Solution of Leray’s problem for stationary Navier-Stokes equations in plane and axially symmetric spatial domains1footnote 11footnote 1Mathematical Subject classification (2010). 35Q30, 76D03, 76D05; Key words: two dimensional bounded domains, axially symmetric domains, stationary Navier–Stokes equations, boundary–value problem.

Solution of Leray’s problem for stationary Navier-Stokes equations in plane and axially symmetric spatial domains111Mathematical Subject classification (2010). 35q30, 76d03, 76d05; Key words: two dimensional bounded domains, axially symmetric domains, stationary Navier–Stokes equations, boundary–value problem.

Mikhail V. Korobkov1, Konstantin Pileckas2   and Remigio Russo3
22Sobolev Institute of Mathematics, Acad. Koptyug pr. 4, and Novosibirsk State University, Pirogova str., 2, 630090 Novosibirsk, Russia; korob@math.nsc.ru
33Faculty of Mathematics and Informatics, Vilnius University, Naugarduko Str., 24, Vilnius, 03225 Lithuania; konstantinas.pileckas@mif.vu.lt
44 Department of Mathematics and Physics, Second University of Naples, Italy; remigio.russo@unina2.it
Abstract

We study the nonhomogeneous boundary value problem for the Navier–Stokes equations of steady motion of a viscous incompressible fluid in arbitrary bounded multiply connected plane or axially-symmetric spatial domains. We prove that this problem has a solution under the sole necessary condition of zero total flux through the boundary. The problem was formulated by Jean Leray 80 years ago. The proof of the main result uses Bernoulli’s law for a weak solution to the Euler equations.


1 Introduction

Let

(1.1)

be a bounded domain in , , with -smooth boundary consisting of disjoint components , . Consider the stationary Navier–Stokes system with nonhomogeneous boun-dary conditions

(1.2)

The continuity equation implies the compatibility condition

(1.3)

necessary for the solvability of problem (1.2), where is a unit outward (with respect to ) normal vector to and . Condition (1.3) means that the total flux of the fluid through is zero.

In his famous paper of 1933 [21] Jean Leray proved that problem (1.2) has a solution provided555Condition (1.4) does not allow the presence of sinks and sources.

(1.4)

The case when the boundary value satisfies only the necessary condition (1.3) was left open by Leray and the problem whether (1.2), (1.3) admit (or do not admit) a solution is know in the scientific community as Leray’s problem.

Leray’s problem was studied in many papers. However, in spite of all efforts, the existence of a weak solution to problem (1.2) was established only under assumption (1.4) (see, e.g., [21], [19], [20], [32], [12]), or for sufficiently small fluxes 666This condition does not assumes the norm of the boundary value to be small. (see, e.g., [7], [8], [10], [11], [2], [28], [29], [17]), or under certain symmetry conditions on the domain and the boundary value (see, e.g., [1], [30], [9], [24], [26], [27]). Recently [14] the existence theorem for (1.2) was proved for a plane domain with two connected components of the boundary assuming only that the flux through the external component is negative (inflow condition). Similar result was also obtained for the spatial axially symmetric case [16]. In particular, the existence was established without any restrictions on the fluxes , under the assumption that all components of intersect the axis of symmetry. For more detailed historical surveys one can see the recent papers [14] or [26][27].

In the present paper we solve Leray’s problem for the plane case and for the axially symmetric domains in . The main result for the plane case is as follows.

Theorem 1.1.

Assume that is a bounded domain of type (1.1) with -smooth boundary . If and satisfies condition , then problem admits at least one weak solution.

The proof of the existence theorem is based on an a priori estimate which we derive using a reductio ad absurdum argument of Leray [21]. The essentially new part in this argument is the use of Bernoulli’s law obtained in [13] for Sobolev solutions to the Euler equations (the detailed proofs are presented in [14]). The results concerning Bernoulli’s law are based on the recent version of the Morse-Sard theorem proved by J. Bourgain, M. Korobkov and J. Kristensen [3]. This theorem implies, in particular, that almost all level sets of a function are finite unions of -curves. This allows to construct suitable subdomains (bounded by smooth stream lines) and to estimate the -norm of the gradient of the total head pressure. We use here some ideas which are close (on a heuristic level) to the Hopf maximum principle for the solutions of elliptic PDEs (for a more detailed explanation see Subsection 3.3.1). Finally, a contradiction is obtained using the Coarea formula.

The paper is organized as follows. Section 2 contains preliminaries. Basically, this section consists of standard facts, except for the results of Subsection 2.2, where we formulate the recent version [3] of the Morse-Sard Theorem for the space , which plays a key role. In Subsection 3.1 we briefly recall the elegant reductio ad absurdum Leray’s argument. In Subsection 3.2 we discuss properties of the limit solution to the Euler equations, which were known before (mainly, we recall some facts from [14]). In Subsection 3.3 we prove some new properties of this limit solution and get a contradiction. Finally, in Section 4 we adapt these methods to the axially symmetric spatial case.

2 Notation and auxiliary results

2.1 Function spaces and definitions

By a domain we mean a connected open set. Let , , be a bounded domain with -smooth boundary . We use standard notation for function spaces: , , , , , where . In our notation we do not distinguish function spaces for scalar and vector-valued functions; it will be clear from the context whether we use scalar, vector, or tensor-valued function spaces. Denote by the subspace of all solenoidal vector-fields () from equipped with the norm . Observe that for functions the norm is equivalent to .

Working with Sobolev functions, we always assume that the ”best representatives” are chosen. For the best representative is defined as

where and is the ball of radius centered at .

Below we discuss some properties of the best representatives of Sobolev functions.

Lemma 2.1 (see, for example, Theorem 1 of §4.8 and Theorem 2 of §4.9.2 in [6]).

If , , then there exists a set with the following properties:

(i) ;

(ii) for each

(iii) for every there exists a set with and such that the function is continuous on ;

(iv) for every unit vector and almost all straight lines parallel to , the restriction is an absolutely continuous function of one variable.

Here and henceforth we denote by the one-dimensional Hausdorff measure, i.e., , where

Remark 2.1.

The property (iii) of Lemma 2.1 means that is quasicontinuous with respect to the Hausdorff content . Really, Theorem 1 (iii) of §4.8 in [6] asserts that is quasicontinuous with respect to the -capacity. But it is well known that for smallness of the -capacity of a set is equivalent to smallness of (see, e.g., Theorem 3 of §5.6.3 in [6] and its proof).

Remark 2.2.

By the Sobolev extension theorem, Lemma 2.1 is true for functions , where is a bounded Lipschitz domain. By the trace theorem each function is ”well-defined” for -almost all . Therefore, we assume that every function is defined on .

2.2 On the Morse-Sard and Luzin N-properties of Sobolev functions in

First, let us recall some classical differentiability properties of Sobolev functions.

Lemma 2.2 (see Proposition 1 in [5]).

If , then is continuous and there exists a set with such that is differentiable (in the classical sense) at all . Moreover, the classical derivative coincides with , where .

The theorem below is due to J. Bourgain, M. Korobkov and J. Kristensen [3].

Theorem 2.2.

Let be a bounded domain with Lipschitz boundary. If , then

(i) ;

(ii) for every there exists such that for any set with ;

(iii) for every there exists an open set with and a function such that for each if , then and , ;

(iv) for –almost all the preimage is a finite disjoint family of -curves , . Each is either a cycle in i.e., is homeomorphic to the unit circle or a simple arc with endpoints on in this case is transversal to .

2.3 Some facts from topology

We shall need some topological definitions and results. By continuum we mean a compact connected set. We understand connectedness in the sense of general topology. A set is called an arc if it is homeomorphic to the unit interval .

Let us shortly present some results from the classical paper of A.S. Kron-
rod  [18] concerning level sets of continuous functions. Let be a square in and let be a continuous function on . Denote by a level set of the function , i.e., . A component of the level set containing a point is a maximal connected subset of containing . By denote a family of all connected components of level sets of . It was established in [18] that equipped by a natural topology is a tree. Vertices of this tree are the components  which do not separate , i.e., is a connected set. Branching points of the tree are the components such that has more than two connected components. By results of [18], see also [23] and [25], the set of all branching points of  is at most countable. The main property of a tree is that any two points could be joined by a unique arc. Therefore, the same is true for .

Lemma 2.3 ([18]).

If , then for any two different points and , there exists a unique arc joining to . Moreover, for every inner point of this arc the points lie in different connected components of the set .

We can reformulate the above Lemma in the following equivalent form.

Lemma 2.4.

If , then for any two different points , there exists an injective function with the properties

(i) , ;

(ii) for any ,

(iii) for any the sets lie in different connected components of the set  .

Remark 2.3.

If in Lemma 2.4 , then by Theorem 2.2 (iv), there exists a dense subset of such that is a – curve for every . Moreover, is either a cycle or a simple arc with endpoints on .

Remark 2.4.

All results of Lemmas 2.32.4 remain valid for level sets of continuous functions , where is a multi–connected bounded domain of type (1.1), provided on each inner boundary component with . Indeed, we can extend  to the whole by putting for , . The extended function  will be continuous on the set which is homeomorphic to the unit square .

3 The plane case

3.1 Leray’s argument “reductio ad absurdum”

Consider the Navier–Stokes problem (1.2) in the -smooth domain defined by (1.1) with . Without loss of generality, we may assume that with 777By the Helmholtz-Weyl decomposition, for a -smooth bounded domain , , every can be represented as the sum for , and with , and the gradient part is included then into the pressure term (see, e.g., [20])., where . If the boundary value satisfies condition (1.3), then there exists a solenoidal extension of (see [20], [31], [11]). Using this fact and standard results [20], we can find a weak solution to the Stokes problem such that and

(3.1)

Moreover,

(3.2)

By weak solution of problem (1.2) we understand a function such that and

(3.3)

Let us reproduce shortly the contradiction argument of Leray [21] which was later used in many other papers (see, e.g., [19], [20], [12], [1]; see also [14] for details). It is well known (see, e.g., [20]) that integral identity (3.3) is equivalent to an operator equation in the space with a compact operator. Therefore, by the Leray–Schauder theorem, to prove the existence of a weak solution to Navier–Stokes problem (1.2), it is sufficient to show that all the solutions of the integral identity

(3.4)

are uniformly bounded in (with respect to ). Assume that this is false. Then there exist sequences and such that

(3.5)

and

(3.6)

Using well known techniques ([14], [1]), one shows that there exist with888The uniform estimates for the norms follow from well-known results concerning regularity of solutions to the Stokes problem (see [31, Chapter 1, §2.5] or [20]). Observe that in [14] we could have only because has been assumed to be only Lipschitz. However, for domains with -smooth boundary and the corresponding estimates hold globally. , , such that the pair is a solution to the following system

(3.7)

Choose in (3.5) and set . Taking into account that

we have

(3.8)

Since , there exists a subsequence converging weakly in to a vector field . By the compact embedding

the subsequence converges strongly in . Therefore, letting in equality (3.8), we obtain

(3.9)

In particular, , so are separated from zero.

Put . Multiplying identities (3.7) by , we see that the pair satisfies the following system

(3.10)

where ,  , the norms and are uniformly bounded for each , , 999The interior regularity of the solution depends on the regularity of , but not on the regularity of the boundary value ,  see [20]., and . Moreover, the limit functions satisfy the Euler system

(3.11)

In conclusion, we can state the following lemma.

Lemma 3.1.

Assume that is a bounded domain of type (1.1) with -smooth boundary , , , and satisfies condition (1.3). If there are no weak solutions to (1.2), then there exist with the following properties.

(E)    , , , and the pair satisfies the Euler system (3.11).

(E-NS)   Conditions (E) are satisfied and there exist sequences of functions , and numbers , such that the norms , are uniformly bounded for every , the pairs satisfy (3.10) with ,  , and

Moreover, , .

From now on we assume that assumptions (E-NS) are satisfied. Our goal is to prove that they lead to a contradiction. This implies the validity of Theorem 1.1.

3.2 Some previous results on the Euler equations

In this subsection we collect the information on the limit solution to (3.11) obtained in previous papers. The next statement was proved in [12, Lemma 4] and in [1, Theorem 2.2] (see also [14, Remark 3.2]).

Theorem 3.1.

If conditions (E) are satisfied, then there exist constants such that

(3.12)
Corollary 3.1.

If conditions (E-NS) are satisfied, then

(3.13)

Proof. By simple calculations from (3.9) and (3.11) it follows

In virtue of (3.12), this implies (3.13).

Set ,  . From (3.11) and (3.11) it follows that there exists a stream function such that

(3.14)

Here and henceforth we set .

Applying Lemmas 2.1, 2.2 and Remark 2.2 to the functions we get the following

Lemma 3.2.

If conditions (E) are satisfied, then the stream function  is continuous on  and there exists a set such that

(i);

(ii) for all

moreover, the function is differentiable at and ;

(iii) for every there exists a set with such that and the functions are continuous in .

The next version of Bernoulli’s Law for solutions in Sobolev spaces was obtained in [13, Theorem 1] (see also [14, Theorem 3.2] for a more detailed proof).

Theorem 3.2.

Let conditions (E) be satisfied and let be the set from Lemma 3.2. For any compact connected set the following property holds: if

(3.15)

then

(3.16)
Lemma 3.3.

If conditions (E) are satisfied, then there exist constants such that on each component , .

Proof. Consider any boundary component . Since is continuous on  and   is connected, we have that is also a connected set. On the other hand, since for -almost all (see (3.11) and (3.14) ), Theorem 2.2 (i)–(ii) yields . Therefore, is a singleton.

For denote by the connected component of the level set containing the point . By Lemma 3.3, for every and for every  . Thus, Theorem 2.2 (ii), (iv) implies that for almost all and for every  the equality holds and the component is a – curve homeomorphic to the circle. We call such an admissible cycle.

The next lemma was obtained in [14, Lemma 3.3].

Lemma 3.4.

If conditions (E-NS) are satisfied, then the sequence converges to uniformly on almost all 101010“Almost all cycles” means cycles in preimages for almost all values . admissible cycles .

Admissible cycles from Lemma 3.4 will be called regular cycles.

3.3 Obtaining a contradiction

We consider two cases.

(a) The maximum of is attained on the boundary :

(3.17)

(b) The maximum of is not attained111111The case is not excluded. on :

(3.18)

3.3.1 The maximum of is attained on the boundary

Let (3.17) hold. Adding a constant to the pressure we can assume, without loss of generality, that

(3.19)

In particular,

(3.20)

If , then by Corollary 3.1 and the flux condition (1.3), we immediately obtain the required contradiction. Thus, assume that

(3.21)

Change (if necessary) the numbering of the boundary components , , …, in such a way that

(3.22)
(3.23)

First, we introduce the main idea of the proof in a heuristic way. It is well known that every satisfies the linear elliptic equation

(3.24)

If , then by Hopf’s maximum principle, in a subdomain with – smooth boundary  the maximum of is attained at the boundary , and if is a maximum point, then the normal derivative of at is strictly positive. It is not sufficient to apply this property directly. Instead we will use some ”integral analogs” that lead to a contradiction by using the the Coarea formula (see Lemmas 3.83.9). For and sufficiently large we construct a set consisting of level lines of such that as and separates the boundary component (where ) from the boundary components with (where ). On the one hand, the length of each of these level lines is bounded from below by a positive constant (since they separate the boundary components), and by the Coarea formula this implies the estimate from below for . On the other hand, elliptic equation (3.24) for , the convergence , and boundary conditions (3.10) allow us to estimate from above (see Lemma 3.8), and this asymptotically contradicts the previous one.

The main idea of the proof for a general multiply connected domain is the same as in the case of annulus–like domains (when  ). The proof has an analytical nature and unessential differences concern only well known geometrical properties of level sets of continuous functions of two variables.

First of all, we need some information concerning the behavior of the limit total head pressure  on stream lines. We do not know whether the function  is continuous or not on . But we shall prove that  has some continuity properties on stream lines.

By Remark 2.4 and Lemma 3.3, we can apply Kronrod’s results to the stream function . Define the total head pressure on the Kronrod tree (see Subsection 2.3 ) as follows. Let with . Take any and put . This definition is correct by Bernoulli’s Law (see Theorem 3.2).

Lemma 3.5.

Let , . Consider the corresponding arc joining to see Lemmas  . Then the restriction is a continuous function.

Proof. Put . Let and in . By construction, each is a connected component of the level set of and the sets lie in different connected components of . Therefore,

(3.25)

By the definition of convergence in , we have

(3.26)

By Theorem 3.2, there exist constants such that for all , where . Analogously, for all . If , then we can assume, without loss of generality, that

(3.27)

and the components converge as in the Hausdorff metric121212The Hausdorff distance between two compact sets is defined as follows: (see, e.g., §7.3.1 in [4]). By Blaschke selection theorem [ibid], for any uniformly bounded sequence of compact sets there exists a subsequence which converges to some compact set with respect to the Hausdorff distance. Of course, if all are compact connected sets and for some , then the limit set is also connected and . to some set . Clearly, . Take a straight line such that the projection of on is not a singleton. Since is a connected set, this projection is a segment. Let be the interior of this segment. For by denote the straight line such that and . From Lemma 3.2 (i), (iii) it follows that for -almost all , and the restriction is continuous. Fix a point with above properties. Then by construction for sufficiently large . Now, take a sequence and extract a convergent subsequence . Since is continuous, we have as . This contradicts (3.27).

For the velocities and denote by and the corresponding vorticities: ,  . The following formulas are direct consequences of , :

(3.28)

We say that a set has -measure zero if . The function has some analogs of Luzin’s -property.

Lemma 3.6.

Let with , . If has -measure zero, then .

Proof. Recall that the Coarea formula