Boundary Regularity of Flows under Perfect Slip Boundary Conditions

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1 Boundary Regularity of Flows under Perfect Slip Boundary Conditions Petr Kaplický, Jakub Tichý Charles University in Prague, Department of Mathematical Analysis, Sokolovská 83, Praha 8, Czech Republic Abstract: We investigate boundary regularity of solutions of generalized Stokes equations. The problem is complemented with perfect slip boundary conditions. The nonlinear elliptic operator satisfies non-standard Φ-growth conditions. We show the existence of second derivatives of velocity and their optimal regularity. Keywords: Boundary regularity; Perfect Slip Boundary Condition; Generalized Stokes System. Mathematics Subject Classification (2000: 35B65, 35J60, 35Q35, 76D03. 1 Introduction This paper is concerned with steady flows of an incompressible fluid in a bounded domain R n, n N, n 2 described by the system of equations div S(D(u + π = f in, (1.1 div u = 0 in, (1.2 where u = (u 1,..., u n is the velocity, π represents the pressure, f stands for the density of volume forces and S denotes the extra stress tensor. D(u is the symmetric part of the velocity gradient, i.e. D(u = 1 2 [ u + ( u ]. We consider the constitutive relation for S of the form S = µ( D(u D(u, (1.3 where µ : [0, [0, is generally non-linear function called generalized viscosity. We can construct scalar potential Φ : [0, [0, to the stress tensor S, i.e. S ij (A = ij Φ( A = Φ ( A A ij A A R n n sym. (1.4 We require following assumption to be fulfilled: Assumption 1.1. We suppose that Φ C 1,1 ((0, C 1 ([0, is N-function, Φ 2, Φ 2 and satisfies the following assumption: There exist constants C 1, C 2 > 0 such that, for s > 0 C 1 Φ (s sφ (s C 2 Φ (s (A1 and Φ (s is almost monotone, i.e. there exists C > 0 such that for all s (0, t] either Φ (s CΦ (t (almost increasing or Φ (s CΦ (t (almost decreasing. Remark 1.1. Every N-function Φ satisfying 2 condition automatically satisfies for some c 1, c 2 > 0. c 1 Φ(s sφ (s c 2 Φ(s (1.5 The relation (1.4 and Assumption 1.1 give us non-standard Φ growth conditions (see [7]. kaplicky@karlin.mff.cuni.cz kuba.tichy@gmail.com 1

2 Corollary 1.1. There are constants C 3, C 4 > 0 that for all A, B R n n sym holds ( S(A S(B ( A B C3 Φ ( A + B A B 2, S(A S(B C 4 Φ ( A + B A B. (A2 Example 1.1. Let us mention that growth conditions ( A2 allow us to consider models with a great deal of disparity, for example power-law models D(u µ( D(u = µ 0 (1 + D(u 2 p 2 2, Φ( D(u = µ0 (1 + s 2 p 2 2 s ds, µ( D(u = µ 0 (1 + D(u p 2, µ 0 R +, p (1,. Also the singular case 0 D(u Φ( D(u = µ 0 (1 + s p 2 s ds, 0 is included. µ( D(u = µ 0 D(u p 2, D(u Φ( D(u = µ 0 s p 1 ds 0 We complement the equations (1.1 and (1.2 by perfect slip boundary conditions u ν = 0, (Sν τ = 0, on. (1.6 By ν we denote an outward normal vector and τ stands for any tangent vector to. Before formulating our main result, we would like to say that the system (1.1 and (1.2 is classical. It is called generalized Stokes system. If we add the term div(u u to the equation (1.1, we talk about generalized Navier-Stokes system. In this paper we don t consider the convective term div(u u, we focus only on the interaction of the elliptic term S with boundary conditions. The generalized model was proposed by O. A. Ladyzhenskaya in [20]. Since that time many results about existence of the weak solutions and their qualitative properties have been proven. To mention only a few of them, we can refer for example to [2]-[6], [12]-[18], [21], [23] and [26]. In the field of regularity of the weak solution, there are many open problems. System (1.1 and (1.2 has been studied especially with homogeneous Dirichlet and periodic boundary conditions. Recently Navier boundary conditions and as a special case perfect slip boundary conditions are very popular. We begin with the definition of the weak solution of the problem (1.1, (1.2 and (1.6. Definition 1.1. We say that the function u is a weak solution of the problem ( 1.1, ( 1.2 and ( 1.6 if u W 1,Φ ν,div (n and weak formulation S(D(u:D(ϕ dx = f, ϕ (1.7 holds for all ϕ W 1,Φ ν,div (n. It is well known that the weak solution exists and is unique. It could be easily proven using the monotone operator theory. Before stating our main result, we define function V and N-function Ψ which are very well suited for expressing differentiability properties of weak solutions. Definition of the function V in the framework of Orlicz spaces was first given in [7]. For given Φ we define the N-function Ψ by and we define V (A such that Ψ( A is a scalar potential to V (A, i.e. Ψ (s := Φ (ss. (1.8 V ij (A := ij Ψ( A = Ψ ( A A ij A A R n n sym. (1.9 It is shown in [7, Lemma 25] that there are C, C > 0 that CΨ (s Φ (s C Ψ (s. (1.10 2

3 Example 1.2. In the case of power-law models we have D(u V ( D(u = µ 0 (1 + D(u 2 p 2 4 D(u, Ψ( D(u = µ0 (1 + s 2 p 2 4 s ds, D(u V ( D(u = µ 0 (1 + D(u p 2 2 D(u, Ψ( D(u = µ0 (1 + s p 2 2 s ds, µ 0 R +, p (1,. Also the singular case 0 0 is included. D(u V ( D(u = µ 0 D(u p 2 2 D(u, Ψ( D(u = µ0 s p 2 ds 0 Now we state the main theorem of the article. Theorem 1.1. Let R n be a non-circular domain, C 3, f W 1,Φ ( n and suppose Assumption 1.1 is fulfilled. Let u be a weak solution of ( 1.1, ( 1.2 and ( 1.6. Then there exists constant C independent of u, such that ( V (D(u 2 dx C Φ ( f dx + Φ ( f dx. (1.11 n If moreover f = div g and g L (Φ q, q [ 1, n 2], then the corresponding pressure π satisfies ( (Φ ( π π q dx C V (D(u 2 dx + (Φ ( g q. (1.12 Remark 1.2 (Assumptions on f. For the special choice of Φ, assumption on f could be weakened. For example if we would consider Φ such that Φ is bounded and decreasing (which corresponds to the power-law model with p < 2 and non-singular case, it is sufficient to take f L Φ ( n (compare [18]. In this part we would like to accentuate the paper [11], because C. Ebmeyer gains with different method very similar results as we have. The author studies the problem (1.1 with the convective term div(u u, together with (1.2 and perfect slip boundary conditions in three dimensions. He suppose that the tensor S has got the p-potential structure. He is interested in the case p < 2. The author gains the regularity results in Sobolev spaces with fractional derivatives and in Nikolskiĭ spaces. Among others, he obtains (κ + D(u p 2 D(u 2 <, κ {0, 1} for p ( 9 5, 2 in the case of power-law Navier-Stokes system and for p (1, 2 in the case of power-law Stokes system. He uses the fact that perfect slip boundary conditions allow to extend the solution beyond the flat boundary. Results are formulated for the flat boundary and at the end by the local change of coordinates the author gains results for general shape of the boundary. In [6] the authors are concerned with the system (1.1 and (1.2 equipped with homogeneous Dirichlet boundary conditions. The extra stress tensor is given by a power-law ansatz with exponent p 2. Among others they show that V (D(u W 1, 2q p+q 2 ( n n for q = np+2 p n 2, if n 3 and for all q <, if n = 2. In tangential directions they are able to improve regularity properties to τ αv (D(u 2 dx <, but in the normal direction there is a loss of regularity due to the absence of some special weighted version of Korn s inequality and the presence of the pressure. The proof of Theorem 1.1 is divided into three main parts. In the first part (Section 3 we show that for the quadratic potential, i.e. Φ is bounded from below and from above (which corresponds to the case p = 2 in the power-law models, the solution u belongs to the space W 2,2 ( n. In the second part (Section 4 we introduce the regularized problem where instead of the generalized viscosity µ we consider truncated viscosity µ ε = min ( max(µ( D(u eε, ε, 1 ε for ε (0, 1. Using the fact that for the regularized problem it holds u eε W 2,2 ( n we show that the term µε ( D(u eε n D(u eε 2 dx is estimated by lower order terms and small same order terms (see Lemma 4.1. The main idea is to test regularized version of (1.1 by second normal derivatives (up to some correction, which is possible due to perfect slip boundary conditions. Further we obtain similar result for the term µε ( D(u eε α D(u eε 2 dx, α {1,..., n 1} (see Lemma 4.2. It can be done by taking tangent derivative of regularized version of (1.1 and testing by suitable function. We finish Section 4 by putting together estimates from Lemma 4.1 and Lemma 4.1, estimating lower order terms and absorbing small same order terms into the left hand side. Although we are using the function µ ε in Section 4, due to the Assumption 1.1 and constitutive relation (1.3 we have cµ(s Φ (s c µ(s for some c, c > 0 and due to the relation (1.10 we easily obtain the result in the terms of the function V ε. In the third part (Section 5 we pass from the regularized problem to the original one. 3

4 In this paper we use standard notation for Lebesgue spaces (L p (, p, Sobolev spaces (W k,p (, k,p, 1 p, k N, Orlicz spaces (L Φ (, Φ and Orlicz-Sobolev spaces (W 1,Φ (, 1,Φ, R n is a domain and C 3. We define 1 W 1,Φ ν,div (n = {ϕ i W 1,Φ (, i = 1,..., n, ϕ ν = 0 on, div ϕ = 0}. Weak partial derivatives in directions x i are denoted by i, i = 1,..., n. The tangential weak derivatives near the boundary are denoted by τ, the normal weak derivatives by ν. We use universal constants C, c > 0 which may vary in different occurrences. 2 Preliminaries In the following we use the notation x = (x, x n. We suppose that C 3, therefore there exists c 0 > 0 such that for all a 0 > 0 there exists n 0 points P, r > 0 and open smooth set 0 that we have 0 P B r (P and for each point P there exists local system of coordinates for which P = 0 and the boundary is locally described by C 3 mapping a P that for x (, and α, β, γ {1,..., n 1} fulfills x x n = a P (x, B (P = {(x, x n B r (P and x n > a P (x } =: P, α a P (0 = 0, α a P (x a 0, α β a P (x + α β γ a P (x c 0. (2.1 Points P can be divided into k groups such that in each group P are disjoint and k depends only on dimension n. By the vector a P (x we consider the vector ( 1 a p,... a p, 0. Let the cut-off function ξ P (x C (B (P and reaches values ξ P (x = 1 x B r (P, (0, 1 x B 2r (P \ B r (P, = 0 R n \ B 2r (P. Let us fix P and drop for simplicity the index P. Next, we assume that we work in the coordinate system corresponding to P. Particularly, P = 0. The tangent vector in the α direction and the outer normal vector to are defined as (2.2 τ α = ( 0,..., 0, 1, 0,..., 0, α a(x, α = 1,..., n 1, ν = ( 1 a(x,..., a(x, 1, (2.3 tangent and normal derivatives as τ α = α + α a(x n, α = 1,..., n 1, ν = α a(x α n. (2.4 We need to work with τ α, ν, τ α and ν not only on, but on the whole, therefore we realize that identities (2.3 and (2.4 actually define τ α, ν, τ α and ν on. Next, we assume that u is sufficiently smooth. It is easy to see, that τ α(u ν = 0 on. Perfect slip condition (Sν τ α = 0 on is equivalent to the condition 0 = τ α D(uν = β a( α u β + β u α ( n u α + α u n + β=1 + (2.5 α a β a( β u n + n u β 2 α a n u n β=1 because of the constitutive relation ( As we use the notation W 1,Φ ν,div (n for vector-valued functions with components in the function space W 1,Φ ν,div (, we use analogically the notation W 1,Φ ν,div (n n for tensor-valued functions. 4

5 Remark 2.1. It would be very interesting to extend results of Theorem 1.1 also for Navier boundary conditions. This method can t be easily modified, because we are using the relation ( 2.5 or similar identities. Navier boundary conditions can t be expressed in this alternative way. Remark 2.2. No boundary term arises from the tangent integration by parts, because if supp g B supp f B then ( τ αf g dx = ( fτ α g dx = where we used that div τ α = 0 and τ α ν = 0 on. + f g div τ α dx f ( gτ α dx+ f gτ α ν dσ = f τ αg dx, or Now we state some basic facts about N-functions. More information about N functions can be found for example in [19] or [25]. Definition 2.1. A real function Φ : R + R + is called N-function if there exists the derivative Φ which is right continuous for s 0, positive for s > 0, non-decreasing and satisfies Φ (0 = 0 and lim s Φ (s =. Definition 2.2. N-function Φ is said to satisfy the 2 condition, denoted Φ 2, if there exists a positive constant C, such that Φ(2s CΦ(s for s > 0. By 2 (Φ we denote the smallest such as constant C. By (Φ 1 : R + R + we denote the function The complementary function of Φ is defined as (Φ 1 (s := sup{t R + : Φ (t s}. (2.6 Φ (s := s 0 (Φ 1 (t dt. (2.7 It is again an N-function and for all δ > 0 there exists c(δ > 0 such that for all s, t 0 holds so called Young s inequality st δφ(s + c(δφ (t. (2.8 For a measurable function f we can define gauge norm as { ( f(x } f Φ := inf λ > 0 : Φ dx 1. (2.9 λ The Orlicz space L Φ ( is defined as the set {f : f Φ, < }. 2 Let f L Φ (, g L Φ (. Then fg L 1 ( and fg dx 2 f Φ g Φ. (2.10 Especially, for f = χ we have g dx There are two constants C, C > 0 such that 2 Φ 1 ( 1 g Φ. (2.11 CΦ (Φ (s Φ(s C Φ (Φ (s. (2.12 Lemma 2.1. Let Φ and Φ be a pair of complementary N-functions. If Φ 2, then Φ(s Cs α, s > 0 for some C > 0 and α > 1, and Φ satisfies Φ (t C t β, t > 0 for some C > 0 and β > 1. Proof. See [25, Section 2.3, Corollary 5]. For a 0 we define shifted N-function Φ a by Φ a(s s := Φ (a + s. (2.13 a + s This basically states that cφ a(s Φ (a + s CΦ a(s for some C, c > 0. Moreover, Φ a 2 and Φ 2 uniformly in a (see [7, Appendix]. 2 In our case Φ always fulfills 2 condition, therefore we have equivalence of following sets: {f; λ > 0 : ( Φ f(x dx < }, λ {f; λ > 0 : ( Φ f(x dx < } and {f; λ Φ( f dx < }. 5

6 Lemma 2.2 (Shift change. Let Φ fulfills Assumption 1.1. Then for any δ > 0 there exists c(δ > 1 such that for all A, B R n n and s 0 Proof. See [9, Lemma 5.15]. Φ A (s c(δφ B (s + δ V (A V (B 2. (2.14 Lemma 2.3 (Korn s inequality. Let R n be a bounded C 0,1 non-circular domain. Let Φ be N-function with Φ 2 and Φ 2. Then for all u Wν 1,Φ ( n it holds that Φ( u dx + Φ( u dx C Φ( D(u dx, (2.15 where C = C(, 2 (Φ, 2 (Φ. Proof. The result follows from Korn s Lemma in [10, Theorem 6.13], Poincaré inequality and 2 condition. The assumption on non-circularity of the domain is necessary, because we need to know that if D(u = 0 a.e. in and u ν = 0 on then u = 0 a.e. in. It holds when is non-circular. See for example [24]. Lemma 2.4 (Bogovskiĭ s Lemma. Let R n be a bounded C 2 domain. Let Φ be N-function with Φ 2 and Φ 2, g W 1,Φ ( n and g dx = 0. Then there exists z W 2,Φ ( n W 1,Φ 0 ( 2 solving div z = g in, (2.16 z = 0 on. (2.17 Moreover, there exists C > 0 depending only on 2 (Φ and 2 (Φ such that Φ( z dx + Φ( z dx C Φ( g dx, (2.18 ( Φ( 2 z dx C Φ( g dx + Φ( g dx. (2.19 Proof. The result can be proven by the method in [10, Theorem 6.6]. First we show by extrapolation technique [10, Theorem 5.2 and Theorem 6.1] that the estimate holds if g W 1,Φ 0. We finish the proof by the application of the inverse trace theorem as in [1, Corollary 3.7 and Corollary 3.8]. 3 Quadratic potential In this section we will confine ourselves to the case Φ is bounded from below and from above. In the definition of the weak solution (Definition 1.1 the space W 1,Φ ν,div (n reduces to W 1,2 ν,div (n. Following lemma claims that the second gradient of u belongs to L 2 ( n. Lemma 3.1. Let R n, C 3, f L 2 ( n. Let Assumption 1.1 be fulfilled for Φ [c 3, c 4 ] (0,. Then for every weak solution of the problem ( 1.1, ( 1.2 and ( 1.6 holds u W 2,2 ( n. (3.1 We omit the proof. Up to some modifications because of the boundary conditions we would follow the method used in [22, Section 3] where the authors are dealing with the evolutional case in 3D under homogeneous Dirichlet boundary conditions. The authors are interested in the power-law model for case p 2. The standard approach is to show the interior regularity first and then the regularity up to the boundary. The interior regularity would be obtained easily using difference quotient technique. The boundary regularity would be split into the tangent direction and normal direction regularity. In tangent direction we would use also the difference quotient technique, but unlike the interior regularity we would have to deal with lots of lower order terms which would appear because of the boundary conditions. In normal direction we would generalize the method used in [22, Section 3] from 3D to n dimensions. This computation for stationary 2D problem is also done in articles [16] and [17]. Remark 3.1 (Reconstruction of the pressure. From De Rham theorem (for formulation and proof see for example [1] we know that the right hand side of the equation π = f + div S (3.2 can be written in the sense of distributions in the gradient form. The right hand side of ( 3.2 is in L 2 ( n. Adding the assumption π dx = 0 we get by Poincaré inequality that there exists the pressure π W 1,2 ( n. 6

7 4 The regularized problem In this section we are concerned with the regularized boundary value problem div S ε (D(u eε + π eε = f e in, (4.1 div u eε = 0 in, (4.2 u eε ν = 0 & (S ε ν τ α = 0 on, (4.3 where the regularization of f is chosen in order to have f e C ( n and f e f in W 1,Φ ( n as e 0 and S ε = µ ε ( D(u eε D(u eε, (4.4 ( µ ε = min max(µ( D(u eε, ε, 1, ε (0, 1. (4.5 ε Scalar potential Φ ε to S ε can be constructed in the following way Φ ε (s := s 0 µ ε (tt dt. (4.6 As one can easily check, the assumption (A1 and therefore also growth conditions (A2 hold if we replace Φ and S by Φ ε and S ε. Proposition 4.1. If Φ 2, Φ 2 then also Φ ε 2, (Φ ε 2 and 2 (Φ ε, (Φ ε doesn t depend od ε. Proof. At first we consider only truncation of µ from below, i.e. µ ε (s := max(µ ε (s, ε, Φ ε (s = s 0 µε (tt dt. It is enough to show µ ε (2s Cµ ε (s. We can distinguish 4 cases: (i µ(s ε and µ(2s ε. Then µ ε (s = µ(s, µ ε (2s = µ(2s and the claim holds, because Φ 2. (ii µ(s ε and µ(2s ε. Then µ ε (s = µ(s ε and µ ε (2s = ε. Trivially µ ε (2s µ ε (s. (iii µ(s ε and µ(2s ε. Then µ ε (s = ε and µ ε (2s = µ(2s, µ(2s Cµ(s Cε Cµ ε (s and the claim holds. (iv µ(s ε and µ(2s ε. Then µ ε (s = µ ε (2s = ε and the claim trivially holds. We could do the same discussion for µ truncated from above, i.e. µ ε (s := min(µ(s, 1 ε. Putting these two considerations together we would obtain the proof for the viscosity defined in (4.5. In the case of the complementary function we would proceed similarly. The first apriori estimate, i.e. testing by the weak solution, gives us S ε (D(u eε : D(u eε dx f e, u eε. (4.7 Using Young s inequality (2.8 and Korn s inequality (2.15 on the right hand side, definition of function V ε or the potential Φ ε on the left hand side, the relation (4.7 can be rewritten to the form V ε (D(u eε 2 dx Φ ε ( D(u eε dx C (Φ ε ( f e dx =: M ap. (4.8 The fact that Φ ε 2 and (Φ ε 2 guarantees the existence of some δ > 0 such that D(u eε L 1+δ ( n n (see Lemma 2.1. Using Korn s inequality 2.3 we obtain u eε L 1+δ ( n n. Now we fix a point P and work in the local system of coordinates for which P = 0. We work in P, but as before, we will drop the index P. Following lemma shows us that the integral r µ ε ( D(u eε n D(u eε 2 dx is estimated by lower order terms and small terms of the same order which can be absorbed in the left hand side at the end. Lemma 4.1. Let R n, C 3 be a non-circular domain. Let u eε be the weak solution of the regularized problem ( ( 4.3. Then there exists a positive constant C independent of u eε and a 0 such that µ ε ( D(u eε n D(u eε 2 dx C ( M + N + (Φ ε ( f e dx + (Φ ε ( f e dx, (4.9 r where M = i,j,k,l=1 ( i Sjk ε l u eε mξ dx + i Sjku ε eε mξ dx + Sij ε k l u eε mξ dx, N = α a µ ε ( D(u eε 2 u eε 2 dx. 7

8 Remark 4.1. The term N contains second derivatives of u, but it is also multiplied by the function α a. We can pick this function as small as we want which allows us to absorb N in the left hand side, see ( 2.1. In M there are only lower order terms. Instead of the cut-off function ξ we should write ξ, where supp ξ supp ξ, because we already estimated some terms containing the derivative of a in M. For simplicity let us write only ξ. Estimates of M and N will be done at the end together with terms from the lemma concerning tangent part of the second gradients. Proof. From results of Section 3 we know that u eε W 2,2 ( n, S ε (D(u eε W 1,2 ( n n, π eε W 1,2 (. We can rewrite (4.1 into components, multiply by a suitable test function and integrate over. The test function has to belong at least to L 2 ( n in order the integrals have sense. k,l=1 l S ε klϕ k dx + k=1 k π eε ϕ k dx = k=1 f e kϕ k dx, (4.10 where ϕ L 2 ( n and supp ϕ. We would like to use the second normal derivatives of solution as a test function in (4.10. One can easily verify that this function isn t divergence free and doesn t fulfill boundary conditions, so we would have to deal with terms containing the pressure. Instead we take as a test function where we denoted ( ϕ = n Θ 1,..., n Θ, α Θ α, (4.11 Θ α := ν (u eε τ α ξ 2 u eε ( ν τ α + τ ανξ 2, α = 1,..., n 1. (4.12 The test function ϕ is constructed in order to fulfill div ϕ = 0. It also has a useful property: Proposition 4.2. It holds Θ α = 0 for all α {1,..., n 1} on in the sense of traces. Proof. We use two facts. First, it holds that τ α(u eε ν = 0 on, which is u eε τ αν = τ αu eε ν. Second, boundary conditions (S ε ν τ α = 0 can be rewritten as τ αu eε ν + ν u eε τ α = 0 which gives us ν u eε τ α = τ αu eε ν. Θ α = ν u eε τ α ξ 2 + u eε ν τ α ξ 2 u eε ξ 2 ν τ α u eε ξ 2 τ αν = = τ αu eε νξ 2 u eε ξ 2 τ αν = u eε τ ανξ 2 u eε ξ 2 τ αν = 0. (4.13 Proposition 4.2 helps us to get rid of terms in (4.10 containing the pressure. In the case we are not on, it is useful to write out Θ α. Θ α = ( i u eε j τj α ν i u eε i j ν i τj α ξ 2 = n u eε α ξ 2 α a n u eε n ξ 2 + i,j=1 ( + β a( β u eε α + α a β u eε n α β au eε β ξ 2, β=1 where we use only the definition of the normal, the normal derivative, the tangent and the tangent derivative. Proposition 4.3. Let ϕ be defined by ( 4.11 and ( Then k=1 Proof. If π eε W 2,2 (, straightforward computation gives us k=1 (4.14 k π eε ϕ k dx = 0. (4.15 k π eε ϕ k dx = α π eε n Θ α dx n π eε α Θ α dx = 0, (4.16 where we integrated by parts twice and use the fact that boundary integrals are equal to zero because Θ α = 0 on due to Proposition 4.2. For π eε W 1,2 ( the statement follows from density of W 2,2 ( in W 1,2 (. For simplicity let us denote A kl := l Sklϕ ε k dx, k, l = 1,..., n, B j kl := j S ε kl j l u eε k ξ 2 dx, j, k, l = 1,..., n. 8

9 Our goal is to express terms Bkl n from A kl. We insert (4.11 into (4.10 and estimate terms A kl, k, l = 1,..., n. It will be done in 4 steps. First we focus on A αβ for fixed α, β {1,..., n 1}. Later we estimate A αn, A nβ and finally A nn. In the first term we integrate by parts twice, use the fact that there are no boundary terms and apply (4.14. A αβ = β Sαβϕ ε α dx = β Sαβ ε n Θ α dx = n Sαβ ε β Θ α dx = = n Sαβ ε β + [ α a n u eε n ξ 2 β=1 n S ε αβ β n u eε α ξ 2 dx M N + B n αβ. In the second term A αn we use only (4.14 to get B n αn. ( β a( β u eε α + α a β u eε n α β au eε β ξ 2] + A αn = n Sαnϕ ε α dx = n Sαn ε n Θ α dx n Sαn ε nu 2 eε α ξ 2 dx M N = Bαn n M N. At the beginning in the isolation of term Bnβ n from A nβ we use (4.14 and n u eε n comes from the divergence-free constraint. A nβ = β Snβϕ ε n dx = = β S ε nβ β S ε nβ 2 nu eε n ξ 2 dx M N. α Θ α dx β Snβ ε (4.17 (4.18 = αu eε α, which α n u eε α ξ 2 dx M N = (4.19 If we integrate by parts twice in the first isolated term on the right hand side of (4.19, we are done. At this moment there would appear boundary integrals. To avoid them we add and subtract some small terms (which could be included in N in order to have (S ε ν τ β instead of S ε nβ in the isolated term. Writing out (Sε ν τ β we get Therefore Snβ ε = (S ε ν τ β + α asαβ ε + α a β asαn ε β asnn. ε (4.20 β Snβ ε nu 2 eε n ξ 2 dx n ξ 2 dx M N = n [(S ε ντ β ] β n u eε n ξ 2 dx M N n Snβ ε n β u eε n ξ 2 dx M N = Bnβ n M N. β [(S ε ντ β ] 2 nu eε In the last term A nn we use only (4.14 and the incompressibility condition n u eε n = αu eε α. (4.21 A nn = n Snnϕ ε n dx = n S ε nn α n u eε α ξ 2 dx M N = n S ε nn 2 nu eε n ξ 2 dx M N = B n nn M N. (4.22 ( fkϕ e k dx f e Θ α dx C (Φ ε ( f e dx + Φ ε ( D(u eε dx. (4.23 k=1 Collecting ( (4.23 and using the first apriori estimate (4.8 in (4.23 we obtain r µ ε ( D(u eε n D(u eε 2 dx ( C M + N + n Skl ε n l u eε k ξ 2 dx = Bkl n k,l=1 k,l=1 (Φ ε ( f e dx + (Φ ε ( f e dx, 9 (4.24

10 which completes the proof. Now we formulate lemma about the boundedness of the term which contains tangential parts of the second gradient. Lemma 4.2. Let R n, C 3 be a non-circular domain. Let u eε be the weak solution of the regularized problem ( ( 4.3. Then there exist an absolute constant C 5 > 0 and constant C > 0 which may depend on a 0 such that for all α {1,..., n 1} holds µ ε ( D(u eε α D(u eε 2 dx C 5 N + C ( M + r (Φ ε ( f e dx + (Φ ε ( f e dx. (4.25 Proof. At first we rewrite the equation (4.1 into components, differentiate in tangent direction, multiply by ϕ W 1,Φε ν,div (n, supp ϕ, integrate over and get k,l=1 τ α l S ε klϕ k dx + k=1 τ α k π eε ϕ k dx = k=1 τ αf e kϕ k dx. (4.26 We would like to integrate by parts in first two terms of (4.26 in direction x l, resp. x k. The tangent derivative doesn t commute with the derivative k, but we can use the relation τ α k π eε = k τ απ eε k α a n π eε (4.27 and similar one for τ α l Skl ε. In the last term of (4.27 we use the equation (4.1 and replace the term nπ eε by (fn e + n l=1 lsnl ε. We obtain τ αskl ε l ϕ k dx = ( τ αfk e + k α afnϕ e k dx+ k,l=1 + k,l=1 k=1 ( k α a l S ε nl l α a n S ε klϕ k dx. The term n k=1 τ απ eε k ϕ k dx disappeared, because div ϕ = 0. As a test function ϕ we take where z is the solution of (4.28 ϕ = τ αu eε ξ 2 + (u τ αν ν ν 2 ξ2 + z = ϕ a + ϕ b + z, (4.29 div z = div( ϕ a ϕ b in, (4.30 z = 0 on. (4.31 The role of z is to ensure that div ϕ = 0. One easily checks that ϕ ν = 0 on : ϕ ν = (ϕ a + ϕ b ν = ( τ αu eε νξ 2 + (u eε τ ανξ 2 = τ α(u eε νξ 2 = 0. (4.32 Therefore the compatibility condition holds 0 = z ν dσ = div z dx = div( ϕ a ϕ b dx = (ϕ a + ϕ b ν dσ = 0 (4.33 and z solving (4.30 and (4.31 exists by Bogovskiĭ s Lemma 2.4 and has the following properties: Φ ε ( z dx + Φ ε ( z dx C Φ ε ( D(u eε dx CM ap. (4.34 After putting (4.29 into (4.26, using results of Lemma 4.1 and Young s inequality we obtain µ ε ( D(u eε α D(u eε 2 ξ 2 dx + J C 5 N + C ( M + (Φ ε ( f e dx + (Φ ε ( f e dx + K + L (4.35 where J = τ αs ε (D(u eε : D(z dx, K = Φ ε ( z dx, L = S ε (D(u eε ( z + z dx. (

11 It remains to estimate terms containing Bogovskiĭ s correction z. The term K, which came from the first integral on the right hand side of (4.28, is estimated directly by (4.34. The term L, which came from the second term on the right hand side of (4.28 after integration by parts and using the fact that there is no boundary integral due to z = 0 on, is estimated using Young s inequality and (4.34: ( L (Φ ε ( D(u eε ( z + z dx C Φ ε ( D(u eε dx+ Φ ε ( z dx+ Φ ε ( z dx CM ap. (4.37 To estimate J the assumption on almost monotonicity of Φ is needed. For almost increasing (Φ ε we move J to the right hand side and apply Young s inequality: J δ (Φ ε ( D(u eε 2 u eε 2 dx + c(δ (Φ ε ( D(u eε z 2 dx = J 1 + J 2. (4.38 If we take δ = α a, then the term J 1 is same as N. In the term J 2 we use the fact that (Φ ε is almost increasing, the definition of shifted N-function (2.13, V ε (D(u eε V ε ( D(u eε 2 dx C V ε (D(u eε V ε (D(u eε 2 dx (see [8, Lemma 2.8], shift change (2.14, Bogovskiĭ s Lemma 2.4 and shift change (2.14 again. J 2 C (Φ ε ( D(u eε + z z 2 dx C Φ ε D(u eε ( z dx C Φ ε D(u eε ( z dx + C V ε (D(u eε V ε ( D(u eε 2 dx (4.39 C Φ ε D(u eε ( ueε dx + CM ap C Φ ε D(u eε ( ueε dx + CM ap = J 3 + CM ap. Using the definition of shifted N-function (2.13, the fact that (Φ ε is almost increasing, 2 condition and Korn s inequality (2.3 we obtain J 3 C (Φ ε ( D(u eε + u eε u eε 2 dx C (Φ ε (2 u eε u eε 2 dx (4.40 C Φ ε ( u eε dx C Φ ε ( D(u eε dx CM ap. If the (Φ ε is almost decreasing we integrate in J by parts using Lemma 2.2 and get: J = S ε (D(u eε : τ αd(z dx. (4.41 Using Young s inequality with Φ ε D(u eε and the fact that (Φε D(u eε ((Φ ε ( D(u eε CΦ ε ( D(u eε we obtain J C (Φ ε ( D(u eε 2 z dx C (Φ ε D(u eε ((Φ ε ( D(u eε dx+ (4.42 +C Φ ε D(u eε ( 2 z dx CM ap + C Φ ε D(u eε ( 2 z dx = CM ap + J 4. In the term J 4 we apply shift change (2.14, Bogovskiĭ s Lemma 2.4, again shift change (2.14 and finally monotonicity of (Φ ε. J 4 δ Φ ε D(u eε ( 2 z dx + c(δ V ε (D(u eε 2 dx δ Φ ε D(u eε ( 2 u dx + CM ap δ Φ ε D(u eε ( 2 u dx + CM ap δ (Φ ε ( D(u eε + 2 u eε 2 u eε 2 dx + CM ap δ (Φ ε ( D(u eε 2 u eε 2 dx + CM ap = J 1 + CM ap. Following lemma combines results from Lemma 4.1 and Lemma 4.2 and estimates terms M and N. (

12 Lemma 4.3. Let R n, C 3 be a non-circular domain. Let u eε be the weak solution of the regularized problem ( ( 4.3. Then there exists a positive constant C independent of u eε such that ( µ ε ( D(u eε 2 u eε 2 dx C (Φ ε ( f e dx + (Φ ε ( f e dx. (4.44 Proof: We put together (4.9 and (4.25. At first we divide the term M into three groups: M 1 = i,j,k,l,m=1 Sij ε k l u eε mξ dx, M 2 = i,j,k,l=1 i Sjku ε eε l ξ dx, M 3 = and show that M 1 - M 3 are estimated by M ap and N. Let s start with M 1 : i,j,k,l,m=1 i Sjk ε l u eε mξ dx M 1 δ µ ε ( D(u eε 2 u eε 2 dx + c(δ Φ ε ( D(u eε dx N + CM ap. (4.45 where we used only Young s inequality with δ > 0. In M 2 we integrate by parts M 2 C S ε ( u eε + u eε dx + C S ε u eε dσ = M M 2.2. (4.46 To estimate M 2.1 we use Young s inequality (2.8, relation (2.12 and Korn s inequality (2.3. M 2.1 C Φ ε ( D(u eε dx + C Φ ε ( u eε dx + C Φ ε ( u eε dx C M ap. (4.47 The boundary term M 2.2 could be rewritten with the help of Young s inequality (2.8 M 2.2 C (Φ ε ( D(u eε u eε dσ C(δ Φ ε ( u eε dσ + δ Φ ε ( D(u eε dσ = M M 2.4. (4.48 To estimate M 2.3 we show that Φ ε ( u eε W 1,1 ( n and therefore Φ ε ( u eε L 1 ( n by the trace theorem. M 2.3 = C Φ ε ( u eε dσ C Φ ε ( u eε dx + C Φ ε ( u eε C (Φ ε ( u eε u eε dx + C Φ ε ( u eε C Φ ε ( u eε + 2C Φ ε ( u eε CM ap. (4.49 where we used Young s inequality (2.8 and Korn s inequality 2.3. To estimate M 2.4 we start with Φ ε ( u eε W 1,1 ( and use the imbedding W 1,1 ( L 1 (. M 2.4 = C Φ ε ( D(u eε dσ C Φ ε ( D(u eε dx + C Φ ε ( D(u eε dx C (Φ ε ( D(u eε D(u eε 2 u eε dx + C Φ ε ( D(u eε dx (4.50 δ (Φ ε ( D(u eε 2 u eε 2 dx + CM ap = N + CM ap. In the term M 3 almost monotonicity of (Φ ε needs to be used. For almost increasing (Φ ε we use Young s inequality at first. M 3 (Φ ε ( D(u eε 2 u eε u eε dx δ (Φ ε ( D(u eε 2 u eε 2 dx+ ( c(δ (Φ ε ( D(u eε u eε 2 dx = N + CM 3.1. For M 3.1 we use that (Φ ε is almost increasing and Korn s inequality. M 3.1 C (Φ ε ( u eε u eε 2 dx C Φ ε ( u eε dx CM ap. (

13 In the case when (Φ ε is almost decreasing, we integrate by parts in M 3. M 3 C S ε 2 u eε ξ dx + C S ε u eε dx + C S ε u eε ξ dσ = M M M 3.3. (4.53 We treat M 3.1 like M 1, the term M 3.2 is easily estimated with the help of Young s and Korn s inequalities, M 3.3 (after Young s inequality can be treated like M 2.4 where we moreover use the fact, that for almost decreasing (Φ ε we can replace (Φ ε ( u eε by (Φ ε ( D(u eε. We put together (4.9 and (4.25, use estimates of M 1 - M 3 and sum over all points P. We recall that points P are divided into k groups and in each group the sets P are mutually disjoint. We have ( µ ε ( D(u eε 2 u eε 2 dx a 0 kc 5 µ ε ( D(u eε 2 u eε 2 dx + C (Φ ε ( f e dx + (Φ ε ( f e dx, (4.54 If we choose a 0 small enough, we can absorb the first integral on the right hand side of (4.54 into the left hand side. Remark 4.2. In two dimensions we can avoid the assumption on almost monotonicity of Φ. Instead of Lemma 4.2 we would test ( 4.1 by ϕ = ( 2 [ τ (u eε νξ 2 ], 1 [ τ (u eε νξ 2 ], which is sufficient only in 2D to obtain all information in tangent direction. In M 3 we would use the fact, that in boundary integral M 3.3 (which comes after integration by parts we are able to replace full gradient by the symmetric one. This works only in 2D. This technique concerning estimates in Orlicz setting was used first in [7], where one of the main features is that it handles the case of p-laplacean for 1 < p < in a unified way. It would be nice to avoid the assumption on almost monotonicity of Φ in the case where we work with symmetric gradients of velocity in n dimensions. 5 Limit passage At first we fix e > 0. To pass with ε 0 in equations (4.1-(4.3 it is enough to have almost everywhere convergence of symmetric gradients (Lemma 5.1 and uniform integrability (Lemma 5.2. Lemma 5.1. Let 1 < β < 0 < α, c > 0. We define m(s = cs α for s (0, 1, m(s = cs β for s 1. Let there exits C > 0 such that the sequence {A k } k=1, A k : R n n fulfills m 2 ( A k ( A k 2 + A k 2 dx C. (5.1 Then there exists a subsequence {A kl } l=1 and A : Rn n such that A kl A a.e. in as l. Proof. Let Ψ be a N-function such that m( B = Ψ ( B for all B R n n. We define M ij (B := ij Ψ( B = Ψ ( B B ij B for B 0, M ij (B = 0 for B = 0. M is Lipschitz mapping and also homeomorphism of R n n onto R n n. From Ak M(A k = m( A k A k A k A k A k + Ψ ( Id ( A k A k A k A k A k 3 we easily see that Because it holds that Ψ (s s ( Ak M(A k C m( A k + Ψ ( A k A k Cm(s for all s > 0, by simple computation we have M(A k Cm( A k A k, M(A k Cm( A k A k. (5.2. (5.3 From the assumption (5.1 we see that M(A k W 1,2 ( n n uniformly in k, so there exists a subsequence k l such that M(A kl M in W 1,2 ( n n, M(A kl M in L 2 ( n n, A kl M 1 (M a.e. in. (5.4 (5.5 13

14 By denoting M 1 (M =: A we complete the proof. Now we would like to use Lemma 5.1, where instead of the general A k we have D(u eε. From Lemma 2.1 we know that there exists α > 1, c > 0 such that for all s (0, 1 holds µ(s cs 2α and there exists β ( 1 2, 0, c > 0 such that for all s > 1 holds µ(s c s 2β. It can be easily seen that function µ ε also fulfills these conditions. So there exists function m(s defined in Lemma 5.1 satisfying m 2 (s < µ ε (s for all s (0, and for all ε (0, 1. The first apriori estimate (4.8 and Lemma 4.3 give m( D(u eε ( D(u eε 2 + D(u eε 2 dx C, therefore Lemma 5.1 gives us existence of A : R n n such that D(u eε A a.e. in. (5.6 Moreover, if we use the definition of the N-function Ψ, using also Korn s inequality from the first apriori estimate (4.8 we have Ψ( u eε + Ψ( u eε dx C. (5.7 Therefore there exists u e W 1, Ψ(, such that (up to a subsequence Clearly A = D(u e and D(u eε D(u e a.e. Following lemma gives us the uniform integrability. u eε u e in L Ψ( n n, (5.8 u eε u e in L Ψ( n. (5.9 Lemma 5.2. Let Φε ( D(u eε dx C. Then there exists δ > 0 such that for all σ (0, 1 and for all E such that E < δ holds (Φ ε ( D(u eε dx < σ. (5.10 E Proof. Let us denote φ(s := s 0 max(µ(t, 1t dt. Then for all s > 0, ε (0, 1 holds φ(s Φε (s. Therefore We note that χ Φ ε = 1 (Φ ε 1 ( 1 1 (Φ ε 1 (s 1 φ 1 (s. (5.11 by (2.11. Therefore 1 χ Φ ε (φ 1 ( 1 (5.12 and φ 1 is increasing function such that φ 1 (s as s. Using Hölder s inequality (2.11 we obtain (Φ ε ( D(u eε dx 2 χ E Φ ε (Φ ε ( D(u eε (Φ ε E C φ 1 ( E 1 (Φ ε ((Φ ε ( D(u eε dx C φ 1 ( E 1 Φ ε ( D(u eε dx. (5.13 The right hand side of (5.13 tends to 0 as E 0. The constant C depends on 2 (Φ ε, (Φ ε, but as we saw before, 2 (Φ ε, (Φ ε doesn t depend on ε. Lemma 5.1 and Lemma 5.2 allow us to pass to the limit ε 0 in the weak formulation of (4.1 - (4.3. It remains to let ε 0 in (4.44. Since {V ε (D(u eε } ε is bounded in W 1,2 ( n n, it follows that up to a subsequence V ε (D(u eε χ in W 1,2 ( n n, (5.14 V ε (D(u eε χ in L 2 ( n n. (5.15 To identify χ with V (D(u e we show V ε (D(u eε V (D(u e a.e. in. For that we need (besides almost everywhere convergence of symmetric gradients locally uniform convergence of V ε, which is provided by the following lemma. Lemma 5.3. Let K R n n, then V ε V on K as ε

15 Proof: We recall that ψ ε ( A is a potential to V ε (A, i.e. V ε (A = (Ψ ε ( A A A and (Ψ ε ( A = A (Φε ( A = A 2 µ ε ( A. Therefore V ε (A = µ ε ( A A and V (A V ε (A = µ( A µ ε ( A A. (5.16 We know that Φ (s = µ(ss and since Φ (s is bounded on (0, 1] we get that there exists C > 0 such that for all s (0, 1 and ε (0, 1 holds µ ε (s max(µ(s, 1 C s. It follows that for any δ > 0 we find σ > 0 such that for all ε (0, 1 holds V (A V ε (A < δ on B σ (0. Since K \ B σ (0 is compact, µ( A attains there its maximum and minimum. Consequently, there is ε 0 such that for ε (0, ε 0 : V ε = V on K \ B σ (0. Using Lemma 5.3 we get χ = V (D(u e and we can pass on the left hand side of (4.44 as ε 0 by weak lower semicontinuity of norms in W 1,2 ( n n. To pass with e 0 in equations (4.1-(4.3 and in (4.44 is easy, because the right hand side f was approximated in order to have f e f in W 1,Φ ( n. To conclude the proof of Theorem 1.1 it remains to reconstruct the pressure and show the inequality (1.12. In the same way like in Section 3 in Remark 3.1 using De Rham theorem we know that the right hand side of the equation π = div S + div g (5.17 can be written in the sense of distributions in the gradient form. We add the assumption π dx = 0. Since g L (Φ q and [Φ ( S(D(u ] q C[Φ (Φ ( D(u ] q C(Φ( D(u q we obtain (1.12 by application of Nečas Theorem on Negative norms (for formulation and proof based on Bogovskiĭ s Lemma in slightly different situation see [10] and imbedding W 1,2 ( L 2n n 2 (. Acknowledgment: The authors thank the Jindřich Nečas Center for Mathematical Modeling, the project LC06052 financed by MSMT and GAUK /B-MAT/MFF for their support. This work was also partially supported by GACR 201/09/0917. Reference [1] Armouche, C., Girault, V.: Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslovak Math. J. 44 (1994, no.119, [2] Beirão da Veiga, H.: On the regularity of flows with Ladyzhenskaya shear-dependent viscosity and slip or nonslip boundary conditions, Comm. Pure Appl. Math. 58. (2005, no. 4, [3] Beirão da Veiga, H.: Navier-Stokes Equations with shear-thickening viscosity: Regularity up to the boundary, J. Math. Fluid Mech. (2008 [4] Beirão da Veiga, H.: Navier-Stokes Equations with shear-thinning viscosity: Regularity up to the boundary, J. Math. Fluid Mech. (2008 [5] Beirão da Veiga, H.: On the Ladyzhenskaya-Smagorinsky turbulence model of the Navier-Stokes equations in smooth domains. The Regularity Problem, J. Eur. Math. Soc. (2008. [6] Beirão da Veiga, H., Kaplický, P., Růžička, M.: Boundary Regularity of Shear Thickening Flows, Journal of Mathematical Fluid Mechanics 13 (3, , (2011 [7] Diening, L., Ettwein, F.: Fractional estimates for non-differentiable elliptic system with general growth, Forum Mathematicum 20 (2008, no. 3, [8] Diening, L., Kaplický, P.: Higher integrability of the generalized Stokes system, to appear. [9] Diening, L., Růžička, M.:Interpolation operators in Orlicz Sobolev spaces, Num. Math. 107 (2007, no. 1, [10] Diening, L., Růžička, M., Schumacher, K.:A decomposition technique for John domains, Ann. Acad. Sci. Fenn. Math. 35 (2010, no. 1, MR [11] Ebmeyer, C.: Regularity in Sobolev spaces of steady flows of fluids with shear-dependent viscosity, Mathematical Methods in the Applied Sciences, Volume 29, Issue 14, [12] Frehse, J., Málek, J. and Steinhauer, M. An Existence Result for Fluids with Shear Dependent Viscosity - Steady Flows, Non. Anal. Theory Meth. Appl. 30 (1997,

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