On the existence of helical invariant solutions to steady Navier-Stokes equations
OOn the existence of helical invariant solutions to steadyNavier-Stokes equations
Mikhail Korobkov * Wenqi Lyu † Shangkun Weng ‡ March 1, 2021
Abstract
In this paper, we investigate the nonhomogeneous boundary value problem for the steadyNavier-Stokes equations in a helically symmetric spatial domain. When data is assumed to behelical invariant and satisfies the compatibility condition, we prove this problem has at least onehelical invariant solution.
Keyword : Steady Navier-Stokes equations, helically symmetric flow
Let
Ω = Ω \ ∪ nj = ¯ Ω j be a bounded multiply connected domain in R with C -smooth boundary ∂ Ω = (cid:83) Nj = Γ j consisting of N + Γ j = ∂ Ω j , j = , ..., N . Consider the nonhomogeneousboundary value problem for the steady Navier-Stokes equations ( u · ∇ ) u + ∇ p = ∆ u + f in Ω , div u = Ω , u = a on ∂ Ω , (1.1)where u and p are unknown velocity and pressure, a and f are given boundary value and body force,we assume the viscous coe ffi cient ν = (cid:90) ∂ Ω a · n dS = N (cid:88) j = (cid:90) Γ j a · n dS = N (cid:88) j = F j = , (1.2)where n is a unit outward normal vector to ∂ Ω . * School of Mathematical Sciences, Fudan University, Shanghai 200433, China; and Sobolev Institute of Mathematics,pr-t Ac. Koptyug, 4, Novosibirsk, 630090, Russia. E-mail: [email protected]. † Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, Guangdong Province,China. E-mail: [email protected]. ‡ School of Mathematics and Statistics and Computational Science Hubei Key Laboratory, Wuhan University, Wuhan,Hubei Province, 430072, People’s Republic of China. Email: [email protected]. a r X i v : . [ m a t h . A P ] F e b n his remarkable article [27], Leray proved the solvablity of (1 .
1) when the flux of the velocityacross each connected component Γ j of the boundary vanishes: (cid:90) Γ j a · n dS = , j = , ..., N . It remained an open problem as to whether the necessary condition (1 .
2) is su ffi cient for (1 .
1) tobe solvable. This is also called Leray’s problem because it actually goes back to his paper [27].Then, Leray’s problem has been studied in many papers [25, 40, 12, 13, 15, 3, 35, 36, 24, 37,14, 32, 33, 34], only recently its solvability was proved in bounded 2D domains and for 3D axiallysymmetric case under the sole necessary condition (1 .
2) by M.Korobkov, K.Pileckas and R.Russo[21]. As far as we know, this problem for general 3D bounded domain remains open.Given a positive number σ , we define the action of the helical group of transformations G σ on R by S θ,σ ( x ) = x cos θ + x sin θ − x sin θ + x cos θ x + σ π θ , θ ∈ R , that is, a rotation around the x axis with simultaneous translation along the x axis. G σ is uniquelydetermined by σ , which we will call the step . We say that the smooth function f ( x ) is helicallysymmetric or simply helical , if f is invariant under the action of G σ , i.e., f ( S θ,σ ( x )) = f ( x ) , ∀ θ ∈ R .Similarly, we say that the smooth vector field u ( x ) is helically symmetric, or simply helical, if it iscovariant with respect to the action of G σ , i.e., M ( θ ) u ( x ) = u ( S θ,σ ( x )) for all θ ∈ R , where M ( θ ) : = cos θ sin θ − sin θ cos θ
00 0 1 . (1.3)We learn from the definitions of helical function and vector field that they are σ -periodic in x variable. A domain Ω ⊂ R × T σ is called a helical domain if for each point x ∈ Ω , S θ,σ ( x ) ∈ Ω forany θ ∈ R . Here we denote by T σ = R /σ Z the corresponding 1-dimensional torus.There is an alternative definition of helical symmetry as follows. We rewrite a vector field u ( x ) = ( u , u , u )( x , x , x ) with respect to the moving orthonormal frame associated to standard cylindricalcoordinates ( r , θ, z ), e r = (cos θ, sin θ, , e θ = ( − sin θ, cos θ, , e z = (0 , , , as u = u r e r + u θ e θ + u z e z , where u r , u θ , u z are functions of ( r , θ, z ). We introduce two new independentvariables in place of θ and z : η : = σ π θ + z , ξ : = σ π θ − z . (1.4)A smooth function p = p ( r , θ, z ) is a helical function if and only if, when expressed in the ( r , ξ, η )variables, it is independent of ξ : p = q ( r , σ π θ + z ), for some q = q ( r , η ). Indeed, by definition f ishelical if and only if f ( S ρ,σ ( x )) is actually independent of ρ , being equal to f ( x ). Then by the ChainRule dd ρ f ( S ρ,σ ( x )) = dd ξ ˜ f ( r , ξ, η ) =
0, where˜ f ( r , ξ, η ) = f (cid:18) r cos (cid:18) πσ ( η + ξ ) (cid:19) , r sin (cid:18) πσ ( η + ξ ) (cid:19) , η − ξ (cid:19) . u is helical if and only if there exist v r , v θ , v z functions of ( r , η )such that u r ( r , θ, z ) = v r ( r , σ π θ + z ), u θ ( r , θ, z ) = v θ ( r , σ π θ + z ) and u z ( r , θ, z ) = v z ( r , σ π θ + z ). Since u r ( r , θ, z ) = u r ( r , θ + π, z ), then v r ( r , σ π θ + z ) = v r ( r , σ π θ + z + σ ), that is to say, v r ( r , η ) is periodicin η with period σ . Clearly, as σ → ∞ a helical flow becomes 2D flow and when σ = ffi cients u r , u θ , u z depend only on r and z ). Hence, a helical flow can beregarded as an interpolation between 2D and 3D axisymmetric flows.It is well-known that the global regularity of unsteady Navier-Stokes equations is a longstandingopen problem which is one of the seven Millennium Prize problems [11]. But if the initial data is two-dimensional or axisymmetric without swirl ( u θ = u satisfies the 2D Ladyzhenskaya inequality: (cid:107) u (cid:107) L ( Ω ) ≤ C σ (cid:107) u (cid:107) L ( Ω ) (cid:107) u (cid:107) H ( Ω ) , (1.5)where Ω = R × T σ . This coe ffi cient σ also reflects that the helical flow is an interpolation between2D and 3D axisymmetric flows. There are more interesting iteratures concerning helical flows, see[2, 29, 9, 38, 28, 17].This note is devoted to solve the Leray’s problem in a helical domain with helical invariant data.More precisely, we prove the following theorem. Theorem 1.1.
Assume that Ω ⊂ R × T σ is a helical domain with C boundary. If a ∈ W / , ( ∂ Ω ) and f ∈ L ( Ω ) are helical invariant functions, condition (1 . is fulfilled, then problem (1 . admits atleast one helical invariant weak solution. Our proof is motivated by M.Korobkov, K.Pileckas and R.Russo’s approach which successfullysolved the Leray’s problem in 2D and 3D axisymmetric domain in [21]. It is plausible that we canextend this result to the helical case. The proof in [21] was based on Leray’s contradiction argumentsand on the integration using the coarea formula along the level lines of the total head pressure. Alsothey used some consequence of Bernoulli’s law for solutions w to Euler equations with low regularity,namely, in [20, 22], the authors proved that for any compact connected set K ⊂ Ω (cid:18) ψ | K ≡ const (cid:19) ⇒ (cid:18) Φ | K ≡ const (cid:19) , (1.6)where ψ is the corresponding stream function and Φ = p + w is the total head pressure (the identityfor Φ is understood up to a negligible set of 1-dimensional measure zero).The helical invariant functions form a closed subspace with respect to weak convergence, we canstill use the contradiction arguments to derive a helical invariant weak solution ( w , p ) to the Euler3quations w r ∂ w r ∂ r + (cid:18) σ π r w θ + w z (cid:19) ∂ w r ∂η − w θ r + ∂ p ∂ r = , w r ∂ w θ ∂ r + (cid:18) σ π r w θ + w z (cid:19) ∂ w θ ∂η + w r w θ r + σ π r ∂ p ∂η = , w r ∂ w z ∂ r + (cid:18) σ π r w θ + w z (cid:19) ∂ w z ∂η + ∂ p ∂η = ,∂∂ r ( rw r ) + ∂∂η (cid:18) σ π w θ + rw z (cid:19) = , w | ∂ Ω = . (1.7)Unfortunately, although an equation (cid:18) w r ∂∂ r + ( σ π r w θ + w z ) ∂∂η (cid:19)(cid:18) | w | + p (cid:19) = . (1.8)is still valid, the Bernoulli’s law which appeared before is no longer true. To see this, let w r = w θ = r , w z = − σ π , p = r . This is an explicit solution to equations (1 . w r = σ π r w θ + w z =
0, but the total head pressure
Φ = r + σ π isnot a constant. The similar problem appears in axially–symmetric case (see, e.g., [23, page 746] ),nevertheless, the Bernoulli identity (1.6) will be satisfied here, due to zero boundary conditions (1.7 )and since each straight line passing through a point of Ω parallel to the axis of symmetry intersects ∂ Ω .But this is not true anymore for helical domains, therefore, the Bernoulli identity (1.6) fails in generalfor the solutions to (1.7) in helical case.This is an obstacle in modifying the arguments in [22, 21, 23], since the Bernoulli’s law is neededto separate the boundary components on which the total head pressure attains its supremum from theothers, this step played a crucial role in solving Leray’s problem in 3D axisymmetric domains.On the other hand, if a function with a finite Dirichlet integral is helically symmetric, then itsrestriction to a two-dimensional hyperplane orthogonal to the axis of symmetry has a finite Dirichletintegral as well (see, e.g., the formula (3.17) ), i.e., the corresponding plane function has no singular-ities. This circumstance allows us to simplify the arguments and carry out the proof without usingidentities of the form (1.6), which turned out to be so significant in the axisymmetric case (when therestriction of considered functions to the two-dimensional hyperplane have singularities near the sym-metry axis).This paper is organized as follows. In section 2, we deal with the nonhomogeneous boundaryvalues and present some properties of the Sobolev functions. In section 3, we first use Leray’s contra-diction arguments to derive a nontrivial solution to the helical Euler equations. Finally we constructa family of level sets of the total head pressure and deduce a contradiction via coarea formula. Acknowledgment.
The work of M.K. is supported by Mathematical Center in Akademgorodok underagreement No. 075-15-2019-1613 with the Ministry of Science and Higher Education of the RussianFederation. The work of S.W. is partially supported by National Natural Science Foundation of ChinaNo.11701431, 11971307, 12071359. 4
Preliminaries
In this section, we first find a solenoidal and helical invariant extension of the nonhomogeneousboundary values. Then we review the Morse-Sard property of Sobolev functions.By a domain we mean an open connected set. We use standard notations for function spaces: 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 is clear from the context whether we usescalar or vector (or tensor) valued function spaces.Denote by H ( Ω ) the closure of the set of all solenoidal smooth vector-functions having compactsupports in Ω with respect to the norm (cid:107) w (cid:107) H ( Ω ) = (cid:18)(cid:82) Ω |∇ w | (cid:19) . We need to use the following symmetry assumptions:( H ) Ω ⊂ R is a helical domain with C boundary and O x is a symmetry axis of Ω .( H ) The assumptions ( H ) are fulfilled and both the boundary value a ∈ W / , ( ∂ Ω ) and f = ∇ × b ∈ L ( Ω ) are helical invariant. Lemma 2.1.
If conditions ( H ) and (1 . are fulfilled, then there exists a helical solenoidal extension A ∈ W , ( Ω ) of a with (cid:107) A (cid:107) W , ( Ω ) ≤ c (cid:107) a (cid:107) W / , ( ∂ Ω ) . (2.1) Proof.
By the well-known results ([16] Chapter III.3), there exists a solenoidal extension A ∈ W , ( Ω ) such that (cid:107) A (cid:107) W , ( Ω ) ≤ c (cid:107) a (cid:107) W / , ( ∂ Ω ) . (2.2)For any j ∈ N we denote A j ( r , ξ, η ) = j ! j ! (cid:88) i = A (cid:32) r , ξ + ij ! σ, η (cid:33) . (2.3)Obviously, A j ∈ W , ( Ω ) is also a solenoidal extension of a , and it satisfies (cid:107) A j (cid:107) W , ( Ω ) ≤ c (cid:107) a (cid:107) W / , ( ∂ Ω ) . (2.4)Moreover, A j ( r , ξ + kl σ, η ) = A j ( r , ξ, η ) holds for any integer 0 ≤ k ≤ l ≤ j . By (2 . A j (cid:48) converges weakly to a function A ∈ W , ( Ω ). Then the solenoidal weak limit A also satisfies A ( r , ξ + kl σ, η ) = A ( r , ξ, η ) for all integer 0 ≤ k ≤ l , so A is independent of ξ , in anotherword, it is a helical function which satisfies (2 . (cid:3) U ∈ W , ( Ω ) to the Stokes problem such that U − A ∈ H ( Ω ) ∩ W , ( Ω ), and (cid:90) Ω ∇ U · ∇ ϕ dx = (cid:90) Ω f · ϕ dx , ∀ ϕ ∈ H ( Ω ) . (2.5)Moreover, (cid:107) U (cid:107) W , ( Ω ) ≤ c ( (cid:107) a (cid:107) W / , ( ∂ Ω ) + (cid:107) f (cid:107) L ( Ω ) ) . (2.6)Actually, uniqueness ensures the solution U is also a helical invariant function, since A is helicalinvariant. Indeed, for any ξ ∈ [0 , σ ] denote U ξ ( r , η, ξ ) = U ( r , η, ξ − ξ ), then U ξ − A ∈ H ( Ω ) ∩ W , ( Ω )since A is helical invariant. We learn from (2 . (cid:90) Ω ∇ U ξ · ∇ ϕ dx = (cid:90) Ω ∇ U · ∇ ϕ − ξ dx = (cid:90) Ω f · ϕ − ξ dx = (cid:90) Ω f ξ · ϕ dx . (2.7)By our assumption, f is helical invariant, therefore, f ξ = f . We have U ξ = U , in another words, U isindependent of ξ and U is helical invariant.Now we consider w = u − U , equation (1 .
1) is equivalent to − ∆ w + ( U · ∇ ) w + ( w · ∇ ) w + ( w · ∇ ) U = −∇ p − ( U · ∇ ) U in Ω , div w = Ω , w = ∂ Ω . (2.8)By a weak solution to problem (1 . u ∈ W , ( Ω ) such that w = u − U ∈ H ( Ω ) and for any ϕ ∈ H ( Ω ), (cid:104) w , ϕ (cid:105) H ( Ω ) = − (cid:90) Ω ( U · ∇ ) U · ϕ dx − (cid:90) Ω ( U · ∇ ) w · ϕ dx − (cid:90) Ω ( w · ∇ ) w · ϕ dx − (cid:90) Ω ( w · ∇ ) U · ϕ dx . (2.9)By Riesz representation theorem, for any w ∈ H ( Ω ) there exists a unique function T w ∈ H ( Ω )such that the right hand side of (2.9) is equivalent to (cid:104) T w , ϕ (cid:105) H ( Ω ) , for any ϕ ∈ H ( Ω ). Moreover, T isa compact operator by the well-known result in [26]. Lemma 2.2.
For any helical function w ∈ H ( Ω ) , T w ∈ H ( Ω ) is also a helical function.Proof. The proof is the same as above. We omit the details here. (cid:3)
The following lemma is concerned with the classical di ff erentiablity properties of Sobolev functions. Lemma 2.3 ([8]) . If ψ ∈ W , ( R ) , then ψ is continuous and there exists a set A ψ such that H ( A ψ ) = and ψ is di ff erentiable (in the classical sense) at each x ∈ R \ A ψ . Furthermore, the classical derivativeat these points x coincides with ∇ ψ ( x ) = lim r → − (cid:82) B r ( x ) ∇ ψ ( y ) dy , where lim r → − (cid:82) B r ( x ) |∇ ψ ( y ) − ∇ ψ ( x ) | dy = . H the one-dimensional Hausdor ff measure, i.e., H ( F ) = lim t → + H t ( F ), where H t ( F ) = inf { ∞ (cid:80) i = diam F i : diam F i ≤ t , F ⊂ ∞ (cid:83) i = F i } .The following Morse-Sard Theorem for Sobolev function has been proved by J.Bourgain, M.Korobkovand J.Kristensen [4, 5]. Theorem 2.4.
Let D ⊂ R be a bounded domain with Lipschitz boundary and ψ ∈ W , ( D ) . Then: (1) H ( { ψ ( x ) : x ∈ ¯ D \ A ψ & ∇ ψ ( x ) = } ) = ; (2) for every (cid:15) > there exists δ > such that for every set U ⊂ ¯ D with H ∞ ( U ) < δ the inequality H ( ψ ( U )) < (cid:15) holds;(3) for H -almost all y ∈ ψ ( ¯ D ) ⊂ R the preimage ψ − ( y ) is a finite disjoint family of C -curves S j ,j = , , ..., N ( y ) . Each S j is either a cycle in D (i.e., S j ⊂ D is homemorphic to the unit circle S ) or a simple arc with endpoints on ∂ D (in this case S j is transversal to ∂ D ). Without loss of generality, we may assume that f = curl b ∈ L ( Ω ). In particular,div f = w = T w in H ( Ω ). By Leray-Schaudertheorem, to prove the existence of weak solution is su ffi cient to show that the solutions of equation w ( λ ) = λ T w ( λ ) are uniformly bounded with respect to λ ∈ [0 , w ∈ H ( Ω ) satisfies for any ϕ ∈ H ( Ω ), (cid:90) Ω ∇ w · ∇ ϕ dx = − λ (cid:90) Ω ( U · ∇ ) U · ϕ dx − λ (cid:90) Ω ( U · ∇ ) w · ϕ dx − λ (cid:90) Ω ( w · ∇ ) w · ϕ dx − λ (cid:90) Ω ( w · ∇ ) U · ϕ dx are uniformly bounded in H ( Ω ) with respect to λ ∈ [0 , { λ n } n ∈ N ⊂ [0 ,
1] and { ˆw n } n ∈ N ⊂ H ( Ω ) such that, for any ϕ ∈ H ( Ω ), (cid:90) Ω ∇ ˆw n · ∇ ϕ dx = − λ n (cid:90) Ω ( U · ∇ ) U · ϕ dx − λ n (cid:90) Ω ( U · ∇ ) ˆw n · ϕ dx − λ n (cid:90) Ω ( ˆw n · ∇ ) ˆw n · ϕ dx − λ n (cid:90) Ω ( ˆw n · ∇ ) U · ϕ dx (3.2)and λ n → λ ∈ [0 , , J n : = || ˆw n || H ( Ω ) → ∞ By the Helmholtz–Weyl decomposition, f can be represented as the sum f = curl b + ∇ ϕ with curl b ∈ L ( Ω ), and thegradient part is included then into the pressure term (see, e.g., [26], [16]). .
2) is equivalent to (cid:90) Ω ∇ ˆw n · ∇ ϕ dx = λ n (cid:90) Ω ( U · ∇ ) ϕ · U dx + λ n (cid:90) Ω ( U · ∇ ) ϕ · ˆw n dx + λ n (cid:90) Ω ( ˆw n · ∇ ) ϕ · ˆw n dx + λ n (cid:90) Ω ( ˆw n · ∇ ) ϕ · U dx (3.3)Denote w n = ˆw n / J n . Since (cid:107) w n (cid:107) H ( Ω ) = , (3.4)there exists a subsequence w n l converging weakly in H ( Ω ) to a function w ∈ H ( Ω ). Then by compactembedding of Sobolev space, w n l converges strongly in L r ( Ω ), for any r ∈ [1 , ϕ = J − n ˆw n in (3.3), we get (cid:90) Ω |∇ w n | dx = λ n (cid:90) Ω ( w n · ∇ ) w n · U dx + J − n λ n (cid:90) Ω ( U · ∇ ) w n · U dx (3.5)Therefore, taking into account (3.4) and passing to a limit as n l → ∞ in (3.5), we obtain1 = λ (cid:90) Ω ( w · ∇ ) w · U dx (3.6)in particular, this implies λ ∈ (0 , . ζ ∈ W , ( Ω ), consider the linear functional R n ( ζ ) = (cid:90) Ω ∇ ˆw n · ∇ ζ dx − λ n (cid:90) Ω ( U · ∇ ) ζ · U dx − λ n (cid:90) Ω ( U · ∇ ) ζ · ˆw n dx − λ n (cid:90) Ω ( ˆw n · ∇ ) ζ · ˆw n dx − λ n (cid:90) Ω ( ˆw n · ∇ ) ζ · U dx (3.7)It is obvious that R n ( ζ ) has the following estimate | R n ( ζ ) | ≤ C ( (cid:107) ˆw n (cid:107) H ( Ω ) + (cid:107) ˆw n (cid:107) H ( Ω ) + (cid:107) U (cid:107) W , ( Ω )) (cid:107) ζ (cid:107) H ( Ω ) (3.8)Again (3 .
3) implies R n ( ϕ ) = , ∀ ϕ ∈ H ( Ω ) . Therefore, there exists a unique function ˆ p n ∈ ˆ L ( Ω ) = { q ∈ L ( Ω ) : (cid:82) Ω q ( x ) dx = } such that R n ( ϕ ) = (cid:90) Ω ˆ p n div ϕ dx , ∀ ϕ ∈ H ( Ω ) . (3.9)actually ˆ p n is a helical function, this can be established by uniqueness since the functions ˆw n and U on the RHS of (3 .
7) are helical invariant. The proof of this part is similar to (2 . . (cid:107) ˆ p n (cid:107) L ( Ω ) ≤ C (cid:107) R n (cid:107) H − ( Ω ) ≤ C ( (cid:107) ˆw n (cid:107) H ( Ω ) + (cid:107) ˆw n (cid:107) H ( Ω ) + (cid:107) a (cid:107) W , / ( ∂ Ω ) + (cid:107) f (cid:107) L ( Ω ) ) . (3.10)8here we have used (2 . . − (3 . ˆw n , ˆ p n ) satisfies for any ζ ∈ W , ( Ω ) (cid:90) Ω ∇ ˆw n · ∇ ζ dx − λ n (cid:90) Ω ( U · ∇ ) ζ · U dx − λ n (cid:90) Ω ( U · ∇ ) ζ · ˆw n dx − λ n (cid:90) Ω ( ˆw n · ∇ ) ζ · ˆw n dx − λ n (cid:90) Ω ( ˆw n · ∇ ) ζ · U dx = (cid:90) Ω ˆ p n div ζ dx (3.11)Let ˆu n = ˆw n + U , then (3 .
11) is equivalent to for any ζ ∈ W , ( Ω ) (cid:90) Ω ∇ ˆu n · ∇ ζ dx − (cid:90) Ω ˆ p n div ζ dx = − λ n (cid:90) Ω ( ˆu n · ∇ ) ˆu n · ζ dx + (cid:90) Ω f · ζ dx where we have used (2 . ˆu n , ˆ p n ) satisfies the following Navier-Stokes equations − ∆ ˆu n + ∇ ˆ p n = − λ n ( ˆu n · ∇ ) ˆu n + f in Ω , div ˆu n = Ω , ˆu n = a on ∂ Ω . (3.12)By the well-known L q -estimates of Stokes systems (see, e.g., [16, P. 279, Theorem IV.4.1] ), equa-tions (3 .
12) imply (cid:107) ˆu n (cid:107) W , / ( Ω ) + (cid:107) ˆ p n (cid:107) W , / ( Ω ) ≤ C ( (cid:107) ( ˆu n · ∇ ) ˆu n (cid:107) L / ( Ω ) + (cid:107) f (cid:107) L ( Ω ) + (cid:107) a (cid:107) W / , ( ∂ Ω ) ) ≤ C ( (cid:107) ˆu n (cid:107) W , ( Ω ) + (cid:107) f (cid:107) L ( Ω ) + (cid:107) a (cid:107) W / , ( ∂ Ω ) ) ≤ C ( (cid:107) ˆw n (cid:107) H ( Ω ) + (cid:107) a (cid:107) W / , ( ∂ Ω ) + (cid:107) f (cid:107) L ( Ω ) + (cid:107) a (cid:107) W / , ( ∂ Ω ) ) (3.13)note that we have used (3 .
10) in the last step.Let u n = J − n ˆu n and p n = λ − n J − n ˆ p n . It is obvious that (cid:107) u n (cid:107) W , ( Ω ) and (cid:107) p n (cid:107) W , / ( Ω ) are uniformlybounded, and − ν n ∆ u n + ( u n · ∇ ) u n + ∇ p n = f n in Ω , div u n = Ω , u n = a n on ∂ Ω . (3.14)where ν n = λ − n J − n , f n = λ − n J − n f , and a n = J − n a . By the observation above, we can extract weaklyconvergent subsequences u n (cid:42) w in W , ( Ω ) and p n (cid:42) p in W , / loc ( Ω ) ∩ L ( Ω ). The pair ( w , p )satisfies the following Euler equations ( w · ∇ ) w + ∇ p = Ω , div w = Ω , w = ∂ Ω . (3.15)We summarize the above results as follows. Lemma 3.1.
Assume that Ω ⊂ R × T σ is a helical domain with C boundary ∂ Ω , f = ∇ × b , b ∈ W , ( Ω ) and a ∈ W / , ( ∂ Ω ) are both helical functions and (1 . is valid. If the assertion ofTheorem . is false, then there exist w , p with the following properties: E – H ) The helical functions w ∈ W , ( Ω ) , p ∈ W , / ( Ω ) satisfy the Euler equations (3 . , moreoverequation (3 . holds. ( E – NS – H ) Conditions ( E – H ) are satisfied, and there exist sequences of helical functions u n ∈ W , ( Ω ) , p n ∈ W , / ( Ω ) and numbers ν n → + , λ n → λ > such that the norms (cid:107) u n (cid:107) W , ( Ω ) , (cid:107) p n (cid:107) W , / ( Ω ) are uniformly bounded, the pair ( u n , p n ) satisfies (3 . , and (cid:107)∇ u n (cid:107) L ( Ω ) → , u n (cid:42) w in W , ( Ω ) , p n (cid:42) p in W , / ( Ω ) .Moreover, u n ∈ W , ( Ω ) and p n ∈ W , loc ( Ω ) . (3.16)Note, that the last inclusion follows from (3.1) and from the fact that locally the laplacian ∆ p belongs to the Hardy space (see, e.g., [20, Section 5.2] for more detailed description). In this subsection we discuss the properties satisfied by the helical invariant solution to Euler equation(3 . Ω is evolved from a two dimensional multiply connected domain D ⊂ R , hence Ω = (cid:26) x x x : ∃ ( y , y ) ∈ D and θ ∈ R such that x x x = y cos θ + y sin θ − y sin θ + y cos θ σ π θ (cid:27) . Define by P the hyperplane P = { ( x , x ,
0) : x , x ∈ R } . Then by construction D = Ω ∩ P . It follows from the helical symmetry of the vector field w that (cid:107) w (cid:107) L q ( Ω ) = σ (cid:107) w ( · , (cid:107) L q ( D ) , hencewe have ∇ w ∈ L ( D ) , w ∈ L q ( D ) ∀ q < ∞ , (3.17)consequently, from Euler system we have ∇ p ∈ L q ( D ) ∀ q < . (3.18)For any A ⊂ R we define (cid:101) A ⊂ R × T σ to be the three dimensional set which is evolved from A ,i.e (cid:101) A = y cos θ + y sin θ − y sin θ + y cos θ σ π θ : ( y , y ) ∈ A , θ ∈ R . (3.19)Denote Σ j : = P ∩ Γ j . Clearly, (cid:101) Σ j = Γ j , j = , · · · , N , and ∂ D = N (cid:91) j = Σ j . (3.20)The next statement was proved in [18, Lemma 4] and in [1, Theorem 2.2].10 emma 3.2. If ( E – H ) are satisfied, then ∃ ˇ p j ∈ R : p ( x ) ≡ ˇ p j for H -almost all x ∈ Γ j , j = , ..., N . In particular, by helical symmetry, p ( x ) ≡ ˇ p j for H -almost all x ∈ Σ j , j = , ..., N . By simple calculation from (3 . . and Lemma 3 .
2, it follows that
Corollary 3.3.
If conditions ( E – NS – H ) are satisfied, then − λ = N (cid:88) j = ˇ p j (cid:90) Γ j a · n dS = N (cid:88) j = ˇ p j F j . (3.21)Set Φ n = p n + | u n | and Φ = p + | w | . By the properties of Sobolev functions’ best representativeof w , ψ , Φ (see [10]), we get the following. Lemma 3.4.
If conditions ( E – H ) hold, then there exists a set A w ⊂ D such that(1) H ( A w ) = ;(2) for all x ∈ D \ A w , lim ρ → − (cid:90) B ρ ( x ) | w ( y ) − w ( x ) | dy = lim ρ → − (cid:90) B ρ ( x ) | Φ ( y ) − Φ ( x ) | dy =
0; (3.22) (3) for every (cid:15) > , there exists a set U ⊂ D with H ∞ ( U ) < (cid:15) , A w ⊂ U and such that the functions w , Φ are continuous on ¯ D \ U . We consider two possible cases.( a ) The maximum of Φ is attained on the boundary ∂ Ω :max j = ,..., N ˇ p j = ess sup x ∈ Ω Φ ( x ) . (3.23)( b ) The maximum of Φ is not attained on the boundary ∂ Ω :max j = ,..., N ˇ p j < ess sup x ∈ Ω Φ ( x ) . (3.24) ess sup Φ = ∞ is not excluded. .3.1 If ( a ) happens. Add to the pressure a constant such that ess sup x ∈ Ω Φ ( x ) = p = ˇ p = · · · = ˇ p M = , M < N , (3.25)ˇ p j < , j = M + , . . . , N . (3.26)Denote p ∗ = max j = M + , N ˇ p j . Then by construction0 > p ∗ > ˇ p j , ∀ j = M + , . . . , N . (3.27)For su ffi ciently small parameter h > j ∈ { , . . . , N } denote Σ jh = { x ∈ D : dist( x , Σ j ) = h } , D jh = { x ∈ D : dist( x , Σ j ) < h } , and Σ = Σ ∪· · ·∪ Σ M , Σ h = M (cid:91) j = Σ jh = (cid:8) x ∈ D : dist( x , Σ ) = h (cid:9) , D h = M (cid:91) j = D jh = (cid:8) x ∈ D : dist( x , Σ ) < h (cid:9) , (3.28) Σ − = Σ M + ∪ · · · ∪ Σ N , Σ − h = N (cid:91) j = M + Σ jh = (cid:8) x ∈ D : dist( x , Σ − ) = h (cid:9) , D − h = N (cid:91) j = M + D jh = (cid:8) x ∈ D : dist( x , Σ − ) < h (cid:9) . In particular, we have (cid:8) x ∈ D : dist( x , ∂ D ) = h (cid:9) = Σ h ∪ Σ − h . (3.29)Since the distance function dist( x , ∂ D ) is C –regular and the norm of its gradient is equal to one in theneighborhood of ∂ Ω , there is a constant δ > h ≤ δ the set Σ jh is a C -smooth curve homeomorphic to the circle, below we will call such curves as cycles . Respectively, Σ h is a disjoint union of cycles.By direct calculations, (3.15) implies ∇ Φ = w × ω in Ω , (3.30)where ω = curl w , 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 ( x ) = | ω ( x ) | .From conditions ( E – NS – H ) and from helical symmetry by construction we obtain u n (cid:42) w in W , ( D ) , (3.31) p n (cid:42) p in W , / ( D ) . (3.32)Then from Theorem 3.2 in [1] (see also [20, Lemma 3.3]) we obtain12 emma 3.5. There exists a subsequence Φ k l such that Φ k l | Σ jh converges to Φ | Σ jh uniformly for almostall h ∈ (0 , δ ) and for all j = { , , . . . , N } . Below we assume (without loss of generality) that the subsequence Φ k l coincides with the wholesequence Φ k .The value h ∈ (0 , δ ) will be called regular , if it satisfies the assertion of Lemma 3.5, i.e., if Φ k | Σ jh ⇒ Φ | Σ jh ∀ j = { , , . . . , N } . (3.33) Lemma 3.6.
There exists a measurable set H ⊂ (0 , δ ) such that(i) each value h ∈ H is regular;(ii) density of H at zero equals 1: meas (cid:0) H ∩ [0 , h ] (cid:1) h → as h → + ; (3.34) (iii) lim H (cid:51) h → + sup x ∈ Σ jh (cid:12)(cid:12)(cid:12) Φ ( x ) − ˇ p j (cid:12)(cid:12)(cid:12) = ∀ j = { , , . . . , N } . Proof.
Fix j ∈ { , , . . . , N } . Since w | ∂ D ≡ ∇ w ∈ L ( D ), by Hardy inequality we have (cid:90) D jh | w | = o ( h ) . Then by H¨older inequality, (cid:90) D jh | w | · |∇ w | = o ( h ) . From Euler equations we have (cid:90) D jh |∇ Φ | ≤ (cid:90) D h | w | · |∇ w | = o ( h ) . Using Fubini theorems, we can rewrite the last estimate as (cid:90) D jh |∇ Φ | = h (cid:90) (cid:18)(cid:90) Σ jt |∇ Φ | ds (cid:19) dt = o ( h ) . The last estimate implies easily that there exists a measurable set H ⊂ (0 , δ ) such that the densityof H at zero equals 1 (e.g., (3.34) holds ), the restriction Φ | Σ jh is an absolute continuous function ofone variable for all h ∈ H , and lim H (cid:51) h → + (cid:90) Σ jh |∇ Φ | ds = , (3.35)Denote by Φ h the mean value of Φ over the curve Σ jh . Then from (3.35) we obtain immediately thatlim H (cid:51) h → + max x ∈ Σ jh | Φ ( x ) − Φ h | = . (3.36)13ince by our assumptions ∂ D is C -smooth, the curves Σ ih are uniformly C -smooth for all su ffi -ciently small h < δ . Then by well-known classical Sobolev theorems, the continuous trace operator T jh : W , / ( D ) (cid:51) f (cid:55)→ f | Σ jh ∈ L ( Σ jh )is well defined. Moreover, for any f ∈ W , / ( D ) the real-valued function[0 , δ ] (cid:51) h (cid:55)→ (cid:90) Σ jh | f | ds (3.37)is continuous.Recall, that by Lemma 3.2 the trace identity Φ | Σ j ≡ ˇ p j holds. Since pressure is defined up toan additive constant, we can assume, without loss of generality, that ˇ p j =
0, i.e., Φ | Σ j = (cid:90) Σ jh | Φ | ds → h → + . (3.39)This implies Φ h → h → + . The last convergence together with the formula (3.35) give us thedesired asymptotic lim H (cid:51) h → + max x ∈ Σ jh | Φ ( x ) | = . (3.40)The Lemma is proved. (cid:3) By Lemmas 3.5–3.6, decreasing H if necessary and taking su ffi ciently large k , we can assume,without loss of generality, that Corollary 3.7.
For all h ∈ H we have Φ k ( x ) < p ∗ ∀ x ∈ Σ − h . (3.41)Also, by Lemma 3.6 we obtain Corollary 3.8.
For all ε > there exists δ ε > and k ε ∈ N such that ∀ h ∈ H ∩ (0 , δ ε ) and ∀ k ≥ k ε we have Φ k ( x ) > − ε ∀ x ∈ Σ h . (3.42)Denote t ∗ = − p ∗ . Let 0 < t < t ∗ . The next geometrical object plays an important role in theestimates below: for t ∈ ( t , t ∗ ) and for all su ffi ciently large k we define the level set S k ( t , t ) ⊂ { x ∈ D : Φ k ( x ) = − t } separating boundary components Σ from Σ − as follows. Namely, take (cid:15) = t andthe corresponding parameters δ (cid:15) > k ◦ = k (cid:15) ∈ N such that (3.41)–(3.42) holds for all k ≥ k ◦ and h ∈ H ∩ (0 , δ (cid:15) ). Fix a number h ◦ ∈ H ∩ (0 , δ (cid:15) ). In particular, we have ∀ t ∈ ( t , t ∗ ) ∀ k ≥ k ◦ (cid:18) Φ k | Σ h ◦ > − t , Φ k | Σ − h ◦ < − t (cid:19) . (3.43)14or k ≥ k ◦ , j = , , . . . , M , and t ∈ ( t , t ∗ ) denote by W jk ( t ; t ) the connected component of the open set (cid:8) x ∈ D : dist( x , ∂ D ) > h & Φ k ( x ) > − t (cid:9) such that ∂ W jk ( t ; t ) ⊃ Σ jh ◦ (see Fig.1), (3.44)and put W k ( t ; t ) = M (cid:91) j = W jk ( t ; t ) , S k ( t ; t ) = (cid:0) ∂ W k ( t ; t ) (cid:1) \ Σ h ◦ . By construction, (see Fig.1), ∂ W k ( t ; t ) = Σ h ◦ ∪ S k ( t ; t ) (3.45)and ∂ W k ( t ; t ) ⊂ (cid:8) x ∈ D : dist( x , ∂ D ) > h ◦ & Φ k ( x ) = − t (cid:9) ∪ (cid:8) x ∈ D : dist( x , ∂ D ) = h ◦ & Φ k ( x ) ≥ − t (cid:9) . (3.46) Figure 1: M = , N = Therefore, by definition of S k ( t ; t ) and in virtue of the identity (3.29), S k ( t ; t ) ⊂ (cid:8) x ∈ D : dist( x , ∂ D ) > h ◦ & Φ k ( x ) = − t (cid:9) ∪ (cid:8) x ∈ Σ − h ◦ : Φ k ( x ) ≥ − t (cid:9) . (3.47)But the last set { x ∈ Σ − h ◦ : Φ k ( x ) ≥ − t } is empty because of (3.43). Therefore, S k ( t ; t ) ⊂ (cid:8) x ∈ D : dist( x , ∂ D ) > h ◦ & Φ k ( x ) = − t (cid:9) . (3.48)Since by (E–NS) each Φ k belongs to W , ( D ), by the Morse-Sard theorem for Sobolev functions(see assertion (iii) of Theorem 2.4 ) we have that for almost all t ∈ ( t , t ∗ ) the level set S k ( t ; t ) consistsof finitely many C -cycles and Φ k is di ff erentiable (in classical sense) at every point x ∈ S k ( t ; t ) with ∇ Φ k ( x ) (cid:44)
0. The values t ∈ ( t , t ∗ ) having the above property will be called k - regular . (Note that W k ( t ; t )) and S k ( t ; t ) are well defined for all t ∈ ( t , t ∗ ) and k ≥ k ◦ = k ◦ ( t ).)Recall that for a set A ⊂ D we denote by (cid:101) A the three dimensional set in T σ which is evolved from A (see (3.19) ). By construction, for every regular value t ∈ ( t , t ∗ ) the set (cid:101) S k ( t ; t ) is a finite union ofsmooth surfaces (tori), and (cid:90) (cid:101) S k ( t ; t ) ∇ Φ k · n dS = − (cid:90) (cid:101) S k ( t ; t ) |∇ Φ k | dS < , (3.49)where n is the unit outward normal vector to ∂ (cid:101) W k ( t ; t ).Note that W k ( t ; t )) and S k ( t ; t ) are well defined for all t ∈ ( t , t ∗ ) and k ≥ k ◦ = k ◦ ( t ). Now weare ready to prove the key estimate (which is analog of Lemma 3.8 from [21]).15 emma 3.9. Let < t < t ∗ . Then there exists k ∗ = k ∗ ( t ) such that for every k ≥ k ∗ and for almostall t ∈ ( t , t ∗ ) the inequality (cid:90) (cid:101) S k ( t ; t ) |∇ Φ k | dS < F t , (3.50) holds with the constant F independent of t , t , and k . Proof.
Fix positive t < t ∗ , (cid:15) = t , and the corresponding parameters δ (cid:15) > k ◦ : = k (cid:15) ∈ N suchthat (3.41)–(3.42) holds for all k ≥ k ◦ and h ∈ H ∩ (0 , δ (cid:15) ). Fix a number h ◦ ∈ H ∩ (0 , δ (cid:15) ). Thenfor all t ∈ ( t , t ∗ ) and all k ≥ k ◦ we can define the set S k ( t ; t ) as above. Moreover, for almost all t ∈ ( t , t ∗ ) the set S k ( t ; t ) consists of finitely many pairwise disjoint C -cycles ( = C -smooth curveshomeomorphic to the circle) and Φ k is di ff erentiable (in classical sense) at every point x ∈ S k ( t ; t )with ∇ Φ k ( x ) (cid:44)
0. The values t ∈ ( t , t ∗ ) having the above property will be called k - regular .Respectively, for every regular value t ∈ ( t , t ∗ ) the set (cid:101) S k ( t ; t ) is a finite union of smooth surfaces(tori), and the inequality (3.49) holds.The main idea of the proof of (3.50) is quite simple: we will integrate the equation ∆Φ k = ω k + ν k div( Φ k u k ) − ν k f k · u k (3.51)over the suitable domain Ω k ( t ) with ∂ Ω k ( t ) ⊃ (cid:101) S k ( t ; t ) such that the corresponding boundary integrals (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90)(cid:0) ∂ Ω k ( t ) (cid:1) \ (cid:101) S k ( t ; t ) ∇ Φ k · n dS (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.52)1 ν k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90)(cid:0) ∂ Ω k ( t ) (cid:1) \ (cid:101) S k ( t ; t ) Φ k u k · n dS (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.53)are negligible. We split the construction of the domain Ω k ( t ) into two steps.First of all, define the open set D k ( t ) : = W k ( t ; t ) ∪ D h ◦ \ Σ . Then by construction (see, e.g., (3.28), (3.45) ) we have ∂ D k ( t ) = Σ ∪ S k ( t ; t ) . (3.54)Denote further Ω k ( t ) : = (cid:101) D k ( t ) . Then Ω k ( t ) is the open set in the three dimensional space and ∂ Ω k ( t ) = Γ ∪ (cid:102) S k ( t ; t ) , (3.55)where we denote Γ : = Γ ∪ Γ ∪ · · · ∪ Γ M .By direct calculations, (3.14) implies ∇ Φ k = − ν k curl ω k + u k × ω k + f k = − ν k curl ω k + u k × ω k + λ k ν k curl b .
16y the Stokes theorem, for any C -smooth closed surface S ⊂ Ω and g ∈ W , ( Ω ) we have (cid:90) S curl g · n dS = . So, in particular, (cid:90) S ∇ Φ k · n dS = (cid:90) S ( u k × ω k ) · n dS . (3.56)Recall that by the pressure normalization condition, Φ | Γ = . (3.57)Our purpose on the next step is as follows: for arbitrary ε > ffi ciently large k to provethe estimates (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) Γ ∇ Φ k · n dS (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Γ ( u k × ω k ) · n dS (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε, (3.58)1 ν k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) Γ Φ k u k dS (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε. (3.59)Recall that in our notation u k = λ k ν k U + w k , where w k ∈ H ( Ω ), (cid:107) w k (cid:107) H ( Ω ) =
1, and U is a solutionto the Stokes problem with boundary value a (see (2.5) ). In particular, we have u k ( x ) ≡ λ k ν k U ( x ) ∀ x ∈ Γ . (3.60)To establish (3.59), we use the uniform boundedness (cid:107) Φ k (cid:107) L ( Ω ) + (cid:107)∇ Φ k (cid:107) L / ( Ω ) ≤ C . (3.61)From (3.61) and the weak convergence Φ k (cid:42) Φ in W , / ( Ω ) we easily have Φ k → Φ in L q ( Γ ) ∀ q ∈ [1 , . (3.62)Thus by virtue of (3.57), (cid:90) Γ | Φ k | dS → k → ∞ . (3.63)Since (cid:107) U (cid:107) L ∞ ( Ω ) ≤ C (cid:107) U (cid:107) W , ( Ω ) < ∞ , (3.64)by identity (3.60) we have1 ν k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) Γ Φ k u k dS (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = λ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) Γ Φ k U dS (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:90) Γ | Φ k | dS → k →∞ , (3.65)that implies the required estimate (3.59) for su ffi ciently large k .To prove (3.58), we need also the uniform estimate (cid:107) ν k u k (cid:107) W , / ( Ω ) ≤ C , (3.66)where C is independent of k (this inequality follows from the construction, see (3.13) ). Thus bySobolev imbedding theorems (cid:107) ν k ∇ u k (cid:107) L ( Γ h ) ≤ C ∀ h ∈ [0 , h ◦ ] ∀ k ∈ N , (3.67)17here C > k , h , here Ω h = { x ∈ Ω : dist( x , Γ ) ≤ h } , Γ h = { x ∈ Ω : dist( x , Γ ) = h } .Moreover, by elementary calculations (3.66) implies the uniform H¨older continuity of the function[0 , h ◦ ] (cid:51) h (cid:55)→ (cid:107) ν k ∇ u k (cid:107) L ( Γ h ) , i.e., there exists a constant σ > k ) such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Γ h (cid:48) (cid:12)(cid:12)(cid:12) ν k ∇ u k (cid:12)(cid:12)(cid:12) dS − (cid:90) Γ h (cid:48)(cid:48) (cid:12)(cid:12)(cid:12) ν k ∇ u k (cid:12)(cid:12)(cid:12) dS (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ σ (cid:12)(cid:12)(cid:12) h (cid:48) − h (cid:48)(cid:48) (cid:12)(cid:12)(cid:12) ∀ h (cid:48) , h (cid:48)(cid:48) ∈ [0 , h ◦ ] ∀ k ∈ N . (3.68)From the last property and from the uniform boundedness of the Dirichlet integral (cid:107)∇ u k (cid:107) L ( Ω ) ≤ + λ k ν k (cid:107)∇ U (cid:107) L ( Ω ) ≤ ffi ciently large k ) one can easily deduce thatsup h ∈ [0 , h ◦ ] (cid:90) Γ h (cid:12)(cid:12)(cid:12) ν k ∇ u k (cid:12)(cid:12)(cid:12) dS → k → ∞ , (3.70)in particular, (cid:90) Γ (cid:12)(cid:12)(cid:12) ν k ∇ u k (cid:12)(cid:12)(cid:12) dS → k → ∞ . (3.71)Then from the identity (3.60) and the estimate (3.64) we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Γ ( u k × ω k ) · n dS (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = λ k ν k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Γ ( U × ω k ) · n dS (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:90) Γ (cid:12)(cid:12)(cid:12) ν k ∇ u k (cid:12)(cid:12)(cid:12) dS → k → ∞ . (3.72)Hence the required estimate (3.58) is proved. Figure 2:
The domain Ω k ( t ) . Recall that by construction we have ∂ Ω k ( t ) = Γ ∪ (cid:102) S k ( t ; t ) (see (3.55) ). Integrating the equation ∆Φ k = ω k + ν k div( Φ k u k ) − ν k f k · u k (3.73)over the domain Ω k ( t ), we obtain (cid:90) (cid:101) S k ( t ; t ) ∇ Φ k · n dS + (cid:90) Γ ∇ Φ k · n dS = (cid:90) Ω k ( t ) ω k dx − ν k (cid:90) Ω k ( t ) f k · u k dx ν k (cid:90) (cid:101) S k ( t ; t ) Φ k u k · n dS + ν k (cid:90) Γ Φ k u k · n dS = (cid:90) Ω k ( t ) ω k dx − ν k (cid:90) Ω k ( t ) f k · u k dx + t ¯ F k + ν k (cid:90) Γ Φ k u k · n dS , (3.74)where ¯ F k = λ k ( F + · · · + F M ) and F j = (cid:82) Γ j a · n dS is the corresponding boundary flux (here we usethe identity Φ k ≡ − t on (cid:101) S k ( t ; t ) ). In view of (3.49) and (3.58)–(3.59) we can estimate (cid:90) (cid:101) S k ( t ; t ) |∇ Φ k | dS ≤ t F k + ε + ν k (cid:90) Ω k ( t ) f k · u k dx − (cid:90) Ω k ( t ) ω k dx (3.75)with F k = | ¯ F k | . By definition, ν k (cid:107) f k (cid:107) L ( Ω ) = λ k ν k (cid:107) f (cid:107) L ( Ω ) → k → ∞ . Therefore, using the uniformestimate (cid:107) u k (cid:107) L ( Ω ) ≤ const, we have (cid:12)(cid:12)(cid:12)(cid:12) ν k (cid:90) Ω k ( t ) f k · u k dx (cid:12)(cid:12)(cid:12)(cid:12) < ε for su ffi ciently large k . Then (3.75) yields (cid:90) (cid:101) S k ( t ; t ) |∇ Φ k | dS < t F k + ε − (cid:90) Ω k ( t ) ω k dx . (3.76)Since F k = λ k |F + · · · + F M | is uniformy bounded, and ε could be taken arbitrary small, the lastinequality implies the required estimate (3.50) for su ffi ciently large k . (cid:3) Lemma 3.10.
Assume that Ω ⊂ R × T σ is a helical domain with C boundary ∂ Ω , f = ∇ × b , b ∈ W , ( Ω ) and a ∈ W / , ( ∂ Ω ) are both helical functions, and (1 . is valid. Then assumptions ( E – NS – H ) with (3.23) lead to a contradiction. Proof.
Fix positive t < t ∗ . Put t i = − i t , i = , , . . . . Take the corresponding paprameters k ∗ i = k ∗ i ( t ) such that the inequality (3.50) holds for all k ≥ k ∗ i and for almost all t ∈ ( t i , t ∗ ). Inparticular, (3.50) holds for all k ≥ k ∗ i and for almost all t ∈ ( t i , t i ).For k ≥ k ∗ i put E ki = (cid:91) t ∈ [ t i , t i ] (cid:101) S k ( t i ; t ) . By the Coarea formula (see, e.g, [31]), for any integrable function g : E ki → R the equality (cid:90) E ki g |∇ Φ k | dx = t i (cid:90) t i (cid:90) (cid:101) S k ( t i ; t ) g ( x ) d H ( x ) dt (3.77)holds. In particular, taking g = |∇ Φ k | and using (3.50), we obtain (cid:90) E ki |∇ Φ k | dx = t i (cid:90) t i (cid:90) (cid:101) S k ( t i ; t ) |∇ Φ k | ( x ) d H ( x ) dt ≤ t i (cid:90) t i F t dt = F (cid:0) (2 t i ) − ( t i ) (cid:1) ≤ F t i (3.78)19ow, taking g = t i (cid:90) t i H (cid:0)(cid:101) S k ( t i ; t ) (cid:1) dt = (cid:90) E ki |∇ Φ k | dx ≤ (cid:18)(cid:90) E ki |∇ Φ k | dx (cid:19) (cid:0) meas( E ki ) (cid:1) ≤ (cid:113) F t i meas( E ki ) ≤ t i (cid:112) F meas( E ki ) . (3.79)By construction (see the arguments after the formula (3.43) ), for all k -regular values t the set (cid:101) S k ( t i ; t )is a finite union of C -smooth surfaces (tori) separating Γ = Γ ∪ . . . Γ M from Γ M + ∪ · · · ∪ Γ N . Itimplies, in partucular, that H (cid:0)(cid:101) S k ( t i ; t ) (cid:1) ≥ C ∗ = C ∗ ( Ω ) >
0. Therefore, by virtue of (3.79) we have C ∗ t i ≤ t i (cid:112) F meas( E ki ) , (3.80)in other words, C ∗ ≤ (cid:112) F meas( E ki ) (3.81)for all i ∈ N and for all k ≥ k ∗ i . By constructions, for all k ≥ k ∗ i the sets E ki , E k ( i − , . . . , E k , E k arepairwise disjoint. Therefore, the measure of some E ki could be made arbitrary small (for su ffi cientlylarge i and k ). This obviously contradicts the estimate (3.81). The Lemma is proved. (cid:3) ( b ) happens. Suppose now that the maximum of Φ is not attained on the boundary ∂ Ω :max j = ,..., N ˇ p j < ess sup x ∈ Ω Φ ( x ) . (3.82)Adding a constant to the pressure, we can assume without loss of generality thatmax j = ,..., N ˇ p j < p ∗ < < ess sup x ∈ Ω Φ ( x ) . (3.83)The proof for this case can be carried out with the same arguments as in the previous subsectionwith obvious simplifications. Let us describe some details. We start from the following simple fact. Lemma 3.11.
Under assumptions (3.83) there exists a straight segment F ⊂ D such that F ∩ A w = ∅ (i.e., F consists of the regular points of Φ and w , see Lemma 3.4 ), and < inf x ∈ F Φ ( x ) , moreover, the uniform convergence Φ k | F ⇒ Φ | F (3.84) holds. This fact follows easily from the definition of Sobolev spaces and from the weak convergence Φ k (cid:42) Φ in W , / ( D ) (see, e.g., the proof of Theorem 3.2 in [1] for details), so we omit its proof here.Fix the segment F from Lemma 3.11. Denote t ∗ = − p ∗ and fix a positive t < t ∗ . Using thearguments from the previous subsection, we can find a small parameter h ◦ > k ◦ ∈ N such that ∀ t ∈ ( t , t ∗ ) ∀ k ≥ k ◦ (cid:18) Φ k | F > , Φ k | Σ − h ◦ < − t (cid:19) , (3.85)where now by definition Σ − h = { x ∈ D : dist( x , ∂ D ) = h } (see Fig. 3). ess sup Φ = ∞ is not excluded. igure 3. Further, using the same arguments, for almost all t ∈ ( t , t ∗ ) we can find a set S k ( t ; t ) consistingof finite disjoint family of C smooth closed curves (cycles) such that Φ k ≡ − t on S k ( t ; t ). Moreover,there is an open set W k ( t ; t ) ⊂ D satisfying the relations W k ( t ; t ) ⊃ F , (3.86) ∂ W k ( t ; t ) = S k ( t ; t ) (3.87)(cf. with (3.45) ), and (cid:90) (cid:101) S k ( t ; t ) ∇ Φ k · n dS = − (cid:90) (cid:101) S k ( t ; t ) |∇ Φ k | dS < , (3.88)where n is the unit outward normal vector to ∂ (cid:101) W k ( t ; t ). Here, by construction, (cid:101) S k ( t ; t ), is a finiteunion of smooth disjoint surfaces (tori).Further we can prove the estimate (3.50) for our case integrating the same identity (3.51) over thedomain Ω k ( t ) = (cid:101) W k ( t ; t ) with ∂ Ω k ( t ) = (cid:101) S k ( t ; t ). Note that now the proof is even much simpler sincewe have no boundary integrals over subsets of ∂ Ω — in other words, now we do not need to proveestimates of type (3.52)–(3.53) and (3.58)–(3.59)After the estimate (3.50) is obtained, we can derive a contradiction in exactly the same way as inLemma 3.10 of the previous section.Now we can summarize the results of the last two subsections in the following statement. Lemma 3.12.
Assume that Ω ⊂ R × T σ is a helical domain with C boundary ∂ Ω , f = ∇ × b , b ∈ W , ( Ω ) and a ∈ W / , ( ∂ Ω ) are both helical functions, and (1 . is valid. Let ( E – NS – H ) befulfilled. Then each assumptions (3.23) or (3.24) lead to a contradiction. Proof of Theorem 1.1.
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