Solution of Leray's problem for stationary Navier-Stokes equations in plane and axially symmetric spatial domains
aa r X i v : . [ m a t h - ph ] F e b Solution of Leray’s problem for stationaryNavier-Stokes equations in plane and axiallysymmetric spatial domains ∗ Mikhail V. Korobkov † , Konstantin Pileckas ‡ and Remigio Russo § February 5, 2013
Abstract
We study the nonhomogeneous boundary value problem for theNavier–Stokes equations of steady motion of a viscous incompress-ible fluid in arbitrary bounded multiply connected plane or axially-symmetric spatial domains. We prove that this problem has a solu-tion under the sole necessary condition of zero total flux through theboundary. The problem was formulated by Jean Leray 80 years ago.The proof of the main result uses Bernoulli’s law for a weak solutionto the Euler equations. ∗ Mathematical Subject classification (2010). 35Q30, 76D03, 76D05;
Key words : two di-mensional bounded domains, axially symmetric domains, stationary Navier–Stokes equa-tions, boundary–value problem. † Sobolev Institute of Mathematics, Acad. Koptyug pr. 4, and Novosibirsk State Uni-versity, Pirogova str., 2, 630090 Novosibirsk, Russia; [email protected] ‡ Faculty of Mathematics and Informatics, Vilnius University, Naugarduko Str., 24,Vilnius, 03225 Lithuania; [email protected] § Department of Mathematics and Physics, Second University of Naples, Italy; [email protected] Introduction
Let Ω = Ω \ (cid:0) N [ j =1 ¯Ω j (cid:1) , ¯Ω j ⊂ Ω , j = 1 , . . . , N, (1.1)be a bounded domain in R n , n = 2 ,
3, with C -smooth boundary ∂ Ω = ∪ Nj =0 Γ j consisting of N + 1 disjoint components Γ j = ∂ Ω j , j = 0 , . . . , N .Consider the stationary Navier–Stokes system with nonhomogeneous boun-dary conditions − ν ∆ u + (cid:0) u · ∇ (cid:1) u + ∇ p = f in Ω , div u = 0 in Ω , u = a on ∂ Ω . (1.2)The continuity equation (1 . ) implies the compatibility condition Z ∂ Ω a · n ds = N X j =0 Z Γ j a · n ds = N X j =0 F j = 0 (1.3)necessary for the solvability of problem (1.2), where n is a unit outward (withrespect to Ω) normal vector to ∂ Ω and F j = R Γ j a · n dS . Condition (1.3)means that the total flux of the fluid through ∂ Ω is zero.In his famous paper of 1933 [21] Jean Leray proved that problem (1.2)has a solution provided F j = Z Γ j a · n dS = 0 , j = 0 , , . . . , N. (1.4)The case when the boundary value a 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’sproblem .Leray’s problem was studied in many papers. However, in spite of allefforts, the existence of a weak solution u ∈ W , (Ω) to problem (1.2) was Condition (1.4) does not allow the presence of sinks and sources. F j (see, e.g., [7], [8], [10], [11], [2], [28], [29],[17]), or under certain symmetry conditions on the domain Ω and the bound-ary value a (see, e.g., [1], [30], [9], [24], [26], [27]). Recently [14] the existencetheorem for (1.2) was proved for a plane domain Ω with two connected com-ponents of the boundary assuming only that the flux through the externalcomponent is negative (inflow condition). Similar result was also obtainedfor the spatial axially symmetric case [16]. In particular, the existence wasestablished without any restrictions on the fluxes F j , under the assumptionthat all components Γ j of ∂ Ω intersect the axis of symmetry. For more de-tailed 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 n = 2and for the axially symmetric domains in R . The main result for the planecase is as follows. Theorem 1.1.
Assume that Ω ⊂ R is a bounded domain of type (1.1) with C -smooth boundary ∂ Ω . If f ∈ W , (Ω) and a ∈ W / , ( ∂ Ω) satisfiescondition (1 . , then problem (1 . admits at least one weak solution. The proof of the existence theorem is based on an a priori estimate whichwe derive using a reductio ad absurdum argument of Leray [21]. The essen-tially 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 presentedin [14]). The results concerning Bernoulli’s law are based on the recent ver-sion of the Morse-Sard theorem proved by J. Bourgain, M. Korobkov andJ. Kristensen [3]. This theorem implies, in particular, that almost all levelsets of a function ψ ∈ W , (Ω) are finite unions of C -curves. This allowsto construct suitable subdomains (bounded by smooth stream lines) and toestimate the L -norm of the gradient of the total head pressure. We usehere some ideas which are close (on a heuristic level) to the Hopf maximumprinciple for the solutions of elliptic PDEs (for a more detailed explanationsee Subsection 3.3.1). Finally, a contradiction is obtained using the Coareaformula.The paper is organized as follows. Section 2 contains preliminaries. Basi-cally, this section consists of standard facts, except for the results of Subsec-tion 2.2, where we formulate the recent version [3] of the Morse-Sard Theoremfor the space W , ( R ), which plays a key role. In Subsection 3.1 we briefly This condition does not assumes the norm of the boundary value a to be small. reductio ad absurdum Leray’s argument. In Subsection 3.2we discuss properties of the limit solution to the Euler equations, which wereknown before (mainly, we recall some facts from [14]). In Subsection 3.3 weprove some new properties of this limit solution and get a contradiction. Fi-nally, in Section 4 we adapt these methods to the axially symmetric spatialcase. By a domain we mean a connected open set. Let Ω ⊂ R n , n = 2 ,
3, be abounded domain with C -smooth boundary ∂ Ω. We use standard notationfor function spaces: C k (Ω), C k ( ∂ Ω), W k,q (Ω), ˚ W k,q (Ω), W α,q ( ∂ Ω), where α ∈ (0 , , k ∈ N , q ∈ [1 , + ∞ ]. In our notation we do not distinguishfunction spaces for scalar and vector-valued functions; it will be clear fromthe context whether we use scalar, vector, or tensor-valued function spaces.Denote by H (Ω) the subspace of all solenoidal vector-fields (div u = 0) from˚ W , (Ω) equipped with the norm k u k H (Ω) = k∇ u k L (Ω) . Observe that forfunctions u ∈ H (Ω) the norm k · k H (Ω) is equivalent to k · k W , (Ω) .Working with Sobolev functions, we always assume that the ”best repre-sentatives” are chosen. For w ∈ L (Ω) the best representative w ∗ is definedas w ∗ ( x ) = ( lim r → − R B r ( x ) w ( z ) dz, if the finite limit exists;0 otherwise , where − R B r ( x ) w ( z ) dz = B r ( x )) R B r ( x ) w ( z ) dz and B r ( x ) = { y : | y − x | < r } is the ball of radius r centered at x .Below we discuss some properties of the best representatives of Sobolevfunctions. Lemma 2.1 (see, for example, Theorem 1 of § § . If w ∈ W ,s ( R ) , s ≥ , then there exists a set A ,w ⊂ R with thefollowing properties: (i) H ( A ,w ) = 0 ; (ii) for each x ∈ Ω \ A ,w lim r → − Z B r ( x ) | w ( z ) − w ( x ) | dz = 0;4iii) for every ε > there exists a set U ⊂ R with H ∞ ( U ) < ε and A ,w ⊂ U such that the function w is continuous on Ω \ U ; (iv) for every unit vector l ∈ ∂B (0) and almost all straight lines L parallelto l , the restriction w | L is an absolutely continuous function ( of one variable ) . Here and henceforth we denote by H the one-dimensional Hausdorff mea-sure, i.e., H ( F ) = lim t → H t ( F ), where H t ( F ) = inf n ∞ X i =1 diam F i : diam F i ≤ t, F ⊂ ∞ [ i =1 F i o . Remark 2.1.
The property (iii) of Lemma 2.1 means that f is quasicontin-uous with respect to the Hausdorff content H ∞ . Really, Theorem 1 (iii) of § f ∈ W ,s ( R ) is quasicontinuous with respect to the s -capacity. But it is well known that for s = 1 smallness of the 1-capacityof a set F ⊂ R is equivalent to smallness of H ∞ ( F ) (see, e.g., Theorem 3 of § Remark 2.2.
By the Sobolev extension theorem, Lemma 2.1 is true forfunctions w ∈ W ,s (Ω), where Ω ⊂ R is a bounded Lipschitz domain. Bythe trace theorem each function w ∈ W ,s (Ω) is ”well-defined” for H -almostall x ∈ ∂ Ω. Therefore, we assume that every function w ∈ W ,s (Ω) is definedon Ω. W , First, let us recall some classical differentiability properties of Sobolev func-tions.
Lemma 2.2 (see Proposition 1 in [5]) . If ψ ∈ W , ( R ) , then ψ is continuousand there exists a set A ψ with H ( A ψ ) = 0 such that ψ is differentiable (inthe classical sense) at all x ∈ R \ A ψ . Moreover, the classical derivativecoincides with ∇ ψ ( x ) , where lim r → − R B r ( x ) |∇ ψ ( z ) − ∇ ψ ( x ) | dz = 0 . The theorem below is due to J. Bourgain, M. Korobkov and J. Kristensen[3]. 5 heorem 2.2.
Let Ω ⊂ R be a bounded domain with Lipschitz boundary.If ψ ∈ W , (Ω) , then (i) H ( { ψ ( x ) : x ∈ Ω \ A ψ & ∇ ψ ( x ) = 0 } ) = 0 ; (ii) for every ε > there exists δ > such that H ( ψ ( U )) < ε for any set U ⊂ Ω with H ∞ ( U ) < δ ; (iii) for every ε > there exists an open set V ⊂ R with H ( V ) < ε anda function g ∈ C ( R ) such that for each x ∈ Ω if ψ ( x ) / ∈ V , then x / ∈ A ψ and ψ ( x ) = g ( x ) , ∇ ψ ( x ) = ∇ g ( x ) = 0 ; (iv) for H –almost all y ∈ ψ (Ω) ⊂ R the preimage ψ − ( y ) is a finitedisjoint family of C -curves S j , j = 1 , , . . . , N ( y ) . Each S j is either a cyclein Ω ( i.e., S j ⊂ Ω is homeomorphic to the unit circle S ) or a simple arc withendpoints on ∂ Ω ( in this case S j is transversal to ∂ Ω ) . We shall need some topological definitions and results. By continuum wemean a compact connected set. We understand connectedness in the senseof general topology. A set is called an arc if it is homeomorphic to the unitinterval [0 , Q = [0 , × [0 , R and let f be a continuous function on Q . Denote by E t a level set of the function f , i.e., E t = { x ∈ Q : f ( x ) = t } . A component K of the level set E t containing a point x is a maximal connected subsetof E t containing x . By T f denote a family of all connected componentsof level sets of f . It was established in [18] that T f equipped by a naturaltopology is a tree. Vertices of this tree are the components C ∈ T f which donot separate Q , i.e., Q \ C is a connected set. Branching points of the treeare the components C ∈ T f such that Q \ C has more than two connectedcomponents. By results of [18], see also [23] and [25], the set of all branchingpoints of T f is at most countable. The main property of a tree is that any twopoints could be joined by a unique arc. Therefore, the same is true for T f . Lemma 2.3 ([18]) . If f ∈ C ( Q ) , then for any two different points A ∈ T f and B ∈ T f , there exists a unique arc J = J ( A, B ) ⊂ T f joining A to B .Moreover, for every inner point C of this arc the points A, B lie in differentconnected components of the set T f \ { C } . We can reformulate the above Lemma in the following equivalent form.6 emma 2.4. If f ∈ C ( Q ) , then for any two different points A, B ∈ T f , thereexists an injective function ϕ : [0 , → T f with the properties (i) ϕ (0) = A , ϕ (1) = B ; (ii) for any t ∈ [0 , , lim [0 , ∋ t → t sup x ∈ ϕ ( t ) dist( x, ϕ ( t )) → for any t ∈ (0 , the sets A, B lie in different connected componentsof the set Q \ ϕ ( t ) . Remark 2.3.
If in Lemma 2.4 f ∈ W , ( Q ), then by Theorem 2.2 (iv), thereexists a dense subset E of (0 ,
1) such that ϕ ( t ) is a C – curve for every t ∈ E .Moreover, ϕ ( t ) is either a cycle or a simple arc with endpoints on ∂Q . Remark 2.4.
All results of Lemmas 2.3–2.4 remain valid for level sets ofcontinuous functions f : Ω → R , where Ω is a multi–connected boundeddomain of type (1.1), provided f ≡ ξ j = const on each inner boundarycomponent Γ j with j = 1 , . . . , N . Indeed, we can extend f to the whole Ω by putting f ( x ) = ξ j for x ∈ Ω j , j = 1 , . . . , N . The extended function f will be continuous on the set Ω which is homeomorphic to the unit square Q = [0 , . Consider the Navier–Stokes problem (1.2) in the C -smooth domain Ω ⊂ R defined by (1.1) with f ∈ W , (Ω). Without loss of generality, we may assumethat f = ∇ ⊥ b with b ∈ W , (Ω) , where ( x, y ) ⊥ = ( − y, x ). If the boundaryvalue a ∈ W / , ( ∂ Ω) satisfies condition (1.3), then there exists a solenoidalextension A ∈ W , (Ω) of a (see [20], [31], [11]). Using this fact and standard By the Helmholtz-Weyl decomposition, for a C -smooth bounded domain Ω ⊂ R n , n = 2 ,
3, every f ∈ W , (Ω) can be represented as the sum f = curl b + ∇ ϕ for n = 3, and f = ∇ ⊥ b + ∇ ϕ with b , b, ϕ ∈ W , (Ω), and the gradient part is included then into thepressure term (see, e.g., [20]). U ∈ W , (Ω) to the Stokes problemsuch that U − A ∈ H (Ω) ∩ W , (Ω) and ν Z Ω ∇ U · ∇ η dx = Z Ω f · η dx ∀ η ∈ H (Ω) . (3.1)Moreover, k U k W , (Ω) ≤ c (cid:0) k a k W / , ( ∂ Ω) + k f k L (Ω) (cid:1) . (3.2)By weak solution of problem (1.2) we understand a function u such that w = u − U ∈ H (Ω) and ν Z Ω ∇ w · ∇ η dx − Z Ω (cid:0) ( w + U ) · ∇ (cid:1) η · w dx − Z Ω (cid:0) w · ∇ (cid:1) η · U dx = Z Ω (cid:0) U · ∇ (cid:1) η · U dx ∀ η ∈ H (Ω) . (3.3)Let us reproduce shortly the contradiction argument of Leray [21] whichwas 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 H (Ω) with a compactoperator. Therefore, by the Leray–Schauder theorem, to prove the existenceof a weak solution to Navier–Stokes problem (1.2), it is sufficient to showthat all the solutions of the integral identity ν Z Ω ∇ w · ∇ η dx − λ Z Ω (cid:0) ( w + U ) · ∇ (cid:1) η · w dx − λ Z Ω (cid:0) w · ∇ (cid:1) η · U dx = λ Z Ω (cid:0) U · ∇ (cid:1) η · U dx ∀ η ∈ H (Ω) (3.4)are uniformly bounded in H (Ω) (with respect to λ ∈ [0 , { λ k } k ∈ N ⊂ [0 ,
1] and { b w k } k ∈ N ∈ H (Ω)such that ν Z Ω ∇ b w k · ∇ η dx − λ k Z Ω (cid:0) ( b w k + U ) · ∇ (cid:1) η · b w k dx − λ k Z Ω (cid:0) b w k · ∇ (cid:1) η · U dx λ k Z Ω (cid:0) U · ∇ (cid:1) η · U dx ∀ η ∈ H (Ω) , (3.5)and lim k →∞ λ k = λ ∈ [0 , , lim k →∞ J k = lim k →∞ k b w k k H (Ω) = ∞ . (3.6)Using well known techniques ([14], [1]), one shows that there exist b p k with k b p k k W ,q (Ω) ≤ C ( q ) J k , q ∈ [1 , (cid:0)b u k = b w k + U , b p k (cid:1) is a solution to the following system − ν ∆ b u k + λ k (cid:0)b u k · ∇ (cid:1)b u k + ∇ b p k = f in Ω , div b u k = 0 in Ω , b u k = a on ∂ Ω . (3.7)Choose η = J − k b w k in (3.5) and set w k = J − k b w k . Taking into accountthat Z Ω (cid:0) ( w k + U ) · ∇ (cid:1) w k · w k dx = 0 , we have ν Z Ω |∇ w k | dx = λ k Z Ω (cid:0) w k · ∇ (cid:1) w k · U dx + J − k λ k Z Ω (cid:0) U · ∇ (cid:1) w k · U dx. (3.8)Since k w k k H (Ω) = 1, there exists a subsequence { w k l } converging weakly in H (Ω) to a vector field v ∈ H (Ω). By the compact embedding H (Ω) ֒ → L r (Ω) ∀ r ∈ [1 , ∞ ) , the subsequence { w k l } converges strongly in L r (Ω). Therefore, letting k l →∞ in equality (3.8), we obtain ν = λ Z Ω (cid:0) v · ∇ (cid:1) v · U dx. (3.9) The uniform estimates for the norms k p k k W ,q (Ω) follow from well-known results con-cerning regularity of solutions to the Stokes problem (see [31, Chapter 1, § p k ∈ W ,q loc (Ω) because ∂ Ω has been assumed to beonly Lipschitz. However, for domains Ω with C -smooth boundary and a ∈ W / , ( ∂ Ω)the corresponding estimates hold globally.
9n particular, λ >
0, so λ k are separated from zero.Put ν k = ( λ k J k ) − ν . Multiplying identities (3.7) by λ k J k = λ k ν k ν , we seethat the pair (cid:0) u k = J k b u k , p k = λ k J k b p k (cid:1) satisfies the following system − ν k ∆ u k + (cid:0) u k · ∇ (cid:1) u k + ∇ p k = f k in Ω , div u k = 0 in Ω , u k = a k on ∂ Ω , (3.10)where f k = λ k ν k ν f , a k = λ k ν k ν a , the norms k u k k W , (Ω) and k p k k W ,q (Ω) areuniformly bounded for each q ∈ [1 , u k ∈ W , (Ω), p k ∈ W , (Ω) , and u k ⇀ v in W , (Ω) , p k ⇀ p in W ,q (Ω). Moreover, the limit functions( v , p ) satisfy the Euler system (cid:0) v · ∇ (cid:1) v + ∇ p = 0 in Ω , div v = 0 in Ω , v = 0 on ∂ Ω . (3.11)In conclusion, we can state the following lemma. Lemma 3.1.
Assume that Ω ⊂ R is a bounded domain of type (1.1) with C -smooth boundary ∂ Ω , f = ∇ ⊥ b , b ∈ W , (Ω) , and a ∈ W / , ( ∂ Ω) satisfiescondition (1.3) . If there are no weak solutions to (1.2) , then there exist v , p with the following properties.(E) v ∈ W , (Ω) , p ∈ W ,q (Ω) , q ∈ (1 , , and the pair (cid:0) v , p (cid:1) satisfiesthe Euler system (3.11).(E-NS) Conditions (E) are satisfied and there exist sequences of functions u k ∈ W , (Ω) , p k ∈ W ,q (Ω) and numbers ν k → , λ k → λ > such thatthe norms k u k k W , (Ω) , k p k k W ,q (Ω) are uniformly bounded for every q ∈ [1 , ,the pairs ( u k , p k ) satisfy (3.10) with f k = λ k ν k ν f , a k = λ k ν k ν a , and k∇ u k k L (Ω) → , u k ⇀ v in W , (Ω) , p k ⇀ p in W ,q (Ω) ∀ q ∈ [1 , . Moreover, u k ∈ W , (Ω) , p k ∈ W , (Ω) . From now on we assume that assumptions (E-NS) are satisfied. Our goalis to prove that they lead to a contradiction. This implies the validity ofTheorem 1.1. The interior regularity of the solution depends on the regularity of f ∈ W , (Ω), butnot on the regularity of the boundary value a , see [20]. .2 Some previous results on the Euler equations In this subsection we collect the information on the limit solution (cid:0) v , p (cid:1) 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 b p , . . . , b p N such that p ( x ) ≡ b p j for H − almost all x ∈ Γ j . (3.12) Corollary 3.1.
If conditions (E-NS) are satisfied, then − νλ = N X j =0 b p j Z Γ j a · n ds = N X j =0 b p j F j . (3.13) Proof.
By simple calculations from (3.9) and (3.11 ) it follows νλ = − Z Ω ∇ p · U dx = − Z Ω div( p U ) dx = − Z ∂ Ω p a · n ds. In virtue of (3.12), this implies (3.13).Set Φ k = p k + | u k | , Φ = p + | v | . From (3.11 ) and (3.11 ) it followsthat there exists a stream function ψ ∈ W , (Ω) such that ∇ ψ ≡ v ⊥ in Ω . (3.14)Here and henceforth we set ( a, b ) ⊥ = ( − b, a ).Applying Lemmas 2.1, 2.2 and Remark 2.2 to the functions v , ψ, Φ weget the following
Lemma 3.2.
If conditions (E) are satisfied, then the stream function ψ iscontinuous on Ω and there exists a set A v ⊂ Ω such that (i) H ( A v ) = 0 ; (ii) for all x ∈ Ω \ A v lim r → − Z B r ( x ) | v ( z ) − v ( x ) | dz = lim r → − Z B r ( x ) | Φ( z ) − Φ( x ) | dz = 0; moreover, the function ψ is differentiable at x and ∇ ψ ( x ) = ( − v ( x ) , v ( x )) ; (iii) for every ε > there exists a set U ⊂ R with H ∞ ( U ) < ε such that A v ⊂ U and the functions v , Φ are continuous in Ω \ U . Theorem 3.2.
Let conditions (E) be satisfied and let A v ⊂ Ω be the set fromLemma 3.2. For any compact connected set K ⊂ Ω the following propertyholds: if ψ (cid:12)(cid:12) K = const , (3.15) then Φ( x ) = Φ( x ) for all x , x ∈ K \ A v . (3.16) Lemma 3.3.
If conditions (E) are satisfied, then there exist constants ξ , . . . , ξ N ∈ R such that ψ ( x ) ≡ ξ j on each component Γ j , j = 0 , . . . , N . Proof.
Consider any boundary component Γ j . Since ψ is continuouson Ω and Γ j is connected, we have that ψ (Γ j ) is also a connected set. Onthe other hand, since ∇ ψ ( x ) = 0 for H -almost all x ∈ Γ j (see (3.11 ) and(3.14) ), Theorem 2.2 (i)–(ii) yields H ( ψ (Γ j )) = 0. Therefore, ψ (Γ j ) is asingleton.For x ∈ Ω denote by K x the connected component of the level set { z ∈ Ω : ψ ( z ) = ψ ( x ) } containing the point x . By Lemma 3.3, K x ∩ ∂ Ω = ∅ for every y ∈ ψ (Ω) \{ ξ , . . . , ξ N } and for every x ∈ ψ − ( y ). Thus, Theorem 2.2 (ii), (iv)implies that for almost all y ∈ ψ (Ω) and for every x ∈ ψ − ( y ) the equality K x ∩ A v = ∅ holds and the component K x ⊂ Ω is a C – curve homeomorphicto the circle. We call such K x 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 { Φ k | S } converges to Φ | S uniformly Φ k | S ⇒ Φ | S on almost all admissible cycles S . Admissible cycles S from Lemma 3.4 will be called regular cycles . “Almost all cycles” means cycles in preimages ψ − ( y ) for almost all values y ∈ ψ (Ω). .3 Obtaining a contradiction We consider two cases.(a) The maximum of Φ is attained on the boundary ∂ Ω:max j =0 ,...,N b p j = ess sup x ∈ Ω Φ( x ) . (3.17)(b) The maximum of Φ is not attained on ∂ Ω:max j =0 ,...,N b p j < ess sup x ∈ Ω Φ( x ) . (3.18) Φ is attained on the boundary ∂ ΩLet (3.17) hold. Adding a constant to the pressure we can assume, withoutloss of generality, thatmax j =0 ,...,N b p j = ess sup x ∈ Ω Φ( x ) = 0 . (3.19)In particular,Φ( x ) ≤ . (3.20)If b p = b p = · · · = b p N , then by Corollary 3.1 and the flux condition (1.3),we immediately obtain the required contradiction. Thus, assume thatmin j =0 ,...,N b p j < . (3.21)Change (if necessary) the numbering of the boundary components Γ , Γ ,. . . , Γ N in such a way that b p j < , j = 0 , . . . , M, (3.22) b p M +1 = · · · = b p N = 0 . (3.23)First, we introduce the main idea of the proof in a heuristic way. It iswell known that every Φ k satisfies the linear elliptic equation∆Φ k = ω k + 1 ν k div(Φ k u k ) − ν k f k · u k (3.24) The case ess sup x ∈ Ω Φ( x ) = + ∞ is not excluded. f k = 0, then by Hopf’s maximum principle, in a subdomain Ω ′ ⋐ Ω with C – smooth boundary ∂ Ω ′ the maximum of Φ k is attained at the boundary ∂ Ω ′ , and if x ∗ ∈ ∂ Ω ′ is a maximum point, then the normal derivative of Φ k at x ∗ is strictly positive. It is not sufficient to apply this property directly.Instead we will use some ”integral analogs” that lead to a contradictionby using the the Coarea formula (see Lemmas 3.8–3.9). For i ∈ N andsufficiently large k ≥ k ( i ) we construct a set E i ⊂ Ω consisting of levellines of Φ k such that Φ k | E i → i → ∞ and E i separates the boundarycomponent Γ N (where Φ = 0) from the boundary components Γ j with j =0 , . . . , M (where Φ < R E i |∇ Φ k | . On the other hand, elliptic equation (3.24) for Φ k ,the convergence f k →
0, and boundary conditions (3.10 ) allow us to estimate R E i |∇ Φ k | from above (see Lemma 3.8), and this asymptotically contradictsthe previous one.The main idea of the proof for a general multiply connected domain is thesame as in the case of annulus–like domains (when ∂ Ω = Γ ∪ Γ ). The proofhas an analytical nature and unessential differences concern only well knowngeometrical properties of level sets of continuous functions of two variables.First of all, we need some information concerning the behavior of thelimit total head pressure Φ on stream lines. We do not know whether thefunction Φ is continuous or not on Ω. But we shall prove that Φ has somecontinuity properties on stream lines.By Remark 2.4 and Lemma 3.3, we can apply Kronrod’s results to thestream function ψ . Define the total head pressure on the Kronrod tree T ψ (see Subsection 2.3 ) as follows. Let K ∈ T ψ with diam K >
0. Take any x ∈ K \ A v and put Φ( K ) = Φ( x ). This definition is correct by Bernoulli’sLaw (see Theorem 3.2). Lemma 3.5.
Let
A, B ∈ T ψ , diam A > , diam B > . Consider the corre-sponding arc [ A, B ] ⊂ T ψ joining A to B ( see Lemmas . − . . Then therestriction Φ | [ A,B ] is a continuous function. Proof.
Put (
A, B ) = [
A, B ] \ { A, B } . Let C i ∈ ( A, B ) and C i → C in T ψ . By construction, each C i is a connected component of the level set of ψ and the sets A, B lie in different connected components of R \ C i . Therefore,diam( C i ) ≥ min(diam( A ) , diam( B )) > . (3.25)14y the definition of convergence in T ψ , we havesup x ∈ C i dist( x, C ) → i → ∞ . (3.26)By Theorem 3.2, there exist constants c i ∈ R such that Φ( x ) ≡ c i for all x ∈ C i \ A v , where H ( A v ) = 0. Analogously, Φ( x ) ≡ c for all x ∈ C \ A v .If c i c , then we can assume, without loss of generality, that c i → c ∞ = c as i → ∞ (3.27)and the components C i converge as i → ∞ in the Hausdorff metric to someset C ′ ⊂ C . Clearly, diam( C ′ ) >
0. Take a straight line L such thatthe projection of C ′ on L is not a singleton. Since C ′ is a connected set,this projection is a segment. Let I be the interior of this segment. For z ∈ I by L z denote the straight line such that z ∈ L z and L z ⊥ L . FromLemma 3.2 (i), (iii) it follows that L z ∩ A v = ∅ for H -almost all z ∈ I , andthe restriction Φ | Ω ∩ L z is continuous. Fix a point z ∈ I with above properties.Then by construction C i ∩ L z = ∅ for sufficiently large i . Now, take a sequence y i ∈ C i ∩ L z and extract a convergent subsequence y i j → y ∈ C ′ . SinceΦ | Ω ∩ L z is continuous, we have Φ( y i j ) = c i j → Φ( y ) = c as j → ∞ . Thiscontradicts (3.27).For the velocities u k = ( u k , u k ) and v = ( v , v ) denote by ω k and ω thecorresponding vorticities: ω k = ∂ u k − ∂ u k , ω = ∂ v − ∂ v = ∆ ψ . Thefollowing formulas are direct consequences of (3 . . ∇ Φ ≡ ω v ⊥ = ω ∇ ψ, ∇ Φ k ≡ − ν k ∇ ⊥ ω k + ω k u ⊥ k + f k in Ω . (3.28)We say that a set Z ⊂ T ψ has T -measure zero if H ( { ψ ( C ) : C ∈ Z} ) = 0.The function Φ | T ψ has some analogs of Luzin’s N -property. Lemma 3.6.
Let
A, B ∈ T ψ with diam( A ) > , diam( B ) > . If Z ⊂ [ A, B ] has T -measure zero, then H ( { Φ( C ) : C ∈ Z} ) = 0 . The Hausdorff distance d H between two compact sets A, B ⊂ R n is defined as fol-lows: d H ( A, B ) = max (cid:0) sup a ∈ A dist( a, B ) , sup b ∈ B dist( b, A ) (cid:1) (see, e.g., § A i ⊂ R n there exists a subsequence A i j which converges to some compact set A with respect tothe Hausdorff distance. Of course, if all A i are compact connected sets and diam A i ≥ δ for some δ >
0, then the limit set A is also connected and diam A ≥ δ . roof. Recall that the Coarea formula Z E |∇ f | dx = Z R H ( E ∩ f − ( y )) dy (3.29)holds for a measurable set E and the best representative (see Lemma 2.1) ofany Sobolev function f ∈ W , (Ω) (see, e.g., [22]).Now, let Z ⊂ [ A, B ] have T -measure zero. Set E = ∪ C ∈Z C . Then bydefinition H ( ψ ( E )) = 0. Take a Borel set G ⊃ ψ ( E ) with H ( G ) = 0 andput Z ′ = { C ∈ [ A, B ] : ψ ( C ) ∈ G } , E ′ = ∪ C ∈Z ′ C . Then E ′ is a Borel setas well and E ′ ⊃ E . Hence, by Coarea formula (3.29) applied to ψ | E ′ we seethat ∇ ψ ( x ) = 0 for H -almost all x ∈ E ′ . Then by (3.28), ∇ Φ( x ) = 0 for H -almost all x ∈ E . Applying the Coarea formula to Φ | E ′ , we obtain0 = Z E ′ |∇ Φ | dx = Z R X C ∈Z ′ : Φ( C )= y H ( C ) dy. Since H ( C ) ≥ min (cid:0) diam( A ) , diam( B ) (cid:1) > C ∈ [ A, B ], we have H ( { Φ( C ) : C ∈ Z ′ } ) = 0 and this implies the assertion of Lemma 3.6.From Lemmas 3.4 and 3.6 we have Corollary 3.2. If A, B ∈ T ψ with diam( A ) > , diam( B ) > , then H (cid:0) { Φ( C ) : C ∈ [ A, B ] and C is not a regular cycle } (cid:1) = 0 . Denote by B , . . . , B N the elements of T ψ such that B j ⊃ Γ j , j = 0 , . . . , N .By virtue of Lemma 3.3, every element C ∈ [ B i , B j ] \ { B i , B j } is a connectedcomponent of a level set of ψ such that the sets B i , B j lie in different con-nected components of R \ C .Put α = max j =0 ,...,M min C ∈ [ B j ,B N ] Φ( C ) . By (3.22), α <
0. Take a sequence of positive values t i ∈ (0 , − α ), i ∈ N ,with t i +1 = t i and such that the implicationΦ( C ) = − t i ⇒ C is a regular cycleholds for every j = 0 , . . . , M and for all C ∈ [ B j , B N ]. The existence of theabove sequence follows from Corollary 3.2.16onsider the natural order on the arc [ C j , B N ], namely, C ′ ≤ C ′′ if C ′′ iscloser to B N than C ′ . For j = 0 , . . . M and i ∈ N put A ji = max { C ∈ [ B j , B N ] : Φ( C ) = − t i } . In other words, A ji is an element of the set { C ∈ [ B j , B N ] : Φ( C ) = − t i } which is closest to Γ N . By construction, each A ji is a regular cycle (see Fig. 1for the case of annulus type domains ( N = 1) ).Denote by V i the connected component of the open set Ω \ (cid:0) ∪ Mj =0 A ji (cid:1) suchthat Γ N ⊂ ∂V i . By construction, the sequence of domains V i is decreasing,i.e., V i ⊃ V i +1 . Hence, the sequence of sets ( ∂ Ω) ∩ ( ∂V i ) is nonincreasing:( ∂ Ω) ∩ ( ∂V i ) k ( ∂ Ω) ∩ ( ∂V i +1 ) . Every set ( ∂ Ω) ∩ ( ∂V i ) consists of several components Γ l with l > M (sincearcs ∪ Mj =0 A ji separate Γ N from Γ , . . . , Γ M , but not necessary from other Γ l ).Since there are only finitely many components Γ l , we conclude that for suf-ficiently large i the set ( ∂ Ω) ∩ ( ∂V i ) is independent of i . So we may as-sume, without loss of generality, that ( ∂ Ω) ∩ ( ∂V i ) = Γ K ∪ · · · ∪ Γ N , where K ∈ { M + 1 , . . . , N } . Therefore, ∂V i = A i ∪ · · · ∪ A Mi ∪ Γ K ∪ · · · ∪ Γ N . (3.30)From Lemma 3.4 we have the uniform convergence Φ k | A ji ⇒ Φ( A ji ) = − t i as k → ∞ . Thus for every i ∈ N there exists k i such that for all k ≥ k i Φ k | A ji < − t i , Φ k | A ji +1 > − t i ∀ j = 0 , . . . , M. (3.31)Then ∀ t ∈ (cid:2) t i , t i (cid:3) ∀ k ≥ k i Φ k | A ji < − t, Φ k | A ji +1 > − t ∀ j = 0 , . . . , M. (3.32)For k ≥ k i and t ∈ [ t i , t i ] denote by W ik ( t ) the connected componentof the open set { x ∈ V i \ V i +1 : Φ k ( x ) > − t } such that ∂W ik ( t ) ⊃ A i +1 andput S ik ( t ) = ( ∂W ik ( t )) ∩ V i \ V i +1 . Clearly, Φ k ≡ − t on S ik ( t ). Since theset S ik ( t ) cannot separate A i +1 from A ji +1 for j = 1 , . . . M (indeed, by (3.30)applied to V i +1 we can join A i +1 and A ji +1 by arcs in V i +1 ⊂ R \ S ik ( t ) ), wehave in addition ∂W ik ( t ) ⊃ A ji +1 . Finally, we get ∂W ik ( t ) = S ik ( t ) ∪ A i +1 ∪ · · · ∪ A Mi +1 (3.33)17see Fig. 1). Since by (E–NS) each Φ k belongs to W , (Ω), by the Morse-Sardtheorem for Sobolev functions (see Theorem 2.2) we have that for almost all t ∈ [ t i , t i ] the level set S ik ( t ) consists of finitely many C -cycles and Φ k isdifferentiable (in classical sense) at every point x ∈ S ik ( t ) with ∇ Φ k ( x ) = 0.The values t ∈ [ t i , t i ] having the above property will be called ( k, i )- regular .By construction, Z S ik ( t ) ∇ Φ k · n ds = − Z S ik ( t ) |∇ Φ k | ds < , (3.34)where n is the unit outward (with respect to W ik ( t )) normal vector to ∂W ik ( t ).For h > h = { x ∈ Ω : dist( x, Γ K ∪ · · · ∪ Γ N ) = h ) } , Ω h = { x ∈ Ω : dist( x, Γ K ∪ · · · ∪ Γ N ) < h ) } . By elementary results of analysis, there is aconstant δ > h ≤ δ the set Γ h is a union of N − K + 1 C -smooth curves homeomorphic to the circle, and H (Γ h ) ≤ C ∀ h ∈ (0 , δ ] , (3.35)where C = 3 H (Γ K ∪ · · · ∪ Γ N ) is independent of h . G i A t k -
For any i ∈ N there exist constants ε i > , δ i ∈ (0 , δ ) and k ′ i ∈ N such that R V i +1 \ Ω δi ω k dx > ε i for all k ≥ k ′ i . The key step is the following estimate.
Lemma 3.8.
For any i ∈ N there exists k ( i ) ∈ N such that the inequality Z S ik ( t ) |∇ Φ k | ds < F t (3.36) holds for every k ≥ k ( i ) and for almost all t ∈ [ t i , t i ] , where the constant F is independent of t, k and i . Proof.
Fix i ∈ N and assume k ≥ k i (see (3.31) ). Take a sufficientlysmall σ > σ will be specified below). We choose theparameter δ σ ∈ (0 , δ i ] (see Lemma 3.7) small enough to satisfy the followingconditions:Ω δ σ ∩ A ji = Ω δ σ ∩ A ji +1 = ∅ , j = 0 , . . . , M, (3.37) Z Γ h Φ ds < σ ∀ h ∈ (0 , δ σ ] , (3.38) − σ < Z Γ h ′ Φ k ds − Z Γ h ′′ Φ k ds < σ ∀ h ′ , h ′′ ∈ (0 , δ σ ] ∀ k ∈ N . (3.39)The last estimate follows from the fact that for any q ∈ (1 ,
2) the norms k Φ k k W ,q (Ω) are uniformly bounded. Consequently, the norms k Φ k ∇ Φ k k L q (Ω) are uniformly bounded as well. In particular, for q = 6 / (cid:12)(cid:12)(cid:12)(cid:12)Z Γ h ′ Φ k ds − Z Γ h ′′ Φ k ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Ω h ′′ \ Ω h ′ | Φ k | · |∇ Φ k | dx ≤ (cid:18) Z Ω h ′′ \ Ω h ′ | Φ k ∇ Φ k | / dx (cid:19) meas(Ω h ′′ \ Ω h ′ ) → h ′ , h ′′ → . k ⇀ Φ in the space W ,q (Ω), q ∈ (1 , k | Γ h ⇒ Φ | Γ h as k → ∞ for almost all h ∈ (0 , δ σ )(see [1], [14] )From the last fact and (3.38)–(3.39) we see that there exists k ′ ∈ N such that Z Γ h Φ k ds < σ ∀ h ∈ (0 , δ σ ] ∀ k ≥ k ′ . (3.40)Obviously, for a function g ∈ W , (Ω) and for an arbitrary C -cycle S ⊂ Ωwe have Z S ∇ ⊥ g · n ds = Z S ∇ g · l ds = 0 , where l is the tangent vector to S . Consequently, by (3.28), Z S ∇ Φ k · n ds = Z S ω k u ⊥ k · n ds (recall, that by our assumptions f = ∇ ⊥ b ).Now, fix a sufficiently small ε > ε will be specifiedbelow). For a given sufficiently large k ≥ k ′ we make a special procedure tofind a number ¯ h k ∈ (0 , δ σ ) such that the estimates (cid:12)(cid:12)(cid:12)(cid:12) Z Γ ¯ hk ∇ Φ k · n ds (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) Z Γ ¯ hk ω k u ⊥ k · n ds (cid:12)(cid:12)(cid:12)(cid:12) < ε, (3.41) Z Γ ¯ hk | u k | ds < C ( ε ) ν k (3.42)hold, where the constant C ( ε ) is independent of k and σ . To this enddefine a sequence of numbers 0 = h < h < h < . . . by the recurrentformulas Z U j |∇ u k | · | u k | dx = ν k , (3.43) In [1] Amick proved the uniform convergence Φ k ⇒ Φ on almost all circles. However,his method can be easily modified to prove the uniform convergence on almost all levellines of every C -smooth function with nonzero gradient. Such modification was done inthe proof of Lemma 3.3 of [14]. U j = { x ∈ Ω : dist( x, Γ K ∪ · · · ∪ Γ N ) ∈ ( h j − , h j ) } .Since R ∂ Ω | u k | ds = ( λ k ν k ) ν k a k L ( ∂ Ω) , where λ k ∈ (0 , Z Γ h | u k | ds ≤ Cjν k ∀ h ∈ ( h j − , h j ) , (3.44)where C is independent of k, j, σ . Consequently, Z U j | u k | dx ≤ ( h j − h j − ) Cjν k . (3.45)Using this estimates and applying the H¨older inequality to (3.43), we obtain ν k = Z U j |∇ u k | · | u k | dx ≤ q ( h j − h j − ) Cjν k (cid:18)Z U j |∇ u k | dx (cid:19) . (3.46)Squaring both sides of the last inequality, we have ν k h j − h j − ≤ C j Z U j |∇ u k | dx. (3.47)We define h j for j = 1 , . . . , j max , where j max is the first index satisfying atleast one of the following two conditions. Stop case 1. h j max − < δ σ , h j max ≥ δ σ ,or Stop case 2. Cj max R U j max |∇ u k | dx < ε .By construction, R U j |∇ u k | dx ≥ Cj ε for every j < j max (since for j
0. Estimate (3.36) is proved.Now, we receive the required contradiction using the Coarea formula.23 emma 3.9.
Assume that Ω ⊂ R is a bounded domain of type (1.1) with C -smooth boundary ∂ Ω , f ∈ W , (Ω) , and a ∈ W / , ( ∂ Ω) satisfies condi-tion (1.3) . Then assumptions (E-NS) and (3.17) lead to a contradiction. Proof.
For i ∈ N and k ≥ k ( i ) (see Lemma 3.8) put E i = [ t ∈ [ t i , t i ] S ik ( t ) . By the Coarea formula (3.29) (see also [22]), for any integrable function g : E i → R the equality Z E i g |∇ Φ k | dx = t i Z t i Z S ik ( t ) g ( x ) d H ( x ) dt (3.57)holds. In particular, taking g = |∇ Φ k | and using (3.36), we obtain Z E i |∇ Φ k | dx = t i Z t i Z S ik ( t ) |∇ Φ k | ( x ) d H ( x ) dt ≤ t i Z t i F t dt = F ′ t i (3.58)where F ′ = F is independent of i . Now, taking g = 1 in (3.57) and usingthe H¨older inequality we have t i Z t i H (cid:0) S ik ( t ) (cid:1) dt = Z E i |∇ Φ k | dx ≤ (cid:18)Z E i |∇ Φ k | dx (cid:19) (cid:0) meas( E i ) (cid:1) ≤ √F ′ t i (cid:0) meas( E i ) (cid:1) . (3.59)By construction, for almost all t ∈ [ t i , t i ] the set S ik ( t ) is a fi-nite union of smooth cycles and S ik ( t ) separates A ji from A ji +1 for j = 0 , . . . , M . Thus, each set S ik ( t ) separates Γ j from Γ N . In particular, H ( S ik ( t )) ≥ min (cid:0) diam(Γ j ) , diam(Γ N ) (cid:1) . Hence, the left integral in (3.59)is greater than Ct i , where C > i . On the otherhand, evidently, meas( E i ) ≤ meas (cid:0) V i \ V i +1 (cid:1) → i → ∞ . The obtainedcontradiction finishes the proof of Lemma 3.9.24 .3.2 The maximum of Φ is not attained at ∂ ΩIn this subsection we consider the case (b), when (3.18) holds. Adding aconstant to the pressure, we assume, without loss of generality, thatmax j =0 ,...,N b p j < ess sup x ∈ Ω Φ( x ) = 0 . (3.60)Denote σ = max j =0 ,...,N b p j < T ψ . In particular, Lemmas 3.5–3.6 hold. Lemma 3.10.
There exists F ∈ T ψ such that diam F > , F ∩ ∂ Ω = ∅ , and Φ( F ) > σ . Proof.
By assumptions, Φ( x ) ≤ σ for every x ∈ ∂ Ω \ A v and there is aset of a positive measure E ⊂ Ω such that Φ( x ) > σ at each x ∈ E . In virtueof Theorem 3.2 (iii), there exists a straight-line segment I = [ x , y ] ⊂ Ωwith I ∩ A v = ∅ , x ∈ ∂ Ω, y ∈ E , such that Φ | I is a continuous function.By construction, Φ( x ) ≤ σ , Φ( y ) ≥ σ + δ with some δ >
0. Take asubinterval I = [ x , y ] ⊂ Ω such that Φ( x ) = σ + δ and Φ( x ) ≥ σ + δ for each x ∈ [ x , y ]. Then by Bernoulli’s Law (see Theorem 3.2) ψ = conston I . Hence, we can take x ∈ I such that the preimage ψ − ( ψ ( x )) consistsof a finite union of regular cycles (see Lemma 3.4). Denote by F the regularcycle containing x . Then by construction Φ( F ) ≥ σ + δ and by definitionof regular cycles diam F > F ∩ ∂ Ω = ∅ .Fix F from above Lemma and consider the behavior of Φ on the Kronrodarcs [ B j , F ], j = 0 , . . . N (recall, that by B j we denote the elements of T ψ such that Γ j ⊂ B j ). The rest part of this subsection is similar to that ofSubsection 3.3.1 with the following difference: F plays now the role whichwas played before by B N , and the calculations become easier since F liesstrictly inside Ω.By construction, Φ( F ) > Φ( B j ) for each j = 0 , . . . , N . So, using Lem-mas 3.5–3.6 and Corollary 3.2 we can find a sequence of positive numbers t i ∈ ( − Φ( F ) , − σ ), i ∈ N , with t i +1 = t i , and the corresponding regularcycles A ji ∈ [ B j , F ], j = 0 , . . . , N , with Φ( A ji ) = − t i . Denote by V i the con-nected component of the set Ω \ ( A i ∪· · ·∪ A Ni ) containing F . By construction, V i ⊂ Ω, V i ⊂ V i +1 and ∂V i = A i ∪ · · · ∪ A Ni . (3.61)25y definition of regular cycles (see Lemma 3.4), we again obtain esti-mates (3.31)–(3.32) for k ≥ k i . Accordingly, for k ≥ k i and t ∈ [ t i , t i ]we can define the domain W ik ( t ) as a connected component of the open set { x ∈ V i \ V i +1 : Φ k ( x ) > − t } with ∂W ik ( t ) = S ik ( t ) ∪ A i +1 ∪ · · · ∪ A Ni +1 , (3.62)where the set S ik ( t ) = ( ∂W ik ( t )) ∩ V i \ V i +1 ⊂ { x ∈ V i : Φ k ( x ) = − t } separates A i ∪ · · · ∪ A Ni from A i +1 ∪ · · · ∪ A Ni +1 . By the Morse-Sard theorem(see Theorem 2.2) applied to Φ k ∈ W , (Ω), for almost all t ∈ [ t i , t i ] thelevel set S ik ( t ) consists of finitely many C -cycles. Moreover, by construction, Z S ik ( t ) ∇ Φ k · n ds = − Z S ik ( t ) |∇ Φ k | ds < , (3.63)where n is the unit outward normal vector to ∂W ik ( t ). As before, we callsuch values t ∈ [ t i , t i ] ( k, i ) -regular .Since Φ = const on V i , from (3.28) it follows that R V i ω dx > i ,and taking into account the weak convergence ω k ⇀ ω in L (Ω) we get Lemma 3.11.
For every i ∈ N there exist constants ε i > , δ i ∈ (0 , δ ) and k ′ i ∈ N such that R V i +1 ω k dx > ε i for all k ≥ k ′ i . Now, we can prove
Lemma 3.12.
Assume that Ω ⊂ R is a bounded domain of type (1.1) with C -smooth boundary ∂ Ω , f ∈ W , (Ω) , and a ∈ W / , ( ∂ Ω) satisfiescondition (1.3) . Then assumptions (E-NS) and (3.18) lead to a contradiction. Proof.
The proof of this Lemma is similar to that of Lemma 3.8. How-ever, the situation now is more easy, since we separate V i from the wholeboundary ∂ Ω. Fix i ∈ N and assume that k ≥ k i (see (3.31) ). For a ( k, i )-regular value t ∈ [ t i , t i ] consider the domainΩ ik ( t ) = W ik ( t ) ∪ V i +1 .
26y construction, ∂ Ω ik ( t ) = S ik ( t ). Integrating identity (3.53) over Ω ik ( t ), weobtain 0 > Z S ik ( t ) ∇ Φ k · n ds = Z Ω ik ( t ) ω k dx + 1 ν k Z S ik ( t ) Φ k u k · n ds − ν k Z Ω ik ( t ) f k · u k dx = Z Ω ik ( t ) ω k dx − tν k Z S ik ( t ) u k · n ds − ν k Z Ω ik ( t ) f k · u k dx = Z Ω ik ( t ) ω k dx − ν k Z Ω ik ( t ) f k · u k dx, (3.64)and, as before, we have a contradiction with Lemma 3.11. Proof of Theorem 1.1.
Let the hypotheses of Theorem 1.1 be satisfied.Suppose that its assertion fails. Then, by Lemma 3.1, there exist v , p anda sequence ( u k , p k ) satisfying (E-NS), and by Lemmas 3.12 and 3.9 theseassumptions lead to a contradiction. First, let us specify some notations. Let O x , O x , O x be coordinate axis in R and θ = arctg( x /x ), r = ( x + x ) / , z = x be cylindrical coordinates.Denote by v θ , v r , v z the projections of the vector v on the axes θ, r, z .A function f is said to be axially symmetric if it does not depend on θ .A vector-valued function h = ( h r , h θ , h z ) is called axially symmetric if h r , h θ and h z do not depend on θ . A vector-valued function h ′ = ( h r , h θ , h z ) iscalled axially symmetric without rotation if h θ = 0 while h r and h z do notdepend on θ .The main result of this section is as follows. Theorem 4.1.
Assume that Ω ⊂ R is a bounded axially symmetric domainof type (1.1) with C -smooth boundary ∂ Ω . If f ∈ W , (Ω) , a ∈ W / , ( ∂ Ω) are axially symmetric and a satisfies condition (1 . , then (1 . admits atleast one weak axially symmetric solution. Moreover, if f and a are axi-ally symmetric without rotation, then (1 . admits at least one weak axiallysymmetric solution without rotation. Using the “reductio ad absurdum” Leray argument (the main idea ispresented in Section 3.1 for the plane case; specific details concerning the27xially symmetric case can be found in [16]), it is possible to prove thefollowing
Lemma 4.1.
Assume that Ω ⊂ R is a bounded axially symmetric domainof type (1.1) with C -smooth boundary ∂ Ω , f = curl b , b ∈ W , (Ω) , a ∈ W / , ( ∂ Ω) are axially symmetric, and a satisfies condition (1 . . If theassertion of Theorem 4.1 is false, then there exist v , p with the followingproperties.(E-AX) The axially symmetric functions v ∈ W , (Ω) , p ∈ W , / (Ω) satisfy the Euler system (3.11) .(E-NS-AX) Condition (E-AX) is satisfied and there exist a sequencesof axially symmetric functions u k ∈ W , (Ω) , p k ∈ W ,q (Ω) and numbers ν k → , λ k → λ > such that the norms k u k k W , (Ω) , k p k k W , / (Ω) areuniformly bounded, the pair ( u k , p k ) satisfies (3.10) with f k = λ k ν k ν f , a k = λ k ν k ν a , and k∇ u k k L (Ω) → , u k ⇀ v in W , (Ω) , p k ⇀ p in W , / (Ω) . (4.1) Moreover, u k ∈ W , (Ω) and p k ∈ W , (Ω) . As in the previous section, in order to prove existence Theorem 4.1, weneed to show that conditions (E-NS-AX) lead to a contradiction.Assume that Γ j ∩ O x = ∅ , j = 0 , . . . , M ′ , Γ j ∩ O x = ∅ , j = M ′ + 1 , . . . , N. Let P + = { (0 , x , x ) : x > , x ∈ R } , D = Ω ∩ P + . Obviously, on P + the coordinates x , x coincide with the coordinates r, z .For a set A ⊂ R put ˘ A := A ∩ P + , and for B ⊂ P + denote by e B the setin R obtained by rotation of B around O z -axis.One can easily see that(S ) D is a bounded plane domain with Lipschitz boundary. Moreover, ˘Γ j is a connected set for every j = 0 , . . . , N . In other words, { ˘Γ j : j = 0 , . . . , N } coincides with the family of all connected components of the set P + ∩ ∂ D .28ence, v and p satisfy the following system in the plane domain D : ∂p∂r − ( v θ ) r + v r ∂v r ∂r + v z ∂v r ∂z = 0 ,∂p∂z + v r ∂v z ∂r + v z ∂v z ∂z = 0 ,v θ v r r + v r ∂v θ ∂r + v z ∂v θ ∂z = 0 ,∂ ( rv r ) ∂r + ∂ ( rv z ) ∂z = 0 (4.2)(these equations are satisfied for almost all x ∈ D ) and v ( x ) = 0 for H -almost all x ∈ P + ∩ ∂ D . (4.3)We have the following integral estimates: v ∈ W , ( D ), Z D r |∇ v ( r, z ) | drdz < ∞ , (4.4)and, by the Sobolev embedding theorem for three–dimensional domains, v ∈ L (Ω), i.e., Z D r | v ( r, z ) | drdz < ∞ . (4.5)Also, the condition ∇ p ∈ L / (Ω) can be written as Z D r |∇ p ( r, z ) | / drdz < ∞ . (4.6) The next statement was proved in [12, Lemma 4] and in [1, Theorem 2.2].
Theorem 4.2.
If conditions (E-AX) are satisfied, then ∀ j ∈ { , , . . . , N } ∃ b p j ∈ R : p ( x ) ≡ b p j for H − almost all x ∈ Γ j . (4.7) In particular, by axial symmetry, p ( x ) ≡ b p j for H − almost all x ∈ ˘Γ j . (4.8)29he following result was obtained in [16]. Theorem 4.3.
If conditions (E-AX) are satisfied, then b p = · · · = b p M ′ ,where b p j are the constants from Theorem 4.2. We need a weak version of Bernoulli’s law for a Sobolev solution ( v , p ) tothe Euler equations (4.2) (see Theorem 4.4 below).From the last equality in (4.2) and from (4.4) it follows that there existsa stream function ψ ∈ W , ( D ) such that ∂ψ∂r = − rv z , ∂ψ∂z = rv r . (4.9)Fix a point x ∗ ∈ D . For ε > D ε the connected component of D ∩ { ( r, z ) : r > ε } containing x ∗ . Since ψ ∈ W , ( D ε ) ∀ ε > , (4.10)by Sobolev embedding theorem, ψ ∈ C ( ¯ D ε ). Hence ψ is continuous at pointsof ¯ D \ O z = ¯ D \ { (0 , z ) : z ∈ R } . Lemma 4.2. [cf. Lemma 3.3]
If conditions (E-AX) are satisfied, then thereexist constants ξ , . . . , ξ N ∈ R such that ψ ( x ) ≡ ξ j on each curve ˘Γ j , j =0 , . . . , N . Proof.
In virtue of (4.3), (4.9), we have ∇ ψ ( x ) = 0 for H -almost all x ∈ ∂ D \ O z . Then the Morse-Sard property (see Theorem 2.2) implies thatfor any connected set C ⊂ ∂ D \ O z ∃ α = α ( C ) ∈ R : ψ ( x ) ≡ α ∀ x ∈ C. Hence, since ˘Γ j are connected (see (S ) ), the lemma follows.Denote by Φ = p + | v | v , p ). Obviously, Z D r |∇ Φ( r, z ) | / drdz < ∞ . (4.11)Hence,Φ ∈ W , / ( D ε ) ∀ ε > . (4.12)Applying Lemmas 2.1, 2.2, and Remark 2.2 to the functions v , ψ, Φ weget the following 30 emma 4.3.
If conditions (E-AX) hold, then there exists a set A v ⊂ D suchthat: (i) H ( A v ) = 0 ; (ii) for all x = ( r, z ) ∈ D \ A v lim ρ → − Z B ρ ( x ) | v ( z ) − v ( x ) | dz = lim ρ → − Z B ρ ( x ) | Φ( z ) − Φ( x ) | dz = 0 , moreover, the function ψ is differentiable at x and ∇ ψ ( x ) =( − rv z ( x ) , rv r ( x )) ; (iii) for every ε > there exists a set U ⊂ R with H ∞ ( U ) < ε , A v ⊂ U ,and such that the functions v , Φ are continuous on D \ ( U ∪ O z ) . The next two results were obtained in [16].
Theorem 4.4 (Bernoulli’s Law) . Let conditions (E-AX) be valid and let A v be a set from Lemma 4.3. For any compact connected set K ⊂ ¯ D \ O z thefollowing property holds: if ψ (cid:12)(cid:12) K = const , (4.13) then Φ( x ) = Φ( x ) for all x , x ∈ K \ A v . (4.14)We also need the following assertion from [16] concerning the behavior ofthe total head pressure near the singularity axis O z . Lemma 4.4.
Assume that conditions (E-AX) are satisfied. Let K i be asequence of compact sets with the following properties: K i ⊂ ¯ D \ O z , ψ | K i =const , and lim i →∞ inf ( r,z ) ∈ K i r = 0 , lim i →∞ sup ( r,z ) ∈ K i r > . Then Φ( K i ) → b p as i → ∞ . Here we denote by Φ( K i ) the corresponding constant c i ∈ R such thatΦ( x ) = c i for all x ∈ K i \ A v (see Theorem 4.4). We consider three possible cases.(a) The maximum of Φ is attained on the boundary component intersect-ing the symmetry axis: b p = max j =0 ,...,N b p j = ess sup x ∈ Ω Φ( x ) . (4.15)31b) The maximum of Φ is attained on a boundary component which doesnot intersect the symmetry axis: b p < b p N = max j =0 ,...,N b p j = ess sup x ∈ ¯Ω Φ( x ) , (4.16)(c) The maximum of Φ is not attained on ∂ Ω:max j =0 ,...,N b p j < ess sup x ∈ Ω Φ( x ) . (4.17) ess sup x ∈ Ω Φ( x ) = b p . Let us consider case (4.15). Adding a constant to the pressure p , we canassume, without loss of generality, that b p = ess sup x ∈ Ω Φ( x ) = 0 . (4.18)Since the identity b p = b p = · · · = b p N is impossible (see Corollary 3.1,which is valid also for the axial-symmetric case), we have that b p j < j ∈ { M ′ + 1 , . . . , N } (recall, that by Theorem 4.3, b p = · · · = b p M ′ = 0 ).Now, we receive a contradiction following the arguments of [16], [15]. Forreader’s convenience, we recall these arguments. From equation (3 . ) weobtain the identity0 = x · ∇ p ( x ) + x · (cid:0) v ( x ) · ∇ (cid:1) v ( x )= div (cid:2) x p ( x ) + (cid:0) v ( x ) · x (cid:1) v ( x ) (cid:3) − p ( x ) div x − | v ( x ) | = div (cid:2) x p ( x ) + (cid:0) v ( x ) · x (cid:1) v ( x ) (cid:3) − x ) + | v ( x ) | . (4.19)Integrating it over Ω, we derive0 ≥ Z Ω (cid:2) x ) − | v ( x ) | (cid:3) dx − = Z ∂ Ω p ( x ) (cid:0) x · n (cid:1) ds = N X j =0 b p j Z Γ j (cid:0) x · n (cid:1) ds = − N X j =1 b p j Z Ω j div x dx = − N X j =1 b p j | Ω j | > . The obtained contradiction finishes the proof for case (4.15).32 .2.2 The case b p < b p N = ess sup x ∈ ¯Ω Φ( x ) . Suppose that (4.16) holds. We may assume, without loss of generality, thatthe maximum value is zero, i.e., b p < b p N = max j =0 ,...,N b p j = ess sup x ∈ ¯Ω Φ( x ) = 0 . (4.20)From Theorem 4.3 we have b p = · · · = b p M ′ < . (4.21)Change (if necessary) the numbering of the boundary components Γ M ′ +1 ,. . . , Γ N − so that b p j < , j = 0 , . . . , M, M ≥ M ′ , (4.22) b p M +1 = · · · = b p N = 0 . (4.23)The first goal is to remove a neighborhood of the singularity line O z from our considerations. Then, we can reduce the proof to the plane caseconsidered in Subsection 3.3.1.Take r > D ε = { ( r, z ) ∈ D : r > ε } is connectedfor every ε ≤ r (i.e., D ε is a domain), and˘Γ j ⊂ D r and inf ( r,z ) ∈ ˘Γ j r ≥ r , j = M ′ + 1 , . . . , N, ˘Γ j ∩ D ε is a connected setand sup ( r,z ) ∈ ˘Γ j ∩ D ε r ≥ r , j = 0 , . . . , M ′ , ε ∈ (0 , r ] . (4.24)Let a set C ⊂ D ε separate ˘Γ i and ˘Γ j in D ε , i.e., ˘Γ i ∩ D ε and ˘Γ j ∩ D ε liein different connected components of D ε \ C . Obviously, for ε ∈ (0 , r ] thereexists a constant δ ( ε ) > ( r,z ) ∈ C r ≥ δ ( ε ) holds(see Fig. 2). Moreover, the function δ ( ε ) is nondecreasing. In particular, δ ( ε ) ≥ δ ( r ) , ε ∈ (0 , r ] . (4.25)By Remark 2.4 and Lemma 4.2, we can apply Kronrod’s results to thestream function ψ | ¯ D ε , ε ∈ (0 , r ]. Accordingly, T ψ,ε means the correspondingKronrod tree for the restriction ψ | ¯ D ε . Define the total head pressure on T ψ,ε as we did in Subsection 3.3.1. Then the following analog of Lemma 3.5 holds33 emma 4.5. Let
A, B ∈ T ψ,ε , where ε ∈ (0 , r ] , diam A > , and diam B > . Consider the corresponding arc [ A, B ] ⊂ T ψ,ε joining A to B ( see Lemmas 2.3, 2.4 ) . Then the restriction Φ | [ A,B ] is a continuous function. The lemma is proved using the argument of Lemma 3.5 and taking intoaccount the above definitions, Theorem 4.4, and the continuity properties ofΦ (see Lemma 4.3 (iii)) ).Denote by B ε , . . . , B εN the elements of T ψ,ε such that B εj ⊃ ˘Γ j ∩ ¯ D ε , j = 0 , . . . , M ′ , and B εj ⊃ ˘Γ j , j = M ′ + 1 , . . . , N . By construction, Φ( B εj ) < j = 0 , . . . , M , and Φ( B εj ) = 0 for j = M + 1 , . . . , N . For r > L r bethe horizontal straight line L r = { ( r, z ) : z ∈ R } . We have Lemma 4.6.
There exist r ∗ ∈ (0 , r ] and C j ∈ [ B r ∗ j , B r ∗ N ] , j = 0 , . . . , M , suchthat Φ( C j ) < and C ∩ L r ∗ = ∅ for all C ∈ [ C j , B r ∗ N ] . Proof.
Suppose that the lemma fails for some j = 0 , . . . , M . Thenit is easy to construct r i → C i ∈ [ B r i j , B r i N ] such that C i ∩ L r i = ∅ and Φ( C i ) →
0. Since by (4.22) b p <
0, we have Φ( C i ) b p . By (4.25),sup ( r,z ) ∈ C i r ≥ δ ( r ). Therefore, we have a contradiction with Lemma 4.4, andthe result is proved.Lemma 4.6 allows us to remove a neighborhood of the singularity line O z from our argument. Thus, we can apply the approach developed in Subsec-tion 3.3.1 for the plane case. Put, for simplicity, T ψ = T ψ,r ∗ and B j = B r ∗ j .Since ∂ D r ∗ ⊂ B ∪ · · · ∪ B N ∪ L r ∗ and the set { B , . . . , B N } ⊂ T ψ is finite,we can change C j (if necessary) so that the assertion of Lemma 4.6 takes thefollowing stronger form: ∀ j = 0 , . . . , M C j ∈ [ B j , B N ] , Φ( C j ) < , (4.26)and C ∩ ∂ D r ∗ = ∅ ∀ C ∈ [ C j , B N ] . (4.27)Observe that Γ j ∩ L r ∗ = ∅ for j = 0 , . . . , M ′ . Therefore, if a cycle C ∈ T ψ separates Γ N from Γ and C ∩ ∂ D r ∗ = ∅ , then C separates Γ N from Γ j for all j = 1 , . . . , M ′ . So we can take C = · · · = C M ′ (see Fig.2) and to consideronly the Kronrod arcs [ C M ′ , B N ], . . . , [ C N , B N ].34 G G G C CC = )( * r d z O * r L Figure 2. The domain D for the case M ′ = 1 , N = 2.Recall that a set Z ⊂ T ψ has T -measure zero, if H ( { ψ ( C ) : C ∈ Z} ) = 0. Lemma 4.7.
For every j = M ′ , . . . , M , T -almost all C ∈ [ C j , B N ] are C -curves homeomorphic to the circle. Moreover, all the functions Φ k | C arecontinuous and the sequence { Φ k | C } converges to Φ | C uniformly: Φ k | C ⇒ Φ | C . The first assertion of the lemma follows from Theorem 2.2 (iv) and(4.27). The validity of the second one for T -almost all C ∈ [ C j , B N ] wasproved in [14, Lemma 3.3].Below we will call regular the cycles C which satisfy the assertion ofLemma 4.7.From Lemmas 4.7 and 3.6 (which is also valid for the axially symmetriccase) we obtain Corollary 4.3.
For each j = M ′ , . . . , M , we have H (cid:0) { Φ( C ) : C ∈ [ C j , B N ] and C is not a regular cycle } (cid:1) = 0 . As in the plane case (see Subsection 3.3.1), we can take a sequence ofpositive values t i with t i +1 = t i , the corresponding regular cycles A ji ∈ [ C j , B N ] with Φ( A ji ) = − t i , and the sequence of domains V i ⊂ D r ∗ with ∂V i = A M ′ i ∪ · · · ∪ A Mi ∪ ˘Γ K ∪ · · · ∪ ˘Γ N , (4.28)35here K ≥ M + 1 is independent of i .By the definition of regular cycles, we have again estimates (3.31)–(3.32)for k ≥ k i . Accordingly, for k ≥ k i and t ∈ [ t i , t i ] we can define the domain W ik ( t ) as a connected component of the open set { x ∈ V i \ V i +1 : Φ k ( x ) > − t } with ∂W ik ( t ) = S ik ( t ) ∪ A M ′ i +1 ∪ · · · ∪ A Mi +1 , (4.29)where the set S ik ( t ) = ( ∂W ik ( t )) ∩ V i \ V i +1 ⊂ { x ∈ V i : Φ k ( x ) = − t } separates A M ′ i ∪ · · · ∪ A Mi from A M ′ i +1 ∪ · · · ∪ A Mi +1 . Since Φ k ∈ W , (Ω) (see (E-NS-AX) ),by the Morse-Sard theorem (see Theorem 2.2), for almost all t ∈ [ t i , t i ] thelevel set S ik ( t ) consists of finitely many C -cycles and Φ k is differentiable (inclassical sense) at every point x ∈ S ik ( t ) with ∇ Φ k ( x ) = 0. Therefore, e S ik ( t )is a finite union of smooth surfaces (tori), and by construction, Z e S ik ( t ) ∇ Φ k · n dS = − Z e S ik ( t ) |∇ Φ k | dS < , (4.30)where n is the unit outward normal vector to ∂ f W ik ( t ) (recall, that for a set B ⊂ P + we denote by e B the set in R obtaining by rotation of B around O z -axis).For h > h = { x ∈ Ω : dist( x, Γ K ∪ · · · ∪ Γ N ) = h ) } , Ω h = { x ∈ Ω : dist( x, Γ K ∪ · · · ∪ Γ N ) < h ) } . Since the distance function dist( x, ∂ Ω) is C –regular and the norm of its gradient is equal to one in the neighborhoodof ∂ Ω, there is a constant δ > h ≤ δ the set Γ h is aunion of N − K + 1 C -smooth surfaces homeomorphic to the torus, and H (Γ h ) ≤ c ∀ h ∈ (0 , δ ] , (4.31)where the constant c = 3 H (Γ K ∪ · · · ∪ Γ N ) is independent of h .By a direct calculation, (4.2) implies ∇ Φ = v × ω in Ω , (4.32)where ω = curl v , i.e., ω = ( ω r , ω θ , ω z ) = (cid:0) − ∂v θ ∂z , ∂v r ∂z − ∂v z ∂r , v θ r + ∂v θ ∂r (cid:1) . Set ω k = curl u k , ω ( x ) = | ω ( x ) | , ω k ( x ) = | ω k ( x ) | . Since Φ = const on V i ,(4.32) implies R e V i ω dx > i . Hence, from the weak convergence ω k ⇀ ω in L (Ω) it follows 36 emma 4.8. For any i ∈ N there exist constants ε i > , δ i ∈ (0 , δ ) and k ′ i ∈ N such that R e V i +1 \ Ω δi ω k dx > ε i for all k ≥ k ′ i . Now we are ready to prove the key estimate.
Lemma 4.9.
For any i ∈ N there exists k ( i ) ∈ N such that for every k ≥ k ( i ) and for almost all t ∈ [ t i , t i ] the inequality Z e S ik ( t ) |∇ Φ k | dS < F t, (4.33) holds with the constant F independent of t, k and i . Proof.
Since the proof of this lemma is similar to that of Lemma 3.8 forthe plane case, we comment only some key steps.Fix i ∈ N . Below we always assume that k ≥ k i (see (3.31) ). Sincewe have removed a neighborhood of the singularity line O z , we can usethe Sobolev embedding theorem in the plane domain D r ∗ . In particular,from the uniform estimate k Φ k k W , / ( D r ∗ ) ≤ const we deduce that the norms k Φ k k L ( D r ∗ ) are uniformly bounded. Consequently, by the H¨older inequality k Φ k ∇ Φ k k L / ( D r ∗ ) ≤ const, and this implies k Φ k ∇ Φ k k L / ( e D r ∗ ) ≤ const . (4.34)Fix a sufficiently small σ > σ will be specified below)and take the parameter δ σ ∈ (0 , δ i ] (see Lemma 4.8) small enough to satisfythe following conditionsΩ δ σ ∩ e A ji = Ω δ σ ∩ e A ji +1 = ∅ , j = M ′ , . . . , M, (4.35) Z Γ h Φ k dS < σ ∀ h ∈ (0 , δ σ ] ∀ k ≥ k ′ . (4.36)(the last estimate follows from the identity Φ | Γ K ∪···∪ Γ N ≡
0, the weak conver-gence Φ k ⇀ Φ in the space W , / (Ω), and (4.34) ).By a direct calculation, (3.10) implies ∇ Φ k = − ν k curl ω k + ω k × u k + f k = − ν k curl ω k + ω k × u k + λ k ν k ν curl b .
37y the Stokes theorem, for any C -smooth closed surface S ⊂ Ω and g ∈ W , (Ω) we have Z S curl g · n dS = 0 . So, in particular, Z S ∇ Φ k · n dS = Z S ( ω k × u k ) · n dS. Now, fix a sufficiently small ε > ε will be specifiedbelow). For a given sufficiently large k ≥ k ′ we make a special procedure tofind a number ¯ h k ∈ (0 , δ σ ) such that the estimates (cid:12)(cid:12)(cid:12)(cid:12) Z Γ ¯ hk ∇ Φ k · n dS (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Γ ¯ hk | u k | · |∇ u k | dS < ε, (4.37) Z Γ ¯ hk | u k | dS ≤ C ( ε ) ν k (4.38)hold, where C ( ε ) is independent of k and σ . This procedure exactly repeatsthe argument lines of the proof of Lemma 3.8.The final part of the proof is identical to that of Lemma 3.8. We haveto integrate formula (3.53) (which is valid for the axially symmetric case aswell) over the three–dimensional domain Ω i ¯ h k ( t ) with ∂ Ω i ¯ h k ( t ) = Γ ¯ h k ∪ e S ik ( t ).This means that we have only to replace the curves S ik ( t ) by the surfaces e S ik ( t ) in the corresponding integrals.Now, we obtain a contradiction by repeating word by word the proofof Lemma 3.9 and replacing the one–dimensional Hausdorff measure by thetwo–dimensional one, and the curves S ik ( t ) by the surfaces e S ik ( t ) in thecorresponding integrals. ess sup x ∈ Ω Φ( x ) > max j =0 ,...,N b p j . Assume that (4.17) is satisfied and set σ = max j =0 ,...,N b p j . Then, as in the proofof Lemma 3.10, we can find a compact connected set F ⊂ D \ A v such that38iam( F ) > ψ | F = const, and Φ( F ) > σ . Without loss of generality, wemay assume that σ < F ) = 0. Since now it is more difficult toseparate F from ∂ D by regular cycles (than in Lemma 3.10), we have toapply the method of Subsection 4.2.2. Namely, take a number r > F ⊂ D r , the open set D ε = { ( r, z ) ∈ D : r > ε } is connected forevery ε ≤ r , and conditions (4.24) are satisfied. Then for ε ∈ (0 , r ] we canconsider the behavior of Φ on the Kronrod trees T ψ,ε corresponding to therestrictions ψ | ¯ D ε . Denote by F ε the element of T ψ,ε containing F . Using thesame procedure as in Subsection 4.2.2, we can find r ∗ ∈ (0 , r ] such that thefollowing lemma holds. Lemma 4.10.
There exist C j ∈ [ B r ∗ j , F r ∗ ] , j = 0 , . . . , N , such that Φ( C j ) < and C ∩ L r ∗ = ∅ for all C ∈ [ C j , F r ∗ ] . Set T ψ = T ψ,r ∗ , F ∗ = F r ∗ , and B j = B r ∗ j , i.e., B j ∈ T ψ and B j ⊃ ˘Γ j ∩ D r ∗ .As above, we can change C j (if necessary) so that Lemma 4.10 takes thefollowing stronger form: ∀ j = 0 , . . . , M C j ∈ [ B j , F ∗ ] , Φ( C j ) < ,C ∩ ∂ D r ∗ = ∅ ∀ C ∈ [ C j , F ∗ ] , and C = · · · = C M ′ . The rest of the procedure of obtaining a contradiction is done in the sameway as in Subsection 3.3.2. Namely, we need to take positive numbers t i =2 − i t , regular cycles A ji ∈ [ C j , F ∗ ] with Φ( A ji ) = − t i , and the set S ik ( t ) withΦ k | S ik ( t ) ≡ − t separating A M ′ i ∪ · · · ∪ A Ni from A M ′ i +1 ∪ · · · ∪ A Ni +1 , etc. Theonly difference is that we have to integrate identity (3.53) over the three–dimensional domains Ω ik ( t ) with ∂ Ω ik ( t ) = e S ik ( t ). Proof of Theorem 4.1.
Let the hypotheses of Theorem 4.1 are satisfied.Suppose that its assertion fails. Then by Lemma 4.1 there exist v , p and asequence ( u k , p k ) satisfying (E-NS-AX). However, in Subsections 4.2.1–4.2.3we have shown that assumptions (E-NS-AX) lead to a contradiction in allpossible cases (4.15)–(4.17). This finishes the proof of Theorem 4.1. Remark 4.1.
Let in Lemma 4.1 the data f and a be axially symmetricwithout rotation. If the corresponding assertion of Theorem 4.1 fails, thenit can be shown (see [16]) that conditions (E-NS-AX) are satisfied with u k Remark 4.2.
It is well know (see [20]) that under hypothesis of Theo-rems 1.1, 4.1, every weak solution u of problem (1.2) is more regular, i.e, u ∈ W , (Ω) ∩ W , (Ω). Acknowledgements
The authors are deeply indebted to S.M. Nedogibchenko and V.V. Pukhnachevfor valuable discussions.The research of M. Korobkov was supported by the Russian Foundation forBasic Research (project No. 12-01-00390-a).The research of K. Pileckas was funded by the Lithuanian-Swiss cooperationprogramme under the project agreement No. CH-˘SMM-01/01.
References [1]
Ch.J. Amick : Existence of solutions to the nonhomogeneous steady Navier–Stokes equations,
Indiana Univ. Math. J. (1984), 817–830.[2] W. Borchers and K. Pileckas : Note on the flux problem for stationaryNavier–Stokes equations in domains with multiply connected boundary,
ActaApp. Math. (1994), 21–30.[3] J. Bourgain, M.V. Korobkov and J. Kristensen : On the Morse– Sardproperty and level sets of Sobolev and BV functions,
Rev. Mat. Iberoam. ,no. 1 (2013), 1–23.[4] D. Burago, Yu. Burago, S. Ivanov : A Course in Metric Geometry , Grad-uate Studies in Mathematics , AMS (2001).[5] J. R. Dorronsoro : Differentiability properties of functions with boundedvariation,
Indiana U. Math. J. , no. 4 (1989), 1027–1045.[6] L.C. Evans, R.F. Gariepy : Measure theory and fine properties of functions ,Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992).[7]
R. Finn : On the steady-state solutions of the Navier–Stokes equations. III,
Acta Math. (1961), 197–244. H. Fujita : On the existence and regularity of the steady-state solutions ofthe Navier-Stokes theorem,
J. Fac. Sci. Univ. Tokyo Sect.
I (1961) , 59–102.[9] H. Fujita : On stationary solutions to Navier-Stokes equation in symmet-ric plane domain under general outflow condition,
Pitman research notes inmathematics, Proceedings of International conference on Navier-Stokes equa-tions. Theory and numerical methods. June 1997. Varenna, Italy (1997) ,16-30.[10]
G.P. Galdi : On the existence of steady motions of a viscous flow with non–homogeneous conditions,
Le Matematiche (1991), 503–524.[11] G.P. Galdi : An Introduction to the Mathematical Theory of the Navier–Stokes Equations , vol. I, II revised edition, Springer Tracts in Natural Philo-sophy (ed. C. Truesdell) , , Springer–Verlag (1998).[12] L.V. Kapitanskii and K. Pileckas : On spaces of solenoidal vector fieldsand boundary value problems for the Navier-Stokes equations in domains withnoncompact boundaries,
Trudy Mat. Inst. Steklov (1983), 5–36 . EnglishTransl.:
Proc. Math. Inst. Steklov (1984), 3–34.[13]
M.V. Korobkov : Bernoulli law under minimal smoothness assumptions,
Dokl. Math. (2011), 107–110.[14] M.V. Korobkov, K. Pileckas and R. Russo , On the flux problem inthe theory of steady Navier–Stokes equations with nonhomogeneous bound-ary conditions,
Arch. Rational Mech. Anal. (2013), 185–213. DOI:10.1007/s00205-012-0563-y.[15]
M.V. Korobkov, K. Pileckas and R. Russo : Steady Navier-Stokes sys-tem with nonhomogeneous boundary conditions in the axially symmetric case,
Comptes rendus – Mecanique (2012), 115–119.[16]
M.V. Korobkov, K. Pileckas and R. Russo : Steady Navier-Stokes sys-tem with nonhomogeneous boundary conditions in the axially symmetric case,arXiv:1110.6301, to appear in
Ann. Scuola Norm. Sup. Pisa Cl. Sci. [17]
H. Kozono and T. Janagisawa : Leray’s problem on the Navier–Stokesequations with nonhomogeneous boundary data,
Math Zeitschrift (2009),27–39.[18]
A.S. Kronrod : On functions of two variables,
Uspechi Matem. Nauk (N.S.) (1950), 24–134 (in Russian). O.A. Ladyzhenskaya : Investigation of the Navier–Stokes equations in thecase of stationary motion of an incompressible fluid,
Uspech Mat. Nauk (1959), 75–97 (in Russian).[20] O.A. Ladyzhenskaya : The Mathematical theory of viscous incompressibleflow , Gordon and Breach (1969).[21]
J. Leray : ´Etude de diverses ´equations int´egrales non lin´eaire et de quelquesprobl`emes que pose l’hydrodynamique,
J. Math. Pures Appl. (1933), 1–82.[22] J. Mal´y, D. Swanson and W.P. Ziemer : The Coarea formula for Sobolevmappings,
Transactions of AMS , No. 2 (2002), 477–492.[23]
Moore R.L : Concerning triods in the plane and the junction points of planecontinua,
Proc. Nat. Acad. Sci. U.S.A. , No. 1 (1928), 85–88.[24] H. Morimoto : A remark on the existence of 2–D steady Navier–Stokesflow in bounded symmetric domain under general outflow condition,
J. Math.Fluid Mech. , No. 3 (2007), 411–418.[25] Pittman C.R. : An elementary proof of the triod theorem,
Proc. Amer. Math. Soc. , No. 4 (1970), 919.[26] V.V. Pukhnachev : Viscous flows in domains with a multiply connectedboundary,
New Directions in Mathematical Fluid Mechanics. The Alexan-der V. Kazhikhov Memorial Volume. Eds. Fursikov A.V., Galdi G.P. andPukhnachev V.V. Basel - Boston - Berlin: Birkhauser (2009) 333–348.[27]
V.V. Pukhnachev : The Leray problem and the Yudovich hypothesis,
Izv.vuzov. Sev.-Kavk. region. Natural sciences. The special issue ”Actual prob-lems of mathematical hydrodynamics” (2009) 185–194 (in Russian).[28]
R. Russo : On the existence of solutions to the stationary Navier–Stokesequations,
Ricerche Mat. (2003), 285–348.[29] A. Russo : A note on the two–dimensional steady-state Navier–Stokes prob-lem,
J. Math. Fluid Mech. (2009), 407–414.[30] L.I. Sazonov , On the existence of a stationary symmetric solution of thetwo–dimensional fluid flow problem,
Mat. Zametki , , No. 6 (1993), 138–141 (in Russian). English Transl.: Math. Notes , , No. 6 (1993), 1280–1283.[31] R. Temam : Navier-Stokes equations: theory and bumerical analysis , North-Holland, Amsterdam, 1979. I.I. Vorovich and V.I. Yudovich : Stationary flows of a viscous incompres-sible fluid,
Mat. Sbornik (1961), 393–428 (in Russian).(1961), 393–428 (in Russian).