,

��

�

��

��#

�

��

Therefore, from (4.29) and the fact that# is bounded, we obtain�

��

��#��

�

��

��#� � G

�# � ��

and so, if (4.28) is satisfied, we conclude# � �, by “classical” uniqueness method.

65

Remark 4.4 In the case when� � (an unrealistic assumption in our present situation) the uniquenessof the zero solution to (4.26) is a simple consequence of the maximum principle applied to the vorticityequation; see [30], Theorem 2.

Remark 4.5 All possible solutions to (4.26) in the class� are smooth and satisfy the followingasymptotic conditions (see Remark 3.5)

��

��#��

��G��

for arbitrary�!� � �. Furthermore, we have

�%�� 5�� 5��

Remark 4.6 Solutions described in Theorem 4.1 are all physically reasonable; see Theorem 3.5.

Let us now come back to the proof of Theorem 4.2. To this end, weneed one more auxiliaryresult.

Lemma 4.9 Let M� � � and let�� be a solution to(4.27)corresponding toM � �� M��. Then, thereexists a positive constantN, depending only on� andM�, such that

�

�� N

Proof. For a givenI � �, let J' � J'�� be a Hopf “cut-off” function, namely, a smooth functionsatisfying the following properties:

(i) �J'�� , for all � � �;

(ii) J'�� , if �� .��;

(iii) J'�� , if �� .�I�;

(iv) ��J'�� I��, for all � � �,

where�� is the distance of� from � and.�I� � �)(��I��; see��=, [20], Lemma III.6.2. Wethen define

' � M

��J'��

��J'��

��

�

and set# � � � '. From (4.27) we thus get that# is a solution to the following problem

�# � # � �#� # � �' � ' � �#� ' � �' ��' ��

� � # � �

#��

#��

(4.30)

66

We now multiply both sides of (4.30)� by #, integrate by parts over��, let3 � and use Theorem3.5. We then obtain the following relation

�

��#��

�

��# � �# � ' � ' � �# � ' ��' � �#� � (4.31)

Employing the properties of the functionJ', and noticing that the support ofJ' is contained in��* � ��, we easily obtain

�#'�� M��J'#�� #� ��J'��

� M

�#��J'�� 5I�#��

�(4.32)

Using the Sobolev inequality (4.19), and the following Hardy inequality

�#�� 5��#��

into (4.32) we find

�#'�� 5M��#�� J'�� I

� �5MI��#�� (4.33)

Therefore, recalling thatM M�, from (4.31), (4.33) and Schwarz inequality, it follows that

�

��#�� 5M�I

�

��#�� 5��I��M� � M��

��

��#��

�� (4.34)

where5��I� � �'�� '��

We now chooseI � ��5M�, so that (4.34) furnishes�

��#�� 5�M�� (4.35)

Since �

��

�

��#��

�

��'��

the lemma follows from (4.35) and this latter inequality. �

We are now in a position to give theproof of Theorem 4.2. Let us denote by� the set ofthoseM � � for which problem (4.27) has a corresponding solution�� . From the results of Part II(see Theorem 2.1), we know that� % �� 5�, for some positive5 � 5��. We shall now show that� %�� where�� enjoys the properties:

(i) �� ;

(ii) �� is unbounded.

Actually, let�� be defined as follows:

M �� if and only if ��

�� M �� uniformly

where�� is a symmetric Leray solution corresponding to a given� � �.

67

Clearly,�� . Also, by Lemma 2.2,�� . Furthermore,�� . In fact, by Lemma 4.1,�� . However,� �� , because, otherwise,�� satisfy the homogeneous equation (4.26),and this, by assumption, would imply� � �. So, by Theorem 4.1, we would conclude� � �, whichleads to a contradiction. This proves property (i) of��. Let us show (ii). Assuming�� boundedmeans� � M M� ��, for someM� � �. Thus, from Lemma 4.9, we have

�

�� 5�M�� (4.36)

for all symmetric Leray solutions corresponding toarbitrary � � �. Using (4.36) into Theorem 4.2,we obtain

� 5��M��for arbitrary � � �, which gives a contradiction. The theorem is, therefore, completely proved. �

Remark 4.7 The assumption of Theorem 4.1 can be somehow weakened, with some interesting con-sequences. Actually, to prove that�� is unbounded what we really need is the existence of at leastone diverging sequence�� for which the corresponding symmetric Leray solutions�� satisfy

��

�� M��

It is worth of emphasizing that if this property is not true, symmetric Leray solutions would present avery anomalous behavior, namely, there would exist a positive �, such that the velocity field of suchsolutions corresponding toany� � � would tend to zero as�� , uniformly pointwise.

Acknowledgments. Good part of this work was accomplished while I was a DeutscheForschungsgemeinschaft(DFG) Mercator Professor at the University of Paderborn in the period May-August 2003. I wish to express mywarm thanks to Professor Hermann Sohr for his hospitality and for several stimulating conversations.

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