Uniform estimates for Stokes equations in domains with small holes and applications in homogenization problems

Transcription

1 Uniform estimates for Stokes equations in domains with small holes and applications in homogenization problems Yong Lu Abstract We consider the Dirichlet problem for the Stokes equations in a domain with a shrinking hole in R d. An almost complete description concerning the uniform W 1,p estimates as the size of the hole goes to zero is shown : for any d < p < d, there hold the uniform W 1,p estimates; for any 1 < p < d or d < p <, there are counterexamples indicating that the uniform W 1,p estimates do not hold. We give applications in homogenization problems; in particular we generalize the restriction operator constructed by Allaire from L 2 framework to L p framework. Keywords: Stokes equations; domain with shrinking hole; homogenization in fluid mechanics; Restriction operator. Contents 1 Introduction Some notations Known results for Stokes equations in a bounded domain Main results Motivation and background Applications in homogenization problems Perforated domain Restriction operator and Bogovskii type operator Proof of Theorem Reformulation Change of variables Dirichlet problem in bounded domain Dirichlet problem in enlarging domain Dirichlet problem in enlarging domain with divergence form source term Dirichlet problem in enlarging domain with compactly supported source term End of the proof Contradiction and compactness argument Homogeneous Sobolev spaces Derivation of contradiction Department of Mathematics, Nanjing University, 22 Hankou Road, Gulou District, Nanjing, China. luyong@nju.edu.cn. The author warmly thanks E. Feireisl and C. Prange for interesting discussions. The author acknowledges the support of the project LL1202 in the program ERC-CZ funded by the Ministry of Education, Youth and Sports of the Czech Republic. 1

2 4 Proof of Theorem Construction of restriction operator Proof of Theorem Proof of Corollary Introduction We consider the following Dirichlet problem of the Stokes equations with a divergence form source term in a bounded domain with a shrinking hole: v + π = div G, in ε := \ εt, div v = 0, in ε = \ εt, 1.1 v = 0, on ε = ε T. This study is mainly motivated by the study of homogenization problems in the framework of fluid mechanics where the main goal is to describe the asymptotic behavior of fluid flows in domains perforated with a large number of tiny holes. As observed by Allaire [2, 3], it is crucial to consider the Stokes equations near each single hole, where there arise problems of type 1.1. In 1.1, v : ε R d is the unknown, G : ε R d d is named source function, 0 R d and T R d are both simply connected, bounded domains of class C 1 Lipschitz domain can be also considered where the corresponding result is generally weaker; see Remark 1.3. Here, and throughout the paper, ε 0, 1 is a small parameter. Without loss of generality, we suppose 1.1 Some notations T B 1/2 := B0, 1/2 B 1 := B0, 1. We recall some notations used in this paper. For any domain R d and a vector-valued function v = v 1,, v d : R d, we define its gradient v to be a d d matrix of which the i, j element is v i,j = xj v i. For a matrix-valued function G = G i,j 1 i,j d : R d d, we define its divergence to be a vector as div G = j xj G i,j 1 i d For any 1 < p <, the notation p denotes the Lebesgue conjugate component of p: 1/p +1/p = 1. For any 1 < p < and any domain R d, we use classical notation L p to denote the space of Lebesgue measurable functions such that 1/p u L p := ux dx p <. We use the classical notation W 1,p to denote the Sobolev space with the norm u W 1,p := u p 1 L p + u p p L p,. 2

3 where the derivatives are defined in the sense of distribution. The notation W 1,p 0 denotes the completion of Cc with respect to the norm W 1,p. The notation W 1,p denotes the dual space of the Sobolev space W 1,p 0. The definition of the W 1,p norm is classical: u W 1,p := sup φ C c, φ W 1,p =1 u, φ. We use the notation L p 0 to denote the space of Lp functions with zero mean value: } L p 0 {f := L p : f dx = Known results for Stokes equations in a bounded domain The well-posedness theory for the Dirichlet problems of the Stokes equations in bounded domains is well developed. We give a remark to collect some known results. We refer to Theorem IV.6.1 in [11] for domains of class C 2, Theorem 5.1 in [7] for domains of class C 1, Section 3 of [18] a review of results for Lipschitz and C 1 domains and Theorem 2.9 in [4] for Lipschitz domains in R 3 for more details and the proof. Remark 1.1. i. For any connected, bounded domain R d, d 2 of class C 1, for any 1 < p < and any h W 1,p ; R d, the following Dirichlet problem of the Stokes equations w + π = h, div w = 0, in ; w = 0, on admits a unique solution w, π W 1,p 0 ; R d L p 0 and there holds the estimate w W 1,p 0 ;R d + π L p Cp, d, h W 1,p ;R d. 1.2 ii. If the bounded domain is only of class Lipschitz instead of C 1, the result in i holds true for p near 2: p 1 d, < p < p 1d, for some p 1 d, > 2. In particular, in three dimensions, one has p 1 3, > 3 see Theorem 2.9 in [4]. 1.3 Main results Now we state our main theorem concerning the uniform W 1,p estimates: Theorem 1.2. For any d < p < d if d 3, or p = 2 if d = 2, and any G L p ε ; R d d, the unique solution v, π W 1,p 0 ε ; R d L p 0 ε to 1.1 satisfies the uniform estimate v L p ε;r d d + π L p ε C G L p ε;r d d 1.3 for some constant C = Cp, d,, T independent of ε. We give two remarks about Theorem 1.2: Remark 1.3. i. For any fixed ε, if G L p ε ; R d d with 1 < p <, the well-posedness of 1.1 is known, see Remark 1.1. The key point of Theorem 1.2 lies in obtaining the uniform estimates as ε 0. When p = 2, it is direct to show that the estimate constant C2, d, ε = 1 in 1.2, while when p 2, the estimate constant C = Cp, d, ε in 1.2 depends on the C 1 character of the domain ε, which is unbounded as ε 0. 3

4 ii. The result in Theorem 1.2 can be generalized to domains and holes with Lipschitz boundary. In the case with Lipschitz domains and holes, the choice range of p has to be restricted to max{p 1, d } < p < min{p 1, d}, where p 1 is introduced in Remark 1.1 ii. This is because we only have the existence of solutions in W 1,p 0 with p 1 < p < p 1. In particular, we obtain optimal results when d = 3: Theorem 1.2 still holds true for simply connected, bounded domains and holes T which are of class Lipschitz. The following theorem indicates that the choice range of p in Theorem 1.2 is critical: Theorem 1.4. Let p > d and {G ε } 0<ε<1 L p ε ; R d d be uniformly bounded: sup G ε L p ε;r d d <. 0<ε<1 Then, up to a substraction of subsequence, the zero extensions G ε of G ε in admit the weak limit: G ε G weakly in L p ; R d d, as ε 0. If the unique solution v to the following Dirichlet problem satisfies v + π = div G, div v = 0, in ; v = 0, on 1.4 v 0 0, 1.5 then the solutions v ε, π ε W 1,p 0 ε ; R d L p 0 ε to 1.1 with source functions G ε satisfy lim inf ε 0 v ε L p ε;r d d =. 1.6 Let 1 < p < d. If there exists G L p ; R d d such that are satisfied, then for any 0 < ε < 1 there exists H ε L p ε ; R d d satisfying H ε L p ε;r d d = 1 such that the solutions w ε, ξ ε W 1,p 0 ε L p 0 ε to 1.1 with source functions H ε satisfy lim inf ε 0 w ε L p ε;r d d =. 1.7 The result in Theorem 1.4 indicates that the uniform W 1,p estimates hold for any p > d necessarily if the solutions are zero at origin. 1.4 Motivation and background The motivation of this paper mainly comes from the study of homogenization problems in the framework of fluid mechanics, that is the study of fluid flows in domains perforated with tiny holes. In the study of homogenization problems, the goal is to describe the asymptotic behavior of fluid flows governed by Stokes or Navier-Stokes equations as the number of holes goes to infinity and the size of the holes goes to zero simultaneously. The limit equations that describe the limit behavior of fluid flows are called homogenized equations which are defined in homogeneous domain without holes. 4

5 Allaire in [2] and [3] gave a systematic study for the homogenization problems of the Stokes equations and the stationary incompressible Navier-Stokes equations. Allaire showed that the homogenized equations are determined by the ratio between the size of holes and the mutual distance of holes. In particular for three-dimensional domains perforated with holes of diameter Oε α with ε the size of the mutual distance of holes, if 1 α < 3 corresponding to the case of large holes, in the limit ε 0, the fluid is governed by the Darcy s law; if α > 3 corresponding to the case of small holes, the homogenization process does not change the motion of the fluid and in the limit there arise the same Stokes equations or the same stationary incompressible Navier-Stokes equations in homogeneous domains; if α = 3 corresponding to the case of critical size of holes, the limit equations are governed by the Brinkman s law a combination of the Darcy s law and the original equations. For the case α = 1 meaning that the size of the holes is comparable to their mutual distance, earlier than Allaire, Tartar [19] recovered the Darcy s law from the homogenization of the Stokes equations. Still for the case α = 1, the study is extended to more complicated equations : Mikelić [16] for the evolutionary incompressible Navier-Stokes equations; Masmoudi [14] for the compressible Navier-Stokes equations; Feireisl, Novotný and Takahashi [10] for the complete Navier- Stokes-Fourier euqations; the Darcy s law is recovered in all these studies. Cases with different size of holes are also considered: Feireisl, Namlyeyeva and Nečasová in [9] studied the case α = 3 critical size of holes for the evolutionary incompressible Navier-Stokes equations and they derived Brinkman s law in the limit; Feireisl, Diening and the author in [8, 6] considered the case α > 3 small holes for the stationary compressible Navier-Stokes equations and we it is shown that the homogenized equations remain the same as the original ones; in both of the studies, three-dimensional domains are considered and the results coincide with the study of Allaire for the Stokes equations. In the study of these homogenization problems, the so-called restriction operator plays an important role. A restriction operator, roughly speaking, is a linear mapping from W 1,p 0 D to W 1,p 0 D ε and preserves the divergence free property, where D is the domain without holes and D ε is the domain perforated with holes. When p = 2, the construction of a restriction operator from W 1,2 0 D to W 1,2 0 D ε is given, by Tartar in Lemma 4 in [19] for the case α = 1, and by Allaire in Section 2.2 in [2] for general α 1. One key step in Allaire s construction is to study the Dirichlet problem for the Stokes equations in a neighborhood of each hole. After rescalling and changes of variables, a Dirichlet problem of type 1.1 arises and the uniform estimate of type 1.3 with p = 2 is needed. Precisely, the domain ε in Allaire s case is B 1 \ ε α 1 T where ε α 1 is the ratio between the diameter of holes which is ε α and the mutual distance of holes which is ε. In the case with p = 2, the uniform estimate 1.3 is rather straightforward with estimate constant 1. A restriction operator from W 1,p 0 D to W 1,p 0 D ε with general p is needed in studying homogenization problems for more complicated systems, such as in [16] concerning incompressible Navier-Stokes equations, in [14] concerning compressible Navier-Stokes equations, in [10] concerning compressible Naiver-Stokes-Fourier system, and particularly in [17] concerning non-newtonian Flows. However, in all these references, only the case α = 1 is considered, such that the model domain B 1 \ ε α 1 T = B 1 \ T is independent of ε. Consequently, the uniform W 1,p estimates can be obtained by applying the classical results see Remark 1.1. As a consequence, when α = 1, such restriction operator from W 1,p 0 D to W 1,p 0 D ε with general p can be constructed by A. Mikelić, see Lemma 3.2 in [17]. 5

6 To extend the study of the homogenization problems to general size of holes, it is motivated to study the uniform W 1,p estimates of the Dirichlet problems of elliptic equations in a domain with a shrinking hole. The author started this study in [13] by focusing on the Laplace equation with divergence source term in three-dimensional domain B 1 \ εt where B 1 R 3 is the unit ball. We showed that if 3/2 < p < 3, the W 1,p estimates are uniform as ε 0; if 1 < p < 3/2 or 3 < p <, there are counterexamples indicating that the uniform W 1,p estimates do not hold. The results in this paper for Stokes equations coincide with the results in [13] for Laplace equations. The study for the Stokes equations case is more complicated and there arises new issues and difficulties. For example, one key step of the proof in [13] relies on the explicit formula of the Green function of Laplace operator in unit ball see Lemma 3.2 in [13]. In this paper we derive rather abstract analysis see Lemma 3.2 instead of using the Green functions such that we can generalize the argument in [13] to make it be suitable for Stokes equations. Moreover, after rescaling we turn to consider the problem in domain /ε \ T see Section 3.1 in this paper and Section 2 in [13]. We then decompose the original problems into two parts by employing a cut-off function defined near the hole : one part is defined in a bounded domain and another part is defined in an enlarging domain without hole. For Stokes equations, this decomposition will destroy the divergence free condition. In Section 3.2, we employ the Bogovskii s operator to remedy this problem. The study for the Stokes equations case in this paper is essentially needed to construct a restriction operator from W 1,p 0 D to W 1,p 0 D ε for general p. Furthermore, this is need in the study of homogenization problems of more complicated situations: such as the homogenization of compressible Navier-Stokes equations and the homogenization of non-newtonian flows in the case α > 1, which are both under our consideration. This paper is organized as follows. In Section 2 we exhibit applications of Theorem 1.2 in homogenization problems. Section 3 and Section 4 are devoted to the proof of Theorem 1.2 and Theorem 1.4 respectively. Throughout this paper, C denotes a positive constant independent of ε unless there is a specification. However, the value of C may change from line to line. 2 Applications in homogenization problems In this section, we give an application of our main result Theorem 1.2 in homogenization problems: we generalize Allaire s construction of restriction operator from L 2 framework to L p framework. As mentioned in Introduction, the restriction operator plays a very important role in the study of homogenization problems one can find it in most literatures in this field. This generalization will be essentially needed in the study of more complicated problems in this field, particularly for the cases where the size of holes are much smaller than the mature distance between holes. 2.1 Perforated domain In this section, we give a precise description for the perforated domains mentioned before. We focus on the case where the holes are periodically distributed in a bounded three dimensional domain. Let D R 3 be a bounded domain of class C 1, and {T ε,k } be a family of C 1 domains satisfying for any k Z 3 : T ε,k T ε,k B ε,k εc δ k εc k 2.1 with C k := 0, k, C δ k := δ, 1 δ3 + k, B ε,k := Bx k, b ε α, 2.2 6

7 for some x k T ε,k, some 0 < δ < 1, b > 0 and α 1 independent of ε. Moreover, we assume that each hole T ε,k is similar to the same model hole T rescaled by ε α : T ε,k = ε α O k T + x k, 2.3 where O k is a rigid rotation. We assume T is a simply connected domain of class C 1 and is independent of ε. We call the small domains T ε,k to be the holes or obstacles, the domain D to be the homogeneous domain without holes. The corresponding perforated domain D ε is defined as D ε := D \ T ε,k, K ε := {k εc k }. k K ε We can see that in D ε, the diameters and mutual distances of the holes are of size Oε α and Oε, respectively, and the number of holes satisfies K ε = D ε o1, as ε Restriction operator and Bogovskii type operator By employing the uniform estimates obtained in Theorem 1.2, we can generalize the construction of restriction operator by Allaire [2] from the L 2 framework to the L p framework : Theorem 2.1. Let D and D ε be given in Section 2.1. For any 3/2 < p < 3, there exists a linear operator R ε from W 1,p 0 D; R 3 to W 1,p 0 D ε ; R 3 such that { u W 1,p 0 D ε ; R 3 u in D ε, = R ε ũ = u in D ε, where ũ := 0 on D \ D ε, u W 1,p 0 D; R 3, div u = 0 in D = div R ε u = 0 in D ε, u W 1,p 0 D; R 3 = R ε u L p D ε;r 3 3 C u L p D;R ε 3 pα 3 p u L p D;R 3. Remark 2.2. We remark that the result in Theorem 2.1 generalizes the result by Allaire see Proposition in [3] where only the case p = 2 is considered, and also generalizes the result by A. Mikelić see Lemma 3.2 in [17] where the restriction α = 1 is needed. Here we find one direct corollary of this generalized restriction operator, that is to obtain some uniformly bounded Bogovskii type operator, which can be seen as the inverse of the divergence operator, on perforated domains. Bogovskii type operators are need in the study of compressible Navier-Stokes equations, in order to improve the integrability of the pressure term. Corollary 2.3. Let D ε be the domain defined in Section 2.1. For any 3/2 < p < 3, there exists a linear operator B ε from L p 0 D ε to W 1,p 0 D ε ; R 3 such that for any f L p 0 D ε, 2.4 div B ε f = f in D ε, B ε f W 1,p 0 D ε;r 3 C 1 + ε 3 pα 3 p f L p D ε, 2.5 for some constant C independent of ε. 7

8 The existence of such an operator satisfying is classical, known as Bogovskii s operator. But the norm of the classical Bogovskii s operator depends on the Lipschitz character of the domain, while the Lipschtiz norm of domain D ε is unbounded as ε 0. The key point of Theorem 2.3 is to give an estimate for the operator norm as shown in In particular, if the holes are suitably small such that 3 pα 3 0, one has uniform estimate for this Bogovskii type operator. In [6], another rather direct construction of a Bogovskii type operator on perforated domains is done and the estimates for these two different constructions are the same. However, these two constructions contain different local properties and may have special advantages in different occasions. The proof for Theorem 2.1 and Corollary 2.3 is given in Section 5. Remark 2.4. We are considering more applications of this fundamental estimate in Theorem 1.2 and this generalized restriction operator in Theorem 2.1, such as the homogenization of non- Newtonian flows when α > 1 for which p Stokes equations under the case α = 1 is considered by A. Mikelić [17] and uniform W 1,p estimates of elliptic equations in perorated domains with α > 1 motivated by the study of Masmoudi in [15] for α = 1. 3 Proof of Theorem 1.2 This section is devoted to the proof of our main result Theorem Reformulation We reformulate the Dirichlet problems 1.1 by introducing the following Dirichlet problem in the rescaled domain ε := ε /ε = /ε \ T : v 1 + π 1 = div G 1, in ε, div v 1 = 0, in ε, 3.1 v 1 = 0, on ε. By scaling technique, it can be shown that to prove Theorem 1.2, it is equivalent to prove the following theorem: Theorem 3.1. For any d < p < d if d 3 or p = 2 if d = 2, and any G 1 L p ε ; R d d, the unique solution v 1, π 1 W 1,p 0 ε ; R d L p 0 ε to 3.1 satisfies the uniform estimate v 1 L p ε;r d d + π 1 L p ε C G 1 L p ε;r d d 3.2 for some constant C = Cp, d,, T independent of ε. We proceed several steps to see the equivalence between Theorem 1.2 and Theorem 3.1. We first prove Theorem 1.2 by supposing that Theorem 3.1 holds true. Let G L p ε ; R d d with d < p < d if d 3 and p = 2 if d = 2, and v, π W 1,p 0 ε ; R d L p 0 ε be the unique solution to 1.1. Thus, the rescaled quantities v 1 := vε, π 1 := ε πε G 1 := ε Gε satisfies 3.1. By Theorem 3.1, there holds the uniform estimate v 1 L p ε;r d d + π 1 L p ε C G 1 L p ε;r d d 8

9 which is equivalent to v L p ε;r d d + π L p ε C G L p ε;r d d. This is exactly our desired estimate 1.3. The proof of Theorem 3.1 by supposing Theorem 1.2 can be done similarly. Hence, to prove Theorem 1.2, it is sufficient to prove Theorem 3.1. The rest of this section is devoted to proving Theorem 3.1. Let G 1 and v 1, π 1 be as in Theorem 3.1. We want to prove the uniform estimate 3.2. The proof for the case p = 2 is straightforward, so in the sequel we focus on the case d < p < d, d Change of variables Inspired by the idea in [12] for the study of the Stokes equations in exterior domains, we introduce the cut-off function ϕ C c B 1, ϕ 1 in B 1/2, 0 ϕ 1, 3.3 and consider the decomposition v 1 = v 2 + v 3, v 2 := ϕ v 1, v 3 := 1 ϕv 1, π 1 = π 2 + π 3, π 2 := ϕ π 1 ϕπ 1, π 3 := 1 ϕπ 1 1 ϕπ 1, where the mean values are defined as ϕπ 1 := 1 ϕπ 1 x dx, 1 ϕπ 1 := 1 1 ϕπ1 x dx ε ε ε ε which satisfy ϕπ ϕπ 1 = π 1 = 0. Then v 2, π 2 can be viewed as the solution of the following Dirichlet problem in a bounded domain: v 2 + π 2 = div ϕg 1 h 1, in B 1 \ T, div v 2 = v 1 ϕ, in B 1 \ T, v 2 = 0, on B 1 T and v 3, π 3 can be seen as the solution of the following Dirichlet problem in a pure rescaled domain without hole: v 3 + π 3 = div 1 ϕg 1 + h1, in /ε, where div v 3 = v 1 ϕ, v 3 = 0, in /ε, on /ε, h 1 := v 1 ϕ + 2 v 1 ϕ + G 1 ϕ π 1 ϕ. 3.4 However, this decomposition destroyed the divergence free condition: the new variables v 2 and v 3 are no longer divergence free. This can be fixed by employing the classical Bogovskii s operator which says that for any bounded Lipschitz domain R d, there exits a linear operator B from L p 1,p 0 to W0 ; R d such that for any f L p 0 : div B f = f in, B f W 1,p 0 ;R d Cd, p, f L p, 9

10 where the constant C depends only on p, d and the Lipschitz character of. For the existence of such an operator, we refer to Chapter III of Galdi s book [11]. By the property of ϕ in 3.3 and the property div v 1 = 0, we have supp v 1 ϕ B 1 \ T and v 1 ϕ dx = div v 1 ϕ dx = div v 1 ϕ dx = 0. B 1 \T B 1 \T B 1 By using the classical Bogovskii s operator, there exists v 4 W 1,p 0 B 1 \ T ; R d such that We then define div v 4 = v 1 ϕ, v 4 W 1,p 0 B 1 \T ;R d C v 1 ϕ L p B 1 \T. 3.5 v 5 := v 2 v 4, v 6 := v 3 + v 4, 3.6 which are both divergence free and solve respectively v 5 + π 2 = div ϕg 1 + v 4 h 1, in B 1 \ T, div v 5 = 0, in B 1 \ T, v 5 = 0, on B 1 T 3.7 and v 6 + π 3 = div 1 ϕg 1 v 4 + h1, div v 6 = 0, v 6 = 0, in /ε, in /ε, on /ε Dirichlet problem in bounded domain For the Dirichlet problem 3.7, the classical theory see Remark 1.1 implies that the unique solution v 5, π 2 satisfies v 5 L p B 1 \T ;R d d + π 2 L p B 1 \T C div ϕg 1 + v 4 h 1 W 1,p B 1 \T ;R d C ϕg 1 + v 4 L p B 1 \T ;R d d + h 1 W 1,p B 1 \T ;R d C G 1 L p B 1 \T ;R d d + v 1 L p B 1 \T ;R d + π 1 W 1,p B 1 \T, where in the last inequality we used 3.4 and 3.5. Moreover, by 3.6 and again by 3.5, we obtain v 2 L p B 1 \T ;R d d + π 2 L p B 1 \T C G 1 L p B 1 \T ;R d d + v 1 L p B 1 \T ;R d + π W 1,p B 1 \T. 3.4 Dirichlet problem in enlarging domain To study 3.8, by observing that the first term of the right-hand side of is of divergence form and the second term is of compact support in B 1 \ T, we consider the decomposition v 6 := v 7 + v 8, π 3 := π 4 + π 5, 10

11 where the two couples v 7, π 4 and v 8, π 5 in W 1,p 0 /ε L p 0 /ε solve respectively v 7 + π 4 = div 1 ϕg 1 v 4, in /ε, div v 7 = 0, in /ε, v 7 = 0, on /ε, 3.10 and v 8 + π 5 = h 1, div v 8 = 0, v 8 = 0, in /ε, in /ε, on /ε Dirichlet problem in enlarging domain with divergence form source term The Dirichlet problem 3.10 has a divergence form source term. We thus consider the following change of variable: v 9 := v 7 /ε, π 6 = ε 1 π 4 /ε, G 2 := ε 1 1 ϕg 1 v 4 /ε. Then v 9, π 6 W 1,p 0 ; R d L p 0 solves v 9 + π 6 = div G 2, in, div v 9 = 0, in, v 9 = 0, on. We employ the known results see Remark 1.1 to obtain which is equivalent to v 9 L p ;R d d + π 6 L p C G 2 L p ;R d d, v 7 L p /ε;r d d + π 4 L p /ε C 1 ϕg 1 v 4 L p /ε;r d d Dirichlet problem in enlarging domain with compactly supported source term For Dirichlet problem 3.11, by the property of ϕ in 3.3, we have supp h 1 B 1 \ B 1/2 B 1 \ T. We introduce the following lemma concerning the Dirichlet problems for Stokes equations in enlarging domains with compactly supported source terms, which may be of independent interest. Lemma 3.2. Let 0 < ε < 1 be a sufficient small parameter, p > d with d 3 and g W 1,p R d ; R d be of compact support independent of ε. Let be a bounded C 1 domain in R d. Then the unique solution w, ξ W 1,p 0 /ε; R d L p 0 /ε to the following Dirichlet problem w + ξ = g, in /ε, div w = 0, in /ε, w = 0, on /ε satisfies w L p /ε;r d d + ξ L p /ε C g W 1,p R d ;R d 3.13 for some constant C = Cp, d, independent of ε. 11

12 Proof of Lemma 3.2. Without loss of generality, we assume that We introduce the changes of variables which satisfies supp g B 1. w 1 = w /ε, ξ 1 = ε 1 ξ /ε g 1 = g /ε 3.14 w 1 + ξ 1 = ε 2 g 1, in, div w 1 = 0, in, w 1 = 0, Then classical results see Remark 1.1 imply on. w 1 L p ;R d d + ξ 1 L p C ε 2 g 1 W 1,p ;R d, 3.15 for some constant C = Cp, d, independent of ε. Now we estimate the quantity on the right-hand side of Let ψ Cc ; R d be any test function and χ Cc B 1 be a cut-off function such that χ = 1 on supp g. Then g 1, ψ W 1,p,W 1,p = g /ε, ψ 0 W 1,p,W 1,p 0 = ε d g, ψε W 1,p /ε,w 1,p 0 /ε 3.16 = ε d g, χ ψε W 1,p /ε,w 1,p We calculate 0 /ε ε d g W 1,p R d χ ψε W 1,p R d. χ ψε p = W 1,p R d χ ψε p W 1,p B 1 C χ + χ ψε p L p B 1 + εp χ ψε p C ε d ψ p + L p B ε εp ε d ψ p L p B ε C ε d ε p ψ p L p, L p B where in the last inequality we used Hölder inequality and Sobolev embedding: where ψ L p B ε C ε ψ L p B ε C ε ψ L p C ε ψ L p, p = 1 p 1 d. In 3.18 we also used the assumption p > d, which indicates p < d. We combine 3.16 and 3.17 to obtain g 1, ψ W 1,p W 1,p C ε 1+d/p g W 1,p R d ψ L p, which implies 0 g 1 W 1,p C ε 1+d/p g W 1,p R d. 12

13 Together with 3.15 we deduce w 1 L p ;R d d + ξ 1 L p ;R d C ε 1+d/p g W 1,p R d ;R d By 3.14, we have w 1 L p ;R d d = ε 1+d/p w L p /ε;r d d, ξ 1 L p = ε 1+d/p ξ L p /ε The estimates in 3.19 and 3.20 imply 3.13 and we complete the proof of Lemma 3.2. We now apply Lemma 3.2 to Dirichlet problem 3.11 to obtain: v 8 L p /ε;r d d + π 5 L p /ε C h 1 W 1,p R d ;R d From the above two estimates 3.12 and 3.21, by 3.3, 3.4 and 3.5, direct calculation gives v 7 L p /ε;r d d + π 4 L p /ε C G 1 L p ε;r d d + v 1 L p B 1 \T ;R d, v 8 L p /ε;r d d + π 5 L p /ε C G 1 L p B 1 \T ;R d d + v 1 L p B 1 \T ;R d + π 1 W 1,p B 1 \T. This implies v 6 L p /ε;r d d + π 3 L p /ε C G 1 L p ε;r d d + v 1 L p B 1 \T ;R d + π 1 W 1,p B 1 \T. Together with 3.5 and 3.6, we finally obtain v 3 L p /ε;r d d + π 3 L p /ε C G 1 L p ε;r d d + v 1 L p B 1 \T ;R d + π 1 W 1,p B 1 \T End of the proof First of all, summing up the estimates in 3.9 and 3.22 implies directly the following result: Proposition 3.3. Let p > d with d 3, G 1 L p ε ; R d d and v 1, π 1 W 1,p 0 ε ; R d L p 0 ε be the unique solution to 3.1. Then there holds v 1 L p ε;r d d + π 1 L p ε C G 1 L p ε;r d d + v 1 L p B 1 \T ;R d + π 1 W 1,p B 1 \T for some constant C = Cp, d,, T independent of ε. 13

14 3.5.1 Contradiction and compactness argument Now we employ contradiction argument to to prove Theorem 3.1. By contradiction we suppose that there exist p d, d, a number sequence {ε k } k N 0, 1 and a function sequence {G k } k N L p εk ; R d d satisfying ε k 0 as k, G k L p εk 0 as k, 3.23 ;R d d such that the unique solution v k, π k W 1,p 0 εk ; R d L p 0 εk to the Dirichlet problem v k + π k = div G k, in εk, div v k = 0, in εk, 3.24 v k = 0, on εk satisfies v k L p εk ;R d d + π k L p εk = 1 for any k N By Proposition 3.3, we have v k L p εk ;R d d + π k L p εk C G k L p ε;r d d + v k L p B 1 \T ;R d + π k W 1,p B 1 \T Since v k is of zero trace on the boundary of εk, the estimate assumed in 3.25 and the classical Sobolev embedding inequality: implies sup k N ψ L q Cq, d ψ L q, for any 1 q < d and ψ C c R d v k L p εk ;R d C sup v k L k N p εk ;R d d C, 1 p := 1 p 1 d, 3.27 for some constant C = Cp, d independent of ε. We remark that in 3.27 the assumption p < d is used. We consider the zero extensions ṽ k, π k, G k defined as: Since v k W 1,p 0 εk ; R d, we have ṽ k = v k in εk, ṽ k = 0 in R d \ εk, π k = π k in εk, π k = 0 in R d \ εk, G k = G k in εk, Gk = 0 in R d \ εk. ṽ k = v k in εk, ṽ k = 0 in R d \ εk. By 3.23, 3.25 and 3.27, we have the uniform estimates for the extensions: G k L p R d \T ;R d d = G k L p εk 0, as k, ;R d d ṽ k L p R d \T ;R d d + π k L p R d \T = 1, sup ṽ k L p R d \T ;R d C. k N 14

15 Furthermore, we have the weak convergence up to substractions of subsequences: ṽ k v weakly in L p R d \ T ; R d, ṽ k v weakly in L p R d \ T ; R d d, π k π weakly in L p R d \ T We would like to show the weak limit couple v, π is zero. Since div ṽ k = 0 for any k, we have div v = For any ψ Cc R d \ T ; R d, there exits k 0 such that for all k k 0, we have ψ Cc εk ; R d. Then ψ is a good test function for 3.24 when k k 0. This implies that for any k k 0, there holds v k : ψ dx εk π k div ψ dx = εk G k : ψ dx. εk This is equivalent to R d \T ṽ k : ψ dx π k div ψ dx = R d \T R d \T G k : ψ dx By 3.23 and 3.28, passing k in 3.30 implies v : ψ dx π div ψ dx = R d \T R d \T Then by 3.29 and 3.31, we deduce { v + π = 0, in R d \ T, div v = 0, in R d \ T In the sequel of this section, we will show that v belongs to some homogeneous Sobolev space that falls in the framework of the study in [12] concerning the Stokes equations in exterior domains. Then we employ the result in [12] to deduce v = 0, π = Homogeneous Sobolev spaces We now recall some concepts of the homogeneous Sobolev spaces in exterior domains. The materials are mainly taken from Chapter II.6 and II.7 of Galdi s book [11]. Let 1 q <, Λ := R d \ ω be an exterior domain with ω a bounded Lipschitz domain in R d. We define the linear space D 1,q Λ = {u L 1 loc Λ : u D 1,q Λ < }, u D 1,q Λ := u L q Λ The space D 1,q is generally not a Banach space. However, if we introduce the equivalent classes for any u D 1,q Λ: [u] = {u + c, c R is a constant}, the space Ḋ1,q Λ of all equivalence classes [u] equipped with the norm is a Banach space. [u] Ḋ1,q Λ := u D 1,q Λ = u L q Λ 15

16 The functional D 1,q Λ introduced in 3.33 defines a norm in Cc Λ. Thus we introduce the Banach space D 1,q 0 Λ which is the completion of C c Λ with respect to the norm D 1,q Λ. By Sobolev embedding, if q < d, for any u D 1,q 0 Λ, we have u Lq Λ where q := dq d q is from the classical Sobolev embedding inequality. In q < d, Galdi in [11, equation II.7.14] gave an equivalent description for the Banach space D 1,q 0 Λ: D 1,q 0 Λ = {u D1,q Λ : u L q Λ such that ψu W 1,q 0 Λ for any ψ C c R d }, 3.34 with the equivalent norm D 1,q 0 Λ := D 1,q Λ + L q Λ Derivation of contradiction We first prove that v obtained in 3.28 is in the homogeneous Sobolev space D 1,p 0 Rd \ T. Since p < d, then by 3.34, it is sufficient to show ψv W 1,q 0 Rd \ T, for any ψ C c R d. This is rather direct. Since ṽ k has zero trace on T, this implies ψṽ k W 1,q 0 Rd \ T, for any ψ C c R d. Then ψv, as the weak limit of ψṽ k in W 1,q 0 Rd \ T, is necessarily in W 1,q 0 Rd \ T. Then, the known result for the Dirichlet problems of Stokes equations in exterior domains see for instance [12, Theorem 1] states that the Dirichlet problem 3.32 admits a unique solution v, π in D 1,p 0 Rd \ T ; R d L p R d \ T ; necessarily this unique solution is v = 0, π = Now we are ready to derive a contradiction. By 3.25 and 3.27, we have sup v k W 1,p B 1 \T C <. k N Then the Rellich-Kondrachov compact embedding theorem implies, up to a substraction of subsequence, that v k v strongly in L p B 1 \ T, π k π strongly in W 1,p B 1 \ T. Finally, by 3.35 and 3.23, we pass k in 3.26 to obtain a contradiction: 1 0. This implies that the uniform estimate 3.2 is true and we complete the proof of Theorem

17 4 Proof of Theorem 1.4 We start by proving the first part of Theorem 1.4. Let p > d and {G ε } 0<ε<1 L p ε ; R d d satisfy the assumptions in the first part of Theorem 1.2. Let v ε, π ε W 1,p 0 ε ; R d L p 0 ε be the unique solution to 1.1 with source function G ε. This means for any ϕ Cc,div ε; R d and any φ Cc ε, there holds v ε : ϕ dx = G ε : ϕ dx, v ε φ dx = ε ε The subscript div means divergence free: C c,div D; Rd := {φ C c D; R d : div φ = 0}, for any domain D R d. By contradiction we assume lim inf ε 0 Then there exists a subsequence {ε k } k N such that v ε L p ε;r d d <. sup v εk L p ε;r d d <. k N This implies the weak convergence of the zero extensions: ṽ εk ṽ weakly in W 1,p 0 ; R d, as k, where ṽ εk are defined as ṽ εk = v εk in ε, ṽ εk = 0 on εt. Since p > d, the Sobolev embedding theorem implies that ṽ C 0,1 d p ; R d, and up to a substraction of subsequence that for any λ < 1 d p, ṽ ε k ṽ strongly in C 0,λ ; R d, as k. 4.2 Hence, the fact ṽ εk = 0 on ε k T 0 and the above strong convergence in 4.2 imply ṽ0 = Passing ε k 0 in 4.1 gives ṽ : ϕ dx = G : ϕ dx, for any ϕ Cc,div \ {0}; Rd, ṽ φ dx = 0, for any φ Cc \ {0}. 4.4 Thus, by equations 4.3 and 4.4, we obtain ṽ + π = div G, div ṽ = 0, in \ {0}; ṽ = 0, on {0}, 4.5 for some π D \ {0} with zero mean value. Moreover, the first equation in 4.5 implies π W 1,p \ {0}; R d. 17

18 Since \ {0} is bounded and is a finite union of star-shaped domains, by employing Theorem III.3.1 in [11], for any 1 < q <, there exists a linear operator B known as Bogovskii s operator from L q 1,q 0 \ {0} to W0 \ {0}; Rd such that for any f L q 0 \ {0}, there holds div Bf = f in \ {0}; Bf W 1,q 0 \{0};R d Cq, d, f L q \{0}. By another point of view, \ {0} is also a John Domain on which a uniformly bounded Bogovskii s operator can be constructed see for instance [1]. Then for any nonzero f Cc \ {0}, we have π, f = π, f f = π, div Bf f = π, Bf f π W 1,p \{0};R d Bf f W 1,p 0 \{0};R d C π W 1,p \{0};R d f f L p \{0} C π W 1,p \{0};R d f L p \{0}, where f := 1 \{0} \{0} fx dx is the mean value of f. This implies π L p \{0} C π W 1,p \{0};R d <. Now we have ṽ, π is the solution to Dirichlet problem 4.5 defined in domain \ {0}. In the following proposition, we show that ṽ, π is also the solution to Dirichlet problem 1.4 in domain : Proposition 4.1. The couple ṽ, π solves the following Dirichlet problem: ṽ + π = div G, div ṽ = 0, in ; ṽ = 0, on. Proof of Proposition 4.1. It is sufficient to show the following two equalities: ṽ : ϕ dx πdiv ϕ dx + G : ϕ = 0 dx for any ϕ Cc ; R d, 4.6 and v φ dx = 0, for any φ C c. 4.7 To this end, we introduce cut-off functions sequence {φ n } n Z+ C R d satisfying 0 φ n 1, φ n = 0 in B 1/n, φ n = 1 on B c 2/n, φ n 4 n. Then for any 1 < q <, direct calculation gives 1 φ n L q R d C n d q, φ n L q R d ;R d C n 1 d q. Let ϕ Cc ; R d be a test function. We calculate ṽ + G : ϕ dx = ṽ + G : ϕφ n dx ṽ + G : ϕ φ n dx + ṽ + G : 1 φ n ϕ dx, π div ϕ dx = π div ϕφ n dx πϕ φ n dx + πdiv ϕ1 φ n dx

19 Since φ n ϕ Cc \ {0}; R d and ṽ, π is the solution to 4.5, we then have ṽ + G : ϕφ n dx π div ϕφ n dx = 0, for any n Z By 4.8 and 4.9, we obtain ṽ : ϕ dx πdiv ϕ dx + G : ϕ dx = ṽ + G : ϕ φ n dx + ṽ + G : 1 φ n ϕ dx + πϕ φ n dx πdiv ϕ1 φ n dx. We then calculate ṽ + G : ϕ φ n dx C ṽ + G L p ;R d d φ n L p ;R d C d n1 p, ṽ + G : 1 φ n ϕ dx C ṽ + G L p ;R d d 1 φ n L p C d n p, πϕ φ n dx C π L p φ n L p ;R d C d n1 p, πdiv ϕ1 φ n dx C π L p 1 φ n L p C d n p Since p > d 3, the power 1 d/p < 0. Therefore, passing n in 4.10 implies the result in 4.6. Similar argument gives 4.7. We thus complete the proof. Hence, by uniqueness of the solution to Dirichlet problem 1.4, we have ṽ = v. We thus obtain a contradiction between 1.5 and 4.3. Then the estimate 1.6 holds true and we finish the proof of the first part of Theorem 1.4. Now we employ duality arguments to prove the second part of Theorem 1.4. Let 1 < p < d and G L p ; R d such that is satisfied. This ensures that the solution v ε to 1.1 with source function G satisfies the estimate 1.6. Let H ε := v ε p 2 v ε p p v ε L p ε;r d d satisfying H ε L p ε;r d d = 1, and w ε, ξ ε W 1,p 0 ε L p 0 ε be the unique solution to 1.1 with source function H ε. Then w ε L p ε;r d d G 1 L p ε;r d d w ε, G = G 1 L p ε;r d d w ε, v ε = G 1 L p ε;r d d H ε, v ε = G 1 L p ε;r d d v ε L p ε;r d d. Together with 1.6, we deduce our desire estimate 1.7. Thus we complete the proof of Theorem

20 5 Construction of restriction operator This section is mainly devoted to the proof of Theorem 2.1. Corollary 2.3 is proved at the end directly using the construction of the restriction operator. 5.1 Proof of Theorem 2.1 We employ the construction of Allaire in Section 2.2 in [2] to prove Theorem 2.1. Thanks to our results in Theorem 1.2, the generalization of Allaire s construction from the L 2 framework to the L p framework is rather straightforward. First of all, we recall the construction of Allaire. Let D and D ε be the domains introduced in Section 2.1. By the distribution of the holes assumed in 2.1, 2.2 and 2.3, in each cube εc k, there exists a ball Bx k, b 1 ε for some b 1 > 0 independent of ε such that T ε,k = ε α O k T + x k Bx k, b 1 ε εc k. 5.1 For any u W 1,p 0 D; R 3 with 3/2 < p < 3, we define R ε u in the following way: R ε ux := ux, for x D \ Bx k, b 1 ε, k K ε R ε ux := u ε,k x, for x Bx k, b 1 ε \ T ε,k, k K ε, where u ε,k solves u ε,k + p ε,k = u, in Bx k, b 1 ε \ T ε,k, 1 div u ε,k = div u + div u dx, in Bx k, b 1 ε \ T ε,k, Bx k, b 1 ε \ T ε,k T ε,k u ε,k = u, on Bx k, b 1 ε, u ε,k = 0, on T ε,k The key point is to prove the following result: Proposition 5.1. Let u W 1,p 0 D; R 3 with 3/2 < p < 3. The Dirichlet problem 5.3 admits a unique solution u ε,k, p ε,k W 1,p Bx k, b 1 ε \ T ε,k ; R 3 L p 0 Bx k, b 1 ε \ T ε,k satisfying u ε,k L p Bx k,b 1 ε\t ε,k ;R p ε,k L p Bx k,b 1 ε\t ε,k C u L p Bx k,b 1 ε;r ε 3 pα 3 p u L p Bx k,b 1 ε;r To prove Proposition 5.1, we introduce the following two lemmas corresponding to L p generalization of Lemma and Lemma in [2]. Lemma 5.2. Let 1 < p < 3, 0 < η < 1/2 and T s B 1 be a simply connected domain of class C 1. There exists a linear operator L from W 1,p B 1 to W 1,p B 1 \ ηt s such that for any u W 1,p B 1 : Lu = u on B 1, Lu = 0 on ηt s, Lu L p B 1 \ηt s ;R 3 u C L p B 1 ;R 3 + η 3 p 1 u L p B 1, where the constant C is independent of η. 20

21 Lemma 5.3. Let 1 < p < 3, 0 < η < 1/2 and T s B 1 be a simply connected domain of class C 1. There exists a linear operator B η from L p 0 B 1 \ ηt s to W 1,p 0 B 1 \ ηt s ; R 3 such that for any f L p 0 B 1 \ ηt s, there holds div B η f = f in B 1 \ ηt s, B η f W 1,p 0 B 1 \ηt s ;R 3 C f L p B 1 \ηt s, where the constant C is independent of η. The proof of above two lemmas can be done by generalizing the proof of Lemma and Lemma in [2] from the L 2 framework to the L p framework, so we omit the details. In particular, Lemma 5.3 can also be proved by observing that B 1 \ ηt s, η > 0 are uniform John domains as η 0 see [1] and [5], such that the result in Lemma 5.3 can also be generalized to any p 1,. Now we apply Lemma 5.2 and Lemma 5.3, together with the results we obtained in Theorem 1.2, to prove the following lemma: Lemma 5.4. Let 3/2 < p < 3, 0 < η < 1/2 and T s B 1 be a simply connected domain of class C 1. There exists a unique solution v η, p η W 1,p B 1 \ ηt s ; R 3 L p 0 B 1 \ ηt s to the following Dirichlet problem for Stokes equations: v η + p η = v, in B 1 \ ηt s, 1 div v η = div v + B 1 \ ηt s div v dx, in B ηt s 1 \ ηt s, 5.5 v η = v, on B 1, v η = 0, on ηt s, and there holds the estimate: v η L p B 1 \ηt s ;R p η L p B 1 \ηt s C v L p B 1 ;R η 3 p 1 v L p B 1 ;R 3, 5.6 where the constant C is independent of η. Proof of Lemma 5.4. First of all, we observe that the compatibility condition holds: div v 1 B 1 \ηt s η dx = div v + B 1 \ηt s B 1 \ ηt s div v dx dx s ηt = div v dx = v n ds = v η n ds. B 1 B 1 B 1 ηt s This actually indicates, for any fixed 0 < η < 1/2, there exits a unique solution v η, p η in W 1,p B 1 \ ηt s ; R 3 L p 0 B 1 \ ηt s to Dirichlet problem 5.5 by employing the classical theory for Stokes equations see Theorem 5.1 in [7], Section 3 in [18] and Theorem 2.9 in [4]. The key point is to show the estimate in 5.6, particularly the dependency on η as η 0. In the following we derive our desired estimate 5.6; at the same time, this also gives a proof for the existence and uniqueness of solution in W 1,p B 1 \ ηt s ; R 3 L p 0 B 1 \ ηt s to Dirichlet problem 5.5. By Lemma 5.2, the new unknown v 1,η := v η Lv solves v 1,η + p η = v + Lv, in B 1 \ ηt s, 1 div v 1,η = div Lv + div v + B 1 \ ηt s div v dx, in B ηt s 1 \ ηt s, v 1,η = 0, on B 1 ηt s. 21

22 Define 1 f := div Lv + div v + B 1 \ ηt s div v dx. s ηt By virtue of Lemma 5.2, direct calculation gives f L p B 1 \ηt s C v L p B 1 ;R η 3 p 1 v L p B 1 ;R 3 and f dx = 1 div Lv + div v + B 1 \ηt s B 1 \ηt s B 1 \ ηt s = div Lv dx + div v dx s B 1 \ηt B 1 = Lv n ds + v n ds B 1 ηt s B 1 = v n ds + v n ds B 1 B 1 = 0. div v dx s ηt dx Then f L p 0 B 1 \ ηt s and B η f is well-defined, where B η is the linear operator defined in Lemma 5.3. Then, by Lemma 5.3, the new unknown solves v 2,η := v 1,η B η f = v η Lv B η f v 2,η + p η = div v + Lv + B η f, in B 1 \ ηt s, div v 2,η = 0, in B 1 \ ηt s, v 2,η = 0, on B 1 ηt s, which is a Dirichlet problem for the Stokes equations with the divergence free condition, with the no-slip boundary condition and with a divergence form source term. This falls in the framework of 1.1. We apply Theorem 1.2 to conclude that there exists a unique solution v 2,η, p η W 1,p 0 B 1 \ ηt s ; R 3 L p 0 B 1 \ ηt s to 5.7 such that 5.7 v 2,η L p B 1 \ηt s ;R p η L p B 1 \ηt s C v + Lv + B η f L p B 1 ;R 3 3 C v L p B 1 ;R η 3 p 1 v L p B 1 ;R 3. Back to v η, we obtain the estimate 5.6 and we complete the proof. Now we are ready to prove Proposition 5.1: Proof of Proposition 5.1. Let u W 1,p 0 D with 3/2 < p < 3 be as in Proposition 5.1. First of all, 22

23 we observe that the Dirchlet problem 5.3 is compatible: div u ε,k dx Bx k,b 1 ε\t ε,k 1 = div u + Bx k,b 1 ε\t ε,k Bx k, b 1 ε \ T ε,k = div u dx = u n ds Bx k,b 1 ε Bx k,b 1 ε = u ε,k n ds. Bx k,b 1 ε T ε,k T ε,k div u dx Thus, for any fixed ε > 0, we have the existence and uniqueness of solution u ε,k, p ε,k in W 1,p Bx k, b 1 ε \ T ε,k ; R 3 L p 0 Bx k, b 1 ε \ T ε,k to Dirichlet problem 5.3 of Stokes equations due to classical theory. The point is to show the estimate in 5.4, particularly the dependency on ε as ε 0. Let u ε,k, p ε,k W 1,p Bx k, b 1 ε \ T ε,k ; R 3 L p 0 Bx k, b 1 ε \ T ε,k be the unique solution to Dirichlet problem 5.3. We consider the following change of unknowns: Then and there holds v ε,k = u ε,k b 1 ε +x k, q ε,k = b 1 εp ε,k b 1 ε +x k. v ε,k, q ε,k W 1,p B 1 \ ηt s ; R 3 L p 0 B 1 \ ηt s, η := ε α 1, T s := b 1 1 O kt, 5.8 where v is defined by v ε,k + q ε,k = v, in B 1 \ ηt s, 1 div v ε,k = div v + B 1 \ ηt s div v dx, in B ηt s 1 \ ηt s, v ε,k = v, on B 1, v ε,k = 0, on ηt s, v = ub 1 ε +x k W 1,p B 1 ; R By 5.1 and 5.8, we have T s T s B 1. If α = 1 such that η = ε α 1 = 1, the domain B 1 \ ηt s = B 1 \ T s is a fixed, bounded domain of class C 1. We employ the classical theory for Stokes equations in bounded domain see for instance [7, 18, 4] to deduce that there exits a unique solution v ε,k, q ε,k W 1,p B 1 \ T s ; R 3 L p 0 B 1 \ T s satisfying v ε,k L p B 1 \T s ;R q ε,k L p B 1 \T s C v W 1,p B 1 ;R Due to 5.8 and 5.9, the estimate in 5.10 is equivalent to u ε,k L p Bx k,b 1 ε\t ε,k ;R p ε,k L p Bx k,b 1 ε\t ε,k C u L p Bx k,b 1 ε;r b 1 ε u L p Bx k,b 1 ε;r 3. dx 5.11 If α > 1, without loss of generality we may assume η := ε α 1 < 1/2. Then by Lemma 5.4, there exits a unique solution v ε,k, q ε,k W 1,p B 1 \ ηt s ; R 3 L p 0 B 1 \ ηt s satisfying v ε,k L p B 1 \ηt s ;R q ε,k L p B 1 \ηt s C v L p B 1 ;R η 3 p 1 v L p B 1 ;R

24 Again by 5.8 and 5.9, we deduce from 5.12 that By observing u ε,k L p Bx k,b 1 ε\t ε,k ;R p ε,k L p Bx k,b 1 ε\t ε,k C u L p Bx k,b 1 ε;r b 1 ε η 3 p 1 u L p Bx k,b 1 ε;r ε 1 η 3 p 1 = ε 1 ε α 1 3 p 1 = ε 3 pα 3 p, 5.14 we obtain our desired result 5.4 by combining the estimates in 5.11 and in The proof is completed. In the end, we verify the linear functional R ε defined by 5.2 and 5.3 fulfills the properties stated in Theorem 2.1. Given u W 1,p 0 D; R 3, by Proposition 5.1, we have R ε u W 1,p 0 D ε ; R 3 for which the zero trace property on the boundaries of holes is guaranteed by the construction 5.3. Given u W 1,p 0 D ε ; R 3, let w := R ε ũ where ũ W 1,p 0 D; R 3 is the zero extension of u in D defined by Then the equation 5.3 in u ε,k = R ε ũ in Bx k, b 1 ε \ T ε,k becomes u ε,k + p ε,k = ũ, in Bx k, b 1 ε \ T ε,k, div u ε,k = div ũ, in Bx k, b 1 ε \ T ε,k, 5.15 u ε,k = ũ, on Bx k, b 1 ε T ε,k. where we used the property ũ = 0, div ũ = 0 on T ε,k. Then, the unique solution u ε,k, p ε,k W 1,p Bx k, b 1 ε \ T ε,k ; R 3 L p 0 Bx k, b 1 ε \ T ε,k to 5.15 is simply u ε,k = ũ = u, p ε,k = 0, in Bx k, b 1 ε \ T ε,k. This means R ε ũ coincides with u near the holes. Since R ε ũ also coincides with u away from the holes, as in 5.2, we have R ε ũ = u in D ε. Given u W 1,p 0 D; R 3 such that div u = 0 in D, it is straightforward to deduce div R ε u = 0 in D ε, due to 5.2 and 5.3. To complete the proof of Theorem 2.1, it is left to verify the estimate in Given u W 1,p 0 D; R 3, we calculate R ε u p L p D ε;r 3 3 D = R ε u p dx ε = D ε\ k Kε Bx k,b 1 ε R εu p dx + R ε u p dx k K Bx k,b 1 ε\t ε,k ε = D ε\ k Kε Bx k,b 1 ε u p dx + u ε,k p, L p Bx k,b 1 ε\t ε,k ;R 3 3 k K ε 5.16 where we used the fact that the balls Bx k, b 1 ε, k K ε, are pairwise disjoint. By Proposition 5.1 and the inequality aθ +b θ 2 a + b θ 2 θ a θ + b θ for any θ > 0, a > 0, b > 0, we obtain u ε,k p L p Bx k,b 1 ε\t ε,k ;R 3 3 C u p L p Bx k,b 1 ε;r ε3 pα 3 u p L p Bx k,b 1. ε;r

25 Applying 5.17 into 5.16, we obtain R ε u p L p D ε;r u 3 3 C p L p D;R ε3 pα 3 u p, L p D;R 3 which implies the desired estimate in We complete the proof of Theorem Proof of Corollary 2.3 We prove the existence of such an operator B ε satisfying the properties stated in Corollary 2.3. Unlike in [6], by employing the restriction operator R ε obtained in Theorem 2.1, we will show the operator defined by B ε := R ε B D E, 5.18 fulfills the properties stated in Corollary 2.3. Here E is the zero extension operator from function spaces defined in D ε to function spaces defined in D, B D is the classical Bogovskii s operator in domain D, and R ε is the restriction operator constructed in the proof of Theorem 2.1 in Section 5.1, precisely in 5.2 and 5.3. This construction can be understood in such a way: extension + classical Bogovskii operator in homogenized domain + restriction. Let 3/2 < p < 3 and f L p 0 D ε. It is direct to find that E is a linear operator from L p 0 D ε to L p 0 D with operator norm 1. The classical Bogovskii s operator B D is a linear operator from L p 0 D to W 1,p 0 D; R 3 with operator norm only depending on the number p and the Lipschitz character of D, therefore interdependent of ε. Finally by the properties of the restriction operator given in Theorem 2.1, we conclude that the operator B ε defined in 5.18 is a linear operator from L p 0 D ε to W 1,p 0 D ε ; R 3 such that B ε f W 1,p 0 D ε;r 3 C 1 + ε 3 pα 3 p f L p D ε. It is left to check the property in Let u := B D Ef, then u W 1,p 0 D; R 3 satisfies div u = f in D ε, div u = 0 on k K ε T ε,k. By the properties of the restriction operator in 5.2 and 5.3, we finally obtain The proof is completed. References div R ε u = div u = f in D ε. [1] G. Acosta, R. G. Durán, M. A. Muschietti. Solutions of the divergence operator on John domains. Adv. Math., [2] G. Allaire. Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes. Arch. Ration. Mech. Anal., , [3] G. Allaire. Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. II. Noncritical sizes of the holes for a volume distribution and a surface distribution of holes. Arch. Ration. Mech. Anal., ,

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