Homogenization of stationary Navier-Stokes equations in domains with tiny holes

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1 Homogenization of stationary Navier-Stokes equations in domains with tiny holes Eduard Feireisl Yong Lu Institute of Mathematics of the Academy of Sciences of the Czech Republic Žitná 25, Praha 1, Czech Republic and Charles University in Prague, Faculty of Mathematics and Physics, Mathematical Institute Sokolovská 83, Praha 8, Czech Republic Abstract We consider the homogenization problem for the stationary compressible Navier-Stokes equations describing a steady flow of a compressible Newtonian fluid in a bounded three dimensional domain. We focus on the case where the domain is perforated with very tiny holes for which the diameters are much smaller than their mutual distances. We show that the homogenization process does not change the motion of the fluids: In the asymptotic limit, we obtain again the same system of equations. This coincides with similar results for the stationary Stokes and stationary incompressible Navier-Stokes system. Key words: 1 Introduction Homogenization problems in the framework of fluid mechanics have gain a lot of interest. For Stokes and stationary incompressible Navier-Stokes equations, Allaire in [1, 2] gave a system study Eduard Feireisl acknowledges the support of the project LL1202 in the programme ERC-CZ funded by the Ministry of Education, Youth and Sports of the Czech Republic. Yong Lu acknowledges the support of the project LL1202 in the programme ERC-CZ funded by the Ministry of Education, Youth and Sports of the Czech Republic. 1

2 for physical domains perforated by a family of holes of different size. More specifically, consider a system of holes of diameter O(ε α ), where ε is their mutual distance. In three spatial dimensions, Allaire showed that when α < 3, the limit fluid behavior is governed by the classical Darcy s law; when α = 3, in the limit yields Brinkman law; when α > 3, the the equations do not change in the homogenization process and the limit problem is determined by the same system of Stokes or Navier-Stokes equations. In the case α = 1, meaning that the size of holes is proportional to their mutual distance, the results have been extended to the incompressible evolutionary Navier-Stokes equations by Mikeli c [14], and to the compressible Navier-Stokes system by Masmoudi [13], and to the complete Navier-Stokes-Fourier system in [9]. In all the aforementioned cases, the asymptotic limit gives rise to Darcy s law. In this paper, we focus on the homogenization process for the stationary compressible Navier- Stokes equations in a domain perforated by very tiny holes, where the diameter of the holes is taken to be of size O(ε α ), with α > 3, where ε denotes the mutual distance between the holes. We start by a precise description of the physical domain. We consider a bounded domain in R 3 of class C 2, and a family of holes (solid obstacles) {T s ε,k}, which are simply connected smooth domains satisfying with T s ε,k T s ε,k B ε,k B ε,k εc k (1.1) C k := (0, 1) 3 + k, k Z 3, B ε,k := B(x k, b 0 ε α ) for some x k T s ε,k, b 0 > 0. (1.2) For the sake of simplicity, we suppose that all holes T s ε,k = ε α T s are similar to the same set T s - a simply connected domain of class C 2. The corresponding family of ε-dependent perforated domains is defined as ε := \ k K ε T s ε,k, K ε := {k εc k }. (1.3) It is easy to check that the number of holes contained in satisfies K ε = (1 + o(1)), as ε 0. (1.4) ε3 We consider the following stationary Navier-Stokes equations in ε : div x (ϱu) = 0, (1.5) div x (ϱu u) + x p(ϱ) = div x S( x u) + ϱf + g, (1.6) ( S( x u) = µ x u + t xu 2 ) 3 div xui + ηdiv x ui, µ > 0, η 0. (1.7) 2

3 Here, ϱ is the fluid mass density, u is the velocity field, p = p(ϱ) denotes the pressure, S = S( x u) stands for the Newtonian viscous stress tensor, and µ, λ are the viscosity coefficients. The density is nonnegative and the total mass of the fluid is fixed to be ε ϱ dx = M > 0. (1.8) In addition, we impose the no-slip boundary condition u = 0 on ε. (1.9) The fluid is driven by external forces here represented by the functions f and g satisfying We consider the pressure law of a typical form f L (R 3 ;R 3 ) + g L (R 3 ;R 3 ) c. (1.10) p(ϱ) = aϱ γ, a > 0, (1.11) with the adiabatic exponent γ 1. The range of γ we can handle by the homogenization technique will be specified below. For a function f defined in ε, we use the symbol f to denote the zero-extension of f in R 3, that is f = f in ε, f = 0 in R 3 \ ε. (1.12) We also use c to a generic positive constant independent of the parameter ε. However, the specific value of c could change from line to line. 1.1 Weak solutions We start by introducing the standard concept of weak solution to the compressible Navier-Stokes system. Definition 1.1 We say that [ϱ, u] is a finite energy weak solution of the Navier-Stokes equations ( ) supplemented with the conditions ( ) in the domain ε if: ϱ 0 a.e. in ε, ϱ dx = M, ϱ L β(γ) ( ε ), for some γ β(γ), u W 1,2 0 ( ε ; R 3 ); ε (1.13) ϱu x ψ dx = 0, (1.14) ε ϱu u : x ϕ + p(ϱ)div x ϕ S( x u) : x ϕ + (ϱf + g) ϕ dx = 0, ε (1.15) 3

4 for any test function ψ Cc ( ε ) and any test function ϕ Cc ( ε ; R 3 ). Moreover, there holds the energy inequality: S( x u) : x u dx (ϱf + g) u dx. (1.16) ε ε Furthermore, we introduce renormalized weak solutions. Definition 1.2 We say a finite energy weak solution [ϱ, u] is a renormalized weak solution if its zero extension [ ϱ, ũ] (see the notation (1.12)) satisfies div x ( ϱũ) = 0, div x (b( ϱ)ũ) + ( ϱb ( ϱ) b( ϱ))div x ũ = 0, in D (R 3 ), (1.17) for any b C 0 ([0, )) C 1 ((0, )) such that with b (s) c s λ 0 for s (0, 1], b (s) c s λ 1 for s [1, ), (1.18) c > 0, λ 0 < 1, 1 < λ 1 β(γ) 2 1. (1.19) Remark 1.1 The existence of finite energy renormalized weak solutions to the Navier-Stokes equations ( ) for fixed ε > 0 is known for certain range of the adiabatic exponent γ. The first global result has been obtained by Lions [12] for the case γ > 5/3. Extensions to lower values of γ were obtained by Březina, Novotný [4], Frehse et al. [10], and Plotnikov, Sokolowski [16]. Moreover, as shown in Theorem 4.3 in the monograph of Novotný, Straškraba [15], any finite energy weak solution [ϱ, u] satisfies ϱ L β(γ), β(γ) = 3(γ 1) if 3/2 < γ < 3, β(γ) = 2γ if γ 3. (1.20) Remark 1.2 In view of DiPerna-Lions s transport theory (see Section II.3 in [5] and the improvement in Lemma 3.3 in [15]), for any r L β (), β 2, and any v W 1,2 0 (), where R 3 is a bounded C 2 domain, a couple of functions [r, v] satisfying satisfies also the renormalized equations div x (rv) = 0 in D (), div x (rv) = 0, div x (b(r)v) + (rb (r) b(r))div x v = 0, in D (R 3 ), for any b satisfying ( ) provided r and v have been extended to be zero outside. Hence, if γ 5/3, any finite energy weak solution in the sense of Definition 1.1 is also a renormalized weak solution in the sense of Definition 1.2. The condition γ 5/3 ensures that ϱ L 2 ( ε ) by (1.20). 4

5 1.2 Main result In this paper, we consider the case, where the adiabatic exponent γ in the pressure law (1.11) satisfies γ 3. (1.21) Note that the same restriction has been imposed by Masmoudi [13] to avoid certain sofar unsurmountable difficulties connected with the compressible Navier-Stokes system. Our main result is the following: Theorem 1.1 Let M > 0, γ 3 be given. Let [ϱ ε, u ε ] 0<ε<1 be a family of finite energy renormalized weak solutions to the compressible Navier-Stokes system ( ) in ε, where f, g obey (1.10). Suppose that the size parameter α of the holes in (1.2) satisfies α > 3. Then { } sup ϱε L 2γ ( ε) + u ε W 1,2 0<ε<1 0 ( ε) < c, (1.22) and, up to a substraction of subsequence, the extensions [ ϱ ε, ũ ε ] satisfy ϱ ε ϱ weakly in L β(γ) (), ũ ε u weakly in W 1,2 0 (), (1.23) where the limit [ϱ, u] is a finite energy renormalized weak solution to the same problem ( ) in the limit domain. We find that the limit equations are the same as the original ones. This means that the homogenization process does not change the motion of the fluid when the holes (obstacles) are very small. The rest of the paper is devoted to the proof of Theorem 1.1. Although the result is formally the same as for the incompressible case studied by Allaire [1], [2], the technique for the compressible system is rather different. Moreover, in contrast with the critical case studied by Masmoudi [13], we do not perform any extra solution scaling and must establish the necessary bounds by means of the so-called Bogovskii s operator, the norm of which must be independent of the parameter ε. The construction of such an operator carried over in Section 2 represents the main step of the proof. The paper is finished by a short discussion in Section 3. 5

6 2 Proof of Theorem 1.1 This section is devoted to the proof of Theorem Bogovskii s operator We introduce a suitable inverse of the divergence, commonly known as Bogovskii s operator, see Bogovskii [3], Galdi [11]. Lemma 2.1 Let ε be defined through (1.1) (1.3) with α 3. Then there exists a linear operator B ε : L 2 ( ε )/R W 1,2 0 ( ε ; R 3 ) such that for any f L 2 ( ε ) and ε f dx = 0, there holds for some c independent of ε. div x (B ε (f)) = f in ε, B ε (f) W 1,2 0 ( ε;r 3 ) c f L 2 ( ε) (2.1) The existence of such a linear operator is nowadays standard, see e.g. Novotný, Straškraba [15, Chapter 3]. The key point here is to show the uniform estimate (2.1) with the constant c independent of ε. To this end, we adapt the construction of the restriction operator R ε by Allaire [1]. We start with the following result [1, Section 2.2]: Lemma 2.2 For ε as in (1.1)-(1.3) with α 3, there exits a linear bounded operator R ε mapping W 1,2 0 (; R 3 ) to W 1,2 0 ( ε ; R 3 ) such that u W 1,2 0 ( ε ; R 3 ) = R ε (ũ) = u in ε, (2.2) div x u = 0 in = div x R ε (u) = 0 in ε, (2.3) R ε (u) W 1,2 0 ( ε;r 3 ) c u W 1,2 0 (;R 3 ), c independent of ε. (2.4) Inspecting the proof of Lemma 2.2 in Section 2.2 of [1], we observe that the restriction operator may be constructed in the following way: Let b 1 > 0 be chosen such that B(x k, b 1 ε) εc k, B ε,k = B(x k, b 0 ε α ) B(x k, b 1 ε). Consider the following decomposition of each cube εc k with k K ε : εc k = T s ε,k E ε,k F ε,k with E ε,k := B(x k, b 1 ε) \ T s ε,k, F ε,k := (εc k ) \ B(x k, b 1 ε). 6

7 For any u W 1,2 0 (; R 3 ), we define R ε through and for k K ε : R ε (u) = u on εc k, for k K ε, (2.5) R ε (u) = u on F ε,k, R ε (u) = 0 on T s ε,k, R ε (u) = v ε,k in E ε,k, (2.6) where v ε,k W 1,2 (εc k ; R 3 ) solves the following Stokes problem p ε,k v ε,k = u in E ε,k, (2.7) div x v ε,k = div x u + 1 div x u dx in E ε,k, (2.8) E ε,k T s ε,k v ε,u = u on E ε,k T s ε,k, v ε,k = 0 on T s ε,k. (2.9) The operator R ε defined by ( ) is a restriction operator satisfying ( ). Now we use Lemma 2.2, along with the properties of the restriction operator stated above, to prove Lemma 2.1. Proof of Lemma 2.1. For f L 2 ( ε ) with ε f dx = 0, we consider the extension f = f in ε, f = 0 on \ ε = Tε,k. s (2.10) k K ε Then, by employing the classical Bogovskii s operator defined on the domain, we obtain u := B(f) W 1,2 0 (; R 3 ) such that div x u = f in and u W 1,2 0 (;R 3 ) c f L 2 () = c f L 2 ( ε) (2.11) for some c that only depends on. Moreover, by virtue of (2.10), we have div x u = f = 0 on T s ε,k. Let R ε be the restriction operator constructed through ( ). In particular, it is easy to check that equation (2.8) gives rise to div x v ε,k = div x u = f in E ε,k, (2.12) whenever u satifies (2.11). In addition, one has R ε (u) = u in ε \ ( k K ε E ε,k ). Together with (2.6) and (2.12), we therefore conclude that div x R ε (u) = f in ε. Thus to prove Lemma 2.1, it is enough to define B ε (f) := R ε (u) = R ε (B( f)), where B is the classical Bogovskii s operator on. The operator norm estimate (2.1) follows from the estimate for the restriction operator in (2.4). 7

8 2.2 Uniform bounds Our goal in this section is to show the uniform estimate (1.22) under the assumption γ 3. Note that, in accordance with what we said in Remark 1.1, we already know that ϱ ε L 2γ ( ε ), u ε W 1,2 0 ( ε ) (2.13) for any fixed ε. However, we have to establish uniform bounds on the norms ϱ ε L 2γ ( ε) and u ε W 1,2 0 ( ε) independent of ε (0, 1). By the energy inequality (1.16), together with Korn s inequality and Hölder s inequality, we have µ x u ε 2 L 2 ( ε) f L ( ε) ϱ ε L 6 5 (ε) u ε L 6 ( ε) + g L ( ε) u ε L 1 ( ε). (2.14) Since u ε W 1,2 0 ( ε ) has zero trace on the boundary, we may use Poincaré s inequality and the Sobolev embedding to obtain u ε L 6 ( ε) c x u ε L 2 ( ε) (2.15) for some constant c independent of the domain ε. Thus we obtain ( ) x u ε L 2 ( ε) + u ε L 6 ( ε) c f L ( ε) ϱ ε 6 + g L 5 L ( (ε) ε) c ( ϱ ε L 65 (ε) + 1 ). (2.16) Next, we introduce a test function ( ϕ(x) := B ε ϱ γ ε(x) 1 ) ϱ γ ε dx, (2.17) ε ε where B ε is the Bogovskii s operator introduced in Lemma 2.1. We remark that such ϕ is well defined since ϱ γ ε L 2 ( ε ) due to (2.13). Then by Lemma 2.1, we have div x ϕ(x) = ϱ γ ε(x) 1 ϱ γ ε dx in ε, (2.18) ε ε ϕ W 1,2 0 ( c ( ) ε) ϱ γ ε L 2 ( ε) + ϱ γ ε L 1 ( ε) c ϱε γ L 2γ (ε). (2.19) We plug this test function ϕ into (1.15) to obtain with I 1 := p(ϱ ε ) dx 1 ϱ γ ε dx, I 2 := µ x u ε : x ϕ dx + ε ε ε ε 4 p(ϱ ε )ϱ γ ε dx = I j (2.20) ε j=1 8 ε ( µ 3 + η ) div x u ε : div x ϕ dx, (2.21)

9 I 3 := ϱ ε u ε u ε : x ϕ dx, ε I 4 := (ϱ ε f + g)ϕ dx. ε Now we estimate I j one by one. By the interpolation between L q spaces, I 1 = a ε ϱ ε 2γ L γ ( ε) a ( ϱε θ 1 L ε 1 ( ϱ ε) ε 1 θ 1 L 2γ ( ε) where M is the total mass and θ 1 (0, 1) is determined by 1 γ = θ θ 1 2γ. ) 2γ = am 2γθ 1 ε ϱ ε 2γ 2γθ 1 L 2γ ( ε), By (2.16) and (2.19), we have for I 2 : ) I 2 c x u ε L 2 ( ε) x ϕ L 2 ( ε) c ( ϱ ε L (ε) ϱ ε γ L 2γ ( c ϱ ε) ε γ ( L 2γ ( ϱε ε) L 2γ ( ε) + 1 ). Again by (2.16) and (2.19), we have for I 3 : I 3 ϱ ε L 6 ( ε) u ε 2 L 6 ( ε) x ϕ L 2 ( ε) c ϱ ε L 6 ( ε) where θ 2 (0, 1) is determined by For I 4, we have c ϱ ε γ+1 ( L 2γ ( ϱε ε) 2θ 2 L 1 ( ϱ ε) ε 2(1 θ 2) L 2γ ( + 1) ε) = c M 2θ 2 ϱ ε γ+3 2θ 2 L 2γ ( ε) + c ϱ ε γ+1 L 2γ ( ε), 5 6 = θ θ 2 2γ. ( ϱ ε 2L 65 (ε) + 1 ) ϱ ε γ L 2γ ( ε) I 4 c ( ϱ ε L 2 + 1) ϱ ε γ L 2γ ( ε) c ϱ ε γ L 2γ ( ε) + c ϱ ε γ+1 L 2γ ( ε). We sum up the estimates for I j and obtain a ϱ ε 2γ L 2γ ( ε) = ε p(ϱ ε )ϱ γ ε dx = for some β 1 > 0. Precisely, we can choose β 1 as 4 j=1 β 1 = min{2γθ 1, γ 1, γ 3 + 2θ 2 }. 9 I j c ( ϱ ε 2γ β 1 L 2γ ( ε) + 1) (2.22)

10 From (2.22), we finally obtain We then go back (2.16) to derive ϱ ε L 2γ ( ε) c, for some c independent of ε. (2.23) u ε W 1,2 0 ( ε) c, for some c independent of ε. (2.24) This completes the proof of the uniform estimate claimed in (1.22). 2.3 Equations in a fixed domain In this section, we derive the equations for the extended functions [ ϱ ε, ũ ε ] in, where [ϱ ε, u ε ] is the finite energy renormalized weak solution in Theorem 1.1. In accordance with the uniform estimates ( ), we have Continuity equation First we claim: ϱ ε L 2γ () + ũ ε W 1,2 0 () c, for some c independent of ε. (2.25) Proposition 2.1 Under the assumptions in Theorem 1.1, the extended functions ϱ ε, ũ ε satisfy div x ( ϱ ε ũ ε ) = 0, div x (b( ϱ ε )ũ ε ) + ( ϱ ε b ( ϱ ε ) b( ϱ ε ))div x ũ ε = 0 in D (R 3 ) (2.26) for any b C 0 ([0, )) C 1 ((0, )) satisfying ( ). This is a direct conclusion from the fact that [ϱ ε, u ε ] is a renomalized weak solution Momentum equation The following proposition is crucial in the proof of convergence. Proposition 2.2 Under the assumptions in Theorem 1.1, we have div x ( ϱ ε ũ ε ũ ε ) + x p( ϱ ε ) = div x S( x ũ ε ) + ϱ ε f + g + Fε, in D (; R 3 ), (2.27) where F ε is a distribution satisfying < F ε, ϕ > D D c ε σ ϕ L r (;R 3 ) + c ε 3(α 1)σ 0 2(2+σ 0) x ϕ L 2+σ 0 (;R 3 3 )), (2.28) 10

11 for any ϕ Cc ((0, T ) ; R 3 ) and some constants c > 0, σ > 0, σ 0 > 0 and 1 < r < independent of ε. In particular, we can choose σ := α 3, r := 4 12(α 1) α 3, σ 0 (0, ). (2.29) Proof of Proposition 2.2. Let ϕ Cc ((0, T ) ; R 3 ) be any test function. It is sufficient to show I ε := ϱ ε ũ ε ũ ε : x ϕ + p( ϱ ε )div x ϕ S( x ũ ε ) : x ϕ + ϱ ε fϕ + gϕ dx (2.30) c ε σ ϕ L r (;R 3 ) + c ε 3(α 1)σ 0 2(2+σ 0) x ϕ L 2+σ 0 (;R 3 3 )), where c, σ, r and σ 0 are the constants enjoying the properties claimed in Proposition 2.2. Using ( ) we can find cut-off functions g ε Cc () satisfying 0 g ε 1, g ε = 0 on Tε,k, s g ε = 1 in \ B ε,k, x g ε L () c ε α. (2.31) k K ε k K ε Together with (1.4), we have for any 1 q : Then direct calculation gives 1 g ε L q () c ε 3(α 1) q, x g ε L q () c ε 3(α 1) q α. (2.32) 3 I ε = ϱ ε u ε u ε : x (g ε ϕ)+p(ϱ ε )div x (g ε ϕ) S( x u ε ) : x (g ε ϕ)+(ϱ ε f +g)(g ε ϕ) dx+ Ij ε, (2.33) ε j=1 where Since g ε ϕ C c I1 ε := I3 ε := ϱ ε ũ ε ũ ε : (1 g ε ) x ϕ ϱ ε ũ ε ũ ε : ( x g ε ϕ) dx, I2 ε := p( ϱ ε )(1 g ε )div x ϕ p( ϱ ε ) x g ε ϕ dx, S( x ũ ε ) : (1 g ε ) x ϕ + S( x ũ ε ) : ( x g ε ϕ) dx. ( ε ) is a test function for the stationary Navier-Stokes equations in ε, we have 3 I ε = Ij ε. j=1 For I 1, using hypothesis γ 3 and Hölder s inequality, we get I 1 c ϱ ε L 2γ () ũ ε 2 L ()( (1 g 6 ε ) x ϕ L 2 () + x g ε ϕ L 2 ()). (2.34) 11

12 By the uniform estimates ( ) and Hölder s inequality, we have I 1 c ( (1 g ε ) 2(2+σ 0 ) x ϕ L 2+σ 0 () + x g ε L r 1 () ϕ L r 2 () L σ 0 () ), (2.35) where Then by (2.32), we deduce σ 0 (0, ), r j (2, ), 1 r r 2 = 1 2. I 1 c ε 3(α 1)σ 0 2(2+σ 0) x ϕ L 2+σ 0 ()) + c ε 3(α 1) r 1 α ϕ L r 2 (), where the number σ := 3(α 1) r 1 α = 3(α 1) 2 α 3(α 1) = α 3 3(α 1) r 2 2 r 2 is strictly positive as long as In particular, we can choose r 2 > 6(α 1) α 3. r 2 = 12(α 1) α 3 such that σ := 3(α 1) α = α 3, r 1 4 which is exactly (2.29). Seeing that I ε 2 and I ε 3 can be handled similarly to I ε 1, we have completed the proof of Proposition Passing to the limit By the uniform estimates (2.25), up to a substraction of subsequence, we have ϱ ε ϱ weakly in L 2γ (), ũ ε u weakly in W 1,2 0 (; R 3 ). (2.36) It is left to show the limit [ϱ, u] represents a finite energy renormalized weak solution to ( ) in. 12

13 2.4.1 Strong convergence of velocity By compact Sobolev embedding, we have the strong convergence ũ ε u strongly in L q (; R 3 ) for any 1 q < 6. (2.37) Together with the weak convergence of the density, we have the weak convergence of nonlinear terms: and ϱ ε ũ ε ϱu weakly in L q (; R 3 ) for any 1 < q < 6γ 3+γ, (2.38) ϱ ε ũ ε ũ ϱu u weakly in L q (; R 3 3 ) for any 1 < q < 6γ 3+2γ. (2.39) Then we let ε 0 in the first equation of (2.26) and in equation (2.27) to deduce the following two equations in D (): div x (ϱu) = 0, (2.40) div x (ϱu u) + x p(ϱ) = div x S( x u) + ϱf + g. (2.41) Here p(ϱ) is the weak limit of p( ϱ ε ) in L 2 (). Moreover, in accordance with Remark 1.2, the pair of functions [ϱ, u] satisfies the renormalized equation div x (ϱu) = 0, div x (b(ϱ)u) + (ϱb (ϱ) b(ϱ))div x u = 0, in D (R 3 ), (2.42) for any b satisfying ( ). Consequently, to finish the proof of Theorem 1.1, we have to show p(ϱ) = p(ϱ). This is done in the next section Convergence of the pressure First of all, we introduce the so-called effective viscous flux which is the quantity p(ϱ) ( 4µ 3 +η)div xu. We shall show that this quantity enjoys some weak compactness property specified in the following lemma. This property, identified first by Lions [12], plays a crucial role in the existence theory in the framework of weak solutions for the compressible Navier-Stokes system. Lemma 2.3 For any ψ Cc (), there holds up to a substraction of subsequence: lim ψ (p( ϱ ε ) ( 4µ ) ( ε η)div xũ ε ϱ ε dx = ψ p(ϱ) ( 4µ ) 3 + η)div xu ϱ dx. (2.43) 13

14 Proof of Lemma 2.3. The proof of Lemma 2.3 is quite technical but nowadays well understood. The idea is to take test functions of the form ψ 1 (1 ϱ ε ), ψ 1 (1 ϱ), where ψ Cc () and 1 is the Fourier multiplier on R 3 with the symbol 1/ ξ 2. It is straightforward to check that 1 = (R i,j ) 1 i,j 3 are the Riesz operators. In particular, using the well known properties of the singular integral operators of Carderón-Zygmund type, we have 1 (r) L q (R 3 ) c r L q (R 3 ). for any r L q (R 3 ), 1 < q <. We take ψ 1 (1 ϱ ε ) as a test functions in the weak formulation of (2.27) and let ε 0. Then we take ψ 1 (1 ϱ) as a test functions in the weak formulation of (2.41), and compare the results of these operations. By using the convergence results ( ), compact Sobolev embedding, the fact γ 3, and the property R i,j = R j,i, we obtain, through long but straightforward calculations, that I := lim ψ (p( ϱ ε ) ( 4µ ) ( ε η)div xũ ε ϱ ε dx ψ p(ϱ) ( 4µ ) 3 + η)div xu ϱ dx (2.44) = lim ϱ ε ũ i εũj ε 0 εψr i,j (1 ϱ ε ) dx ϱu i u j ψr i,j (1 ϱ) dx. On the other hand, by taking 1 1 (ψ ϱ ε ũ ε ) as a test function for the first equation of (2.26) and taking 1 1 (ψϱu) as a test function for (2.40), we obtain 1 ϱ ε ũ i εr i,j (ψ ϱ ε ũ ε ) dx = 0, 1 ϱu i R i,j (ψϱu) dx = 0. (2.45) Plugging (2.45) into (2.44) yields ũ i ε ( ϱε ũ j εψr i,j (1 ϱ ε ) 1 ϱ ε R i,j (ψ ϱ ε ũ ε ) ) dx I = lim ε 0 u i ( ϱu j ψr i,j (1 ϱ) 1 ϱr i,j (ψϱu) ) dx. (2.46) The proof of the strong convergence of the densities is finished by means of the strong convergence of the velocity in (2.37) and the following property of commutators of the type appearing in (2.46): Lemma 2.4 Suppose u ε u weakly in L p (R 3 ), v ε v weakly in L q (R 3 ), 14 1 p + 1 q = 1 r < 1.

15 Then for any 1 i, j 3, See [8, Lemma 3.4] for the proof. u ε R i,j (v ε ) v ε R i,j (u ε ) ur i,j (v) vr i,j (u) weakly in L r (R 3 ). Now, we are ready to prove the following result: Proposition 2.3 Let p(ϱ)ϱ be a weak limit of p( ϱ ε ) ϱ ε in L 2γ γ+1 (), we then have p(ϱ)ϱ = p(ϱ)ϱ in L 2γ γ+1 (). Proof of Proposition 2.3. Using the uniform bound for ϱ ε in (2.25) and the specific form of p(ϱ) in (1.11), we have p( ϱ ε ) ϱ ε p(ϱ)ϱ weakly in L 2γ γ+1 (). Taking b(s) = s log s in the renormalized equations (2.26) and (2.42) we obtain Letting ε 0 in (2.47) yields div x ( ϱ ε log ϱ ε ũ ε ) + ϱ ε div x ũ ε = 0, div x (ϱ log ϱu) + ϱdiv x u = 0. (2.47) Then for any ψ Cc (), we have ψϱ div x u dx = x ψ (ϱ log ϱ u) dx, div x (ϱ log ϱ u) + ϱ div x u = 0. ψϱ div x u dx = Passing to the limit for ε 0 in (2.43) and using (2.48) we get ψp(ϱ)ϱ ( 4µ 3 + η) xψ (ϱ log ϱ) u dx = Now we choose test functions ψ n C c () such that x ψ (ϱ log ϱ)u dx. (2.48) ψp(ϱ)ϱ ( 4µ 3 + η) xψ (ϱ log ϱ)u dx. (2.49) 0 ψ n 1, ψ n (x) = 1 if d(x, ) > 2 n, ψ n(x) = 0 if d(x, ) < 1 n, xψ n L () 2n and for any q [1, ]: 1 ψ n L q c n 1 q, x ψ n L q c n 1 1 q, and consequently d(x, ) x ψ n L q 15 c n 1 q

16 The velocity u W 1,2 0 (; R 3 ), then we have [d(x, )] 1 u L 2 (; R 3 ). Therefore, x ψ n (ϱ log ϱ)u dx d(x, ) x ψ n L 10 (ϱ log ϱ) L 5/2 [d(x, )] 1 u L 2 c n 1/10. Letting ψ = ψ n in (2.49) and passing to the limit n we may infer that p(ϱ)ϱ p(ϱ)ϱ dx = 0. (2.50) Since the mapping ϱ p(ϱ) is strictly increasing, there holds p(ϱ)ϱ p(ϱ)ϱ, a.e. in, which, together with (2.50), gives rise to the desired conclusion p(ϱ)ϱ = p(ϱ)ϱ, a.e. in. We have completed the proof of Proposition 2.3. Our desired result p(ϱ) = p(ϱ) is a direct corollary of Proposition 2.3, due to the monotonicity and convexity of p( ). Accordingly, we have finished the proof of Theorem Concluding remarks The hypothesis concerning the shape of the holes as well as their spatial distribution may be considerably relaxed. As a matter of fact, it is enough to impose the following restriction on the model hole: There exists a constant ω > 0 such that at each point x T s there exists a closed cone C x with vertex at x and of aperture ω such that C x T s = {x}. Moreover, the holes need not be periodically distributed, it is enough that their mutual distance is proportional to ε, see [7] for the relevant homogenization problem in the context of incompressible fluids. 16

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