Homogenization of Stokes Systems with Periodic Coefficients

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1 University of Kentucky UKnowledge Theses and Dissertations--Mathematics Mathematics 2016 Homogenization of Stokes Systems with Periodic Coefficients Shu Gu University of Kentucky, Digital Object Identifier: Click here to let us know how access to this document benefits you. Recommended Citation Gu, Shu, "Homogenization of Stokes Systems with Periodic Coefficients" (2016). Theses and Dissertations--Mathematics This Doctoral Dissertation is brought to you for free and open access by the Mathematics at UKnowledge. It has been accepted for inclusion in Theses and Dissertations--Mathematics by an authorized administrator of UKnowledge. For more information, please contact

2 STUDENT AGREEMENT: I represent that my thesis or dissertation and abstract are my original work. Proper attribution has been given to all outside sources. I understand that I am solely responsible for obtaining any needed copyright permissions. I have obtained needed written permission statement(s) from the owner(s) of each thirdparty copyrighted matter to be included in my work, allowing electronic distribution (if such use is not permitted by the fair use doctrine) which will be submitted to UKnowledge as Additional File. I hereby grant to The University of Kentucky and its agents the irrevocable, non-exclusive, and royaltyfree license to archive and make accessible my work in whole or in part in all forms of media, now or hereafter known. I agree that the document mentioned above may be made available immediately for worldwide access unless an embargo applies. I retain all other ownership rights to the copyright of my work. I also retain the right to use in future works (such as articles or books) all or part of my work. I understand that I am free to register the copyright to my work. REVIEW, APPROVAL AND ACCEPTANCE The document mentioned above has been reviewed and accepted by the student s advisor, on behalf of the advisory committee, and by the Director of Graduate Studies (DGS), on behalf of the program; we verify that this is the final, approved version of the student s thesis including all changes required by the advisory committee. The undersigned agree to abide by the statements above. Shu Gu, Student Dr. Zhongwei Shen, Major Professor Dr. Peter Hislop, Director of Graduate Studies

3 Homogenization of Stokes Systems with Periodic Coefficients DISSERTATION A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the College of Arts and Sciences at the University of Kentucky By Shu Gu Lexington, Kentucky Director: Dr. Zhongwei Shen, Professor of Mathematics Lexington, Kentucky 2016 Copyright c Shu Gu 2016

4 ABSTRACT OF DISSERTATION Homogenization of Stokes Systems with Periodic Coefficients In this dissertation we study the quantitative theory in homogenization of Stokes systems. We study uniform regularity estimates for a family of Stokes systems with rapidly oscillating periodic coefficients. We establish interior Lipschitz estimates for the velocity and L estimates for the pressure as well as Liouville property for solutions in R d. We are able to obtain the boundary W 1,p estimates in a bounded C 1 domain for any 1 < p <. We also study the convergence rates in L 2 and H 1 of Dirichlet and Neumann problems for Stokes systems with rapidly oscillating periodic coefficients, without any regularity assumptions on the coefficients. KEYWORDS: Homogenization, Stokes systems, convergence rates, uniform regularity, Dirichlet problem, Neumann problem Author s signature: Shu Gu Date: July 25, 2016

5 Homogenization of Stokes Systems with Periodic Coefficients By Shu Gu Director of Dissertation: Director of Graduate Studies: Zhongwei Shen Peter Hislop Date: July 25, 2016

6 Dedicated to Jing Wei

7 ACKNOWLEDGMENTS My utmost thanks and deepest gratitude go to my advisor Professor Zhongwei Shen for his enormous patience, guidance, suggestions and encouragement during my Ph.D. years, without which the work would not be possible. His immense knowledge, enthusiasm and work ethic have influenced me remarkably and will continue to guide me throughout my life. Living in a different country was never easy, I am extremely grateful to Professor Shen and his wife Mrs. Liping Peng for their kindness and loads of invitations to their home. I extend my genuine thanks and gratitude to Professor John Lewis, Professor Russell Brown, Professor Yuan Liao and Professor Dong-Sheng Yang for helpful comments and for serving on my thesis committee. Special thanks go to Professor Russell Brown for his teaching of Real Analysis and Harmonic analysis, holding secret seminars, and for generous support throughout my graduate studies. I would like to Professor Qiang Ye and Professor Qun He for all their invaluable advise and help at various stages of my mathematical education. Most of all, I wish to thank my wife, Jing Wei. Her tremendous love and support have always been my guiding light. Finally, I thank my parents for their constant encouragement and support. iii

8 TABLE OF CONTENTS Acknowledgments Table of Contents iii iv Chapter 1 Introduction and Main Results Chapter 2 Notations and Definitions Notations Smoothing in Steklov s sense Chapter 3 Preliminaries Homogenization Theory of Stokes Systems Asymptotic Expansions Compactness Theorem Homogenization for Neumann problems Chapter 4 Convergence Rates of Dirichlet Problems in Homogenization of Stokes Systems Introduction Convergence rates for u ε in H Convergence rates for the pressure term Convergence rates for u ε in L Chapter 5 Convergence Rates of Neumann Problems in Homogenization of Stokes Systems Introduction Convergence rates for u ε in H Convergence rates of the pressure term Convergence rates for u ε in L Chapter 6 Uniform Regularity Estimates in Homogenization of Stokes Systems Introduction Interior Lipschitz estimates for u ε A Liouville property for Stokes systems L estimates for p ε and proof of Theorem Boundary Hölder estimates Interior W 1,p estimates Uniform Boundary W 1,p estimates and Proof of Theorem Bibliography Vita iv

9 Chapter 1 Introduction and Main Results The theory of homogenization was introduced in part to describe the behavior of composite materials in mechanics, physics, chemistry and engineering. Composite materials are usually characterized by two scales, the microscopic one, describing the heterogeneities, and the macroscopic one, describing the global behavior of the composite. In a composite, the heterogeneities are small compared to its global dimension, while from the macroscopic points of view, the composite looks like a homogeneous material. The intent of homogenization theory is to replace the microscopically heterogeneous material by a homogenized material, whose global characteristics are a good approximation of the initial ones. In the study of boundary value problems in media with periodic structure, if the period of the structure is small compared to the size of the region in which the system is to be studied, we will use a small parameter ε to denote the ratio of the period of the structure to a typical length in the region. In mathematics terms, a family of partial differential operators L ε with rapidly oscillating coefficients, depending on the small parameter ε, is given. In a domain, we have a boundary value problem { L ε (u ε ) = F in, u ε subject to appropriate boundary conditions. Homogenization theory has shown that u ε converges to u 0 as ε 0 (with suitable definition of weak type convergence), where u 0 is the solution of { L 0 (u 0 ) = F in, u 0 subject to the same kind of boundary conditions, where L 0 is a partial differential operator with constant coefficients, which is called the homogenized operator of the family L ε. Specifically in qualitative homogenization theory, for the standard elliptic system div(a(x/ε) u ε ) = F in a bounded domain in R d, the proof of homogenization theorem was first obtained by De Diorgi and Spagnolo [16 18,53,54]. Here A(y) is a matrix with periodic measurable coefficients satisfying ellipticity condition. Shortly thereafter, Bakhvalov [8,9] and then Lions [10,40] established the same result based on method of asymptotic expansions. Another approach to the homogenization theory, based on compensated compactness, was developed by Murat [43] and Tartar [57]. 1

10 Quantitative homogenization has been studied extensively in recent years, given these qualitative results in homogenization for various types of equations with various boundary conditions. There are two main and natural tasks in quantitative homogenization theory, 1. uniform regularity estimates of solutions, which are independent of the small parameter ε; 2. sharp convergence rates, which describe the speed of convergence. Uniform Regularity Estimates For uniform regularity estimates, we consider a family of standard second-order elliptic operator L ε in divergence form with rapidly oscillating coefficients, which are defined by L ε = div(a(x/ε) ) = x i [ a αβ ij ( x ε ) x j ], ε > 0. (1.0.1) with 1 i, j, α, β d, the summation convention is used throughout this thesis. The study of uniform regularity estimates in homogenization theory was initiated by M. Avellaneda and F. Lin in In a series of paper [3 7] from 1987 to 1991, Avellaneda and Lin established interior and boundary Lipschitz estimates and also W 1,p estimates for the standard elliptic system L ε (u ε ) = F with Dirichlet boundary condition u ε = g on for C 1,α domains, assuming the coefficient matrix A is elliptic, periodic and Hölder continuous. The approach called compactness method was used in [3] to prove Lipschitz estimates. We should mention that the Lipschitz estimates are sharp; in fact, even with C data, one cannot expect high-order uniform estimates for u ε, since u ε converges to u 0 only weakly. For standard second-order elliptic system L ε (u ε ) = F with Neumann boundary condition uε ν ε = g, Lipschitz estimates has been a longstanding open problem, as the boundary conditions are ε-dependent. It was only until in 2013, in [34] C. Kenig, F. Lin and Z. Shen were able to extend the boundary Lipschitz estimates to Neumann problems in C 1,α domains, with additional symmetry condition A = A. The breakthrough is based on the Rellich estimates obtained in [36,37] and nontangential maximal function estimates in [34]. Sharp W 1,p estimates for Neumann problem were also obtained. The symmetry condition was recently removed by S. Armstrong and Z. Shen. In [1], the uniform Lipschitz estimates and W 1,p estimates in C 1,α were obtained 2

11 for second-order elliptic system in divergence form with rapidly oscillating, almostperiodic coefficients, with either Dirichlet or Neumann data. In contrast to papers [3,34], the results were proved through constructive arguments, and thus the constants are in principle computable. In this thesis, we study the uniform regularity estimates for a family of Stokes systems with rapidly oscillating periodic coefficients. We establish interior Lipschitz estimates for the velocity and L estimates for the pressure as well as a Liouville property for solutions in R d. We also obtain the boundary W 1,p estimates in a bounded C 1 domain for any 1 < p <. More precisely, we consider the Stokes systems in fluid dynamics, { Lε (u ε ) + p ε = F, div(u ε ) = g, (1.0.2) in a bounded domain in R d, where ε > 0 and L ε is defined in (1.0.1). We will assume that the coefficient matrix A(y) = (a αβ ij (y)) is real, bounded measurable, satisfies the ellipticity condition: µ ξ 2 a αβ ij (y)ξα i ξ β j 1 µ ξ 2, for y R d and ξ = (ξ α i ) R d d, (1.0.3) where µ > 0, and the periodicity condition: A(y + z) = A(y) for y R d and z Z d. (1.0.4) We note that the system (1.0.2), which does not fit the standard framework of secondorder elliptic systems considered in [3,34], is used in the modeling of flows in porous media. The following is one of the main results we obtained in [30]. Theorem Suppose that A(y) satisfies the ellipticity condition (1.0.3) and periodicity condition (1.0.4). Let (u ε, p ε ) be a weak solution of the Stokes system (1.0.2) in B(x 0, R) for some x 0 R d and R > ε. Then, for any ε r < R, ( ) 1/2 ( ) 1/2 u ε 2 + p ε p ε 2 B(x 0,r) B(x 0,r) B(x 0,R) {( ) 1/2 C u ε 2 + g L (B(x 0,R)) + R ρ [g] C 0,ρ (B(x 0,R)) B(x 0,R) ( ) 1/q + CR F }, q B(x 0,R) (1.0.5) where 0 < ρ = 1 d q < 1, and the constant C depends only on d, µ, and ρ. 3

12 The scaling-invariant estimate (1.0.5) should be regarded as a Lipschitz estimate for the velocity and L estimate for the pressure down to the microscopic scale ε, even though no smoothness assumption is made on the coefficients A(y). In [30], we also obtain the following boundary W 1,p estimates. Theorem Let be a bounded C 1 domain in R d and 1 < q <. Suppose that A satisfies ellipticity (1.0.3) and periodicity (1.0.4) conditions. Also assume that A VMO(R d ). Let f = (f α i ) L q (; R d d ), g L q () and h B 1 1 q,q ( ; R d ) satisfy the compatibility condition g h n = 0, where n denotes the outward unit normal to. W 1,q (; R d ) L q () to Dirichlet problem L ε (u ε ) + p ε = div(f) in, div(u ε ) = g in, u ε = h on, satisfy the estimate u ε L q () + p ε where C q depends only on d, µ, A, and. Then the solutions (u ε, p ε ) in (1.0.6) { } p ε L q () C q f L q () + g L q () + h 1 1, (1.0.7) B q,q ( ) Sharp Convergence Rates As for the second task concerning sharp convergence rates, the primary purpose is to establish the optimal rate of convergence of solution u ε to homogenized solution u 0 in L 2 (; R d ) for both Dirichlet and Neumann problems. For Dirichlet problems, consider the scalar elliptic equation L ε (u ε ) = F in a Lipschitz domain with Dirichlet condition u ε = f on. By energy estimates and maximum principle, it is well known that u ε u 0 L 2 () Cε { 2 u 0 L 2 () + u 0 L ( )}. More recently, using the method of periodic unfolding, Griso [26, 27] was able to establish the much sharper estimate u ε u 0 L 2 () Cε u 0 H 2 (). (1.0.8) In the case of elliptic systems, the estimates (1.0.8) continue to hold under the additional assumption that A is Hölder continuous. The approach was based on the 4

13 uniform regularity estimates established in [3, 37] and do not apply to operators with bounded measurable coefficients. Recently, by using the Steklov smoothing operator, T.A. Suslina [55] was able to establish the O(ε) estimate (1.0.8) in L 2 for a broader class of elliptic operators in C 1,1 domains without any smoothness assumptions on the coefficient matrix A. There are relatively fewer known results for Neumann problems. Consider the Neumann problem for the scalar elliptic equation L ε (u ε ) = F in with uε ν ε = 0 on, the estimate u ε u 0 L 2 () Cε F H 2 () was proved by Griso [27] for C 1,1 domains with bounded measurable coefficients using the periodic unfolding method [14, 15]. The same result was also proved by Moskow and Vogelius [42] for curvilinear convex polygons in R 2. For the system case, consider elliptic systems L ε (u ε ) = F in with Neumann condition uε ν ε = g on, C. Kenig, F. Lin and Z. Shen [32] have shown that the estimate (1.0.8) holds in bounded Lipschitz domain, under additional assumption that A is Hölder continuous. Also recently, by using the Steklov smoothing operator, T. A. Suslina [56] was able to eliminate the smoothness condition on coefficients to establish the O(ε) estimate (1.0.8) in L 2 for a broader class of elliptic operators with Neumann data. In this thesis, we study the convergence rates in L 2 and H 1 of both Dirichlet and Neumann problems for Stokes systems with rapidly oscillating periodic coefficients in C 1,1 domains, without any smoothness assumptions on the coefficients. More precisely, by the homogenization theory of Stokes systems (see [10, 30]), under suitable conditions on F, g and h, suppose (u ε, p ε ) is a weak solution of Stokes system (1.0.2) with either Dirichlet u ε = h or Neumann uε ν ε p ε n = h boundary conditions on, it is known that u ε u 0 weakly in H 1 (; R d ) and p ε p ε p 0 p 0 weakly in L 2 (), where (u 0, p 0 ) H 1 (; R d ) L 2 () is the weak solution of the homogenized problem with constant coefficients, { L0 (u 0 ) + p 0 = F, in, div(u 0 ) = g, in, (1.0.9) satisfying the same Dirichlet u 0 = h or Neumann u 0 ν 0 p 0 n = h boundary condition on. Our primary purpose is to study the rate of convergence u ε u 0 L 2 () as ε 0. The following is the main result for Dirichlet problem we obtained in [28]. 5

14 Theorem Let be a bounded C 1,1 domain. Suppose that A satisfies the ellipticity condition (1.0.3) and periodicity condition (1.0.4). Given g H 1 () and h H 3/2 ( ; R d ) satisfying the Dirichlet compatibility condition g h n = 0, where n denotes the outward unit normal to. For F L 2 (; R d ), let (u ε, p ε ), (u 0, p 0 ) be weak solutions of Stokes systems (1.0.2), (1.0.9) respectively with Dirichlet boundary conditions u ε = u 0 = h on. Then where the constant C depends only on d, µ, and. u ε u 0 L 2 () Cε u 0 H 2 (), (1.0.10) The next theorem is our main result for Neumann problem in [29]. Theorem Let be a bounded C 1,1 domain. Suppose A satisfies ellipticity condition (1.0.3) and periodicity condition (1.0.4). Given F L 2 (; R d ) and h H 1/2 ( ; R d ) satisfying the Neumann compatibility condition F + h = 0, for g H 1 (), let (u ε, p ε ), (u 0, p 0 ) be weak solutions of Stokes systems (1.0.2), (1.0.9) respectively with Neumann boundary conditions uε ν ε p ε n = u 0 ν 0 p 0 n = h.then where the constant C depends only on µ, d, and. u ε u 0 L 2 () Cε u 0 H 2 (), (1.0.11) The organization of this thesis is as follows. Chapter 2 contains notations and definitions that will be used throughout the thesis. Chapter 3 is devoted to the homogenization theory of Stokes systems, including asymptotic expansions and compactness theorem. Our main results described above are presented in Chapter 4-6. In Chapter 4 and Chapter 5, we give the convergence results in L 2 and H 1 of Stokes system with Dirichlet and Neumann boundary conditions, respectively. Chapter 6 deals with uniform regularity estimates in homogenization of Stokes systems. Copyright c Shu Gu,

15 Chapter 2 Notations and Definitions In this chapter we first give basic notations and definitions that will be used throughout this thesis. Then we introduce the Steklov smoothing operator and its properties, as well as a lemma that plays an vital part in the study of convergence rates. 2.1 Notations - 1-periodic function. We call a function f 1-periodic if it satisfies the periodicity condition we defined in (1.0.4), i.e. f(y + z) = f(y) for a.e. y R d and z Z d. - Conormal derivative. We define the conormal derivative of Stokes system (1.0.2) on by ( uε ν ε ) α p ε n α = n i (x)a αβ ij (x/ε) uβ ε x j p ε (x)n α (x), (2.1.1) where n = (n 1,, n d ) is the outward unit normal to. - L 1 -average. We denote the L 1 average of f over the set E by f = 1 f. E - Hölder condition. We call A(y) Hölder continuous, if it satisfies where τ 0 and λ (0, 1]. E E A(x) A(y) τ x y λ for x, y R d, (2.1.2) - Rescaling property of Stokes systems. The technique of rescaling will be used routinely in the rest of the paper. Indeed, if (u ε, p ε ) is a weak solution of Stokes system (1.0.2) and v(x) = u ε (rx), then { L ε r (v) + q = F, div(v) = g, (2.1.3) where and g(x) = rg(rx), F (x) = r 2 F (rx), (2.1.4) q(x) = rp ε (rx). (2.1.5) 7

16 - r-neighborhood of the boundary. For r > 0, we let r and r to denote the r-neighborhood of as r = {x : dist(x, ) r}, r = {x R d : dist(x, ) r}. (2.1.6) k,λ - Hölder Space. The Hölder space C (Ē) consists of all functions u Ck (Ē) for which the norm u C k,λ (Ē) = D α u C( Ē) + [D α u] C 0,λ (Ē) (2.1.7) α k is finite, where the λ-th semi-norm of g is denoted by { } g(x) g(y) [g] C 0,λ (Ē) = sup : x, y x y Ē and x y, (2.1.8) λ α =k and C k (E) denotes the set of functions having all derivatives of order k continuous in E. - BMO Space. A locally integrable function f will be said to belong to BMO(R d ) if the following norm is finite. f BMO = sup f f dx (2.1.9) Q - VMO Space. A function f in BMO(R d ) is said to be VMO(R d ), the space of functions of vanishing mean oscillation, if lim f Q 0 Q 2.2 Smoothing in Steklov s sense Q Q Q f dx = 0. (2.1.10) We will use this section to introduce the Steklov smoothing operator as well as its properties, which play a crucial role in deriving convergence rates in the following chapters. More details about Steklov smoothing operator can be found in the literatures such as [45, 46, 55, 56, 59]. Let S ε be the operator on L 2 (R d ) given by (S ε u)(x) = u(x εz)dz (2.2.1) Y 8

17 and called the Steklov smoothing operator. Note that S ε u L 2 (R d ) u L 2 (R d ). Obviously, D α S ε u = S ε D α u for u H s (R d ) and any multi-index α such that α s. Therefore, S ε u H s (R d ) u H s (R d ). The following are a few properties of Steklov s operator; see [55, 56]. Proposition For any u H 1 (R d ) we have where C depends only on d. S ε u u L 2 (R d ) Cε u L 2 (R d ), For simplicity, we will use the notation f ε (x) = f(x/ε). And we let Y = [0, 1) d. Proposition Let f(x) be a 1-periodic function in R d such that f L 2 (Y ). Then for any u L 2 (R d ), f ε S ε u L 2 (R d ) f L 2 (Y ) u L 2 (R d ). The following lemma gives us an estimate for integrals near the boundary, see [55, 56] for example. Lemma Let R d be a bounded C 1 domain. Then, for any function u H 1 () and for any 0 < r diam(), ( ) 1/2 u 2 dx C r u 1/2 H 1 () u 1/2 L 2 (). (2.2.2) r Moreover, for any 1-periodic function f L 2 (Y ) and u H 1 (R d ), ( ) 1/2 f ε 2 S ε u 2 dx C ε f 1/2 L 2 (Y ) u 1/2 H 1 (R d ) u 1/2, (2.2.3) L 2 (R d ) 2ε where C depends only on and S ε is the Steklov smoothing operator defined in (2.2.1). Copyright c Shu Gu,

18 Chapter 3 Preliminaries In this chapter we will first give a brief introduction to the homogenization theory of Stokes systems, including the definition of correctors and the homogenization theorem. Then we will formally derive the asymptotic expansion of Stokes system, showing the intuition behind the definition of correctors and effective matrices. Then we prove a compactness theorem for a sequence of Stokes systems with periodic coefficient matrices satisfying the ellipticity condition (1.0.3) with the same µ. At last, we will describe the homogenization of Stokes system with Neumann boundary conditions. 3.1 Homogenization Theory of Stokes Systems In this section we will give a review of homogenization theory of Stokes systems with periodic coefficients. We refer the reader to [10, pp.76-81] for a detailed presentation. Let be a bounded Lipschitz domain in R d. For u, v H 1 (; R d ), we define the bilinear form as the following, a ε (u, v) = ( x ) u a αβ β v α ij dx. (3.1.1) ε x j x i For F H 1 (; R d ) and g L 2 (), we say that (u ε, p ε ) H 1 (; R d ) L 2 () is a weak solution of the Stokes system (1.0.2) { Lε (u ε ) + p ε = F, div(u ε ) = g in, if div(u ε ) = g in and for any ϕ C0(; 1 R d ), a ε (u ε, ϕ) p ε div(ϕ) = F, ϕ. The following theorem gives us the existence and uniqueness (up to constants) of weak solution of Stokes system with Dirichlet boundary condition. Theorem Let be a Lipschitz domain in R d. Suppose A satisfies the ellipticity condition (1.0.3). Let F H 1 (; R d ), g L 2 () and h H 1/2 ( ; R d ) satisfy the Dirichlet compatibility condition g 10 h n = 0, (3.1.2)

19 where n is the outward unit normal to. Then there exist a unique u ε H 1 (; R d ) and p ε L 2 () (unique up to constants) such that (u ε, p ε ) is a weak solution of (1.0.2) in and u ε = h on. Moreover, u ε H 1 () + p ε p ε L 2 () C { F H 1 () + h H 1/2 ( ) + g L ()} 2, (3.1.3) where C depends only on d, µ, and. Proof. This theorem is well known and does not use the periodicity condition of A. First, we choose h H 1 (; R d ) such that h = h on and h H 1 () C h H 1/2 ( ). By considering u ε h, we may assume that h = 0. Next, we choose a function U(x) in H 1 0(; R d ) such that div(u) = g in, and U H 1 () C g L 2 (), detailed proof of the existence of such functions can be found in [20]. By considering u ε U, we may further assume that g = 0. Finally, the case h = 0 and g = 0 may be proved by applying the Lax-Milgram Theorem to the bilinear form a ε (u, v) on the Hilbert space This completes the proof. V = {u H 1 0(; R d ) : div(u) = 0 in }. Remark If is C 1,1 and A is a constant matrix, the weak solution (u, p), given by Theorem 3.1.1, is in H 2 (; R d ) H 1 (), provided that F L 2 (; R d ), g H 1 () and h H 3/2 ( ; R d ). Moreover, u H 2 () + p L 2 () C { F L 2 () + g H 1 () + h H 3/2 ( )}, (3.1.4) where C depends only on d, µ, and (see e.g. [24]). Let Y = [0, 1) d. We denote by H 1 per(y ; R d ) the closure in H 1 (Y ; R d ) of C per(y ; R d ), the set of C 1-periodic and R d -valued functions in R d. Let a per (ψ, φ) = a αβ φ α ij (y) ψβ, y j y i Y where φ = (φ α ) and ψ = (ψ α ). By applying the Lax-Milgram Theorem to the bilinear form a per (ψ, φ) on the Hilbert space V per (Y ) = {u H 1 per(y ; R d ) : div(u) = 0 in Y and 11 Y u = 0},

20 it follows that for each 1 j, β d, there exists a unique χ β j V per(y ) such that a per (χ β j, φ) = a per(p β j, φ) for any φ V per(y ) where P β j = P β j (y) = y je β = y j (0,..., 1,..., 0) with 1 in the β th position. As a result, there exist 1-periodic functions (χ β j, πβ j ) H1 loc (Rd ; R d ) L 2 loc (Rd ), which are called the first-order correctors for the Stokes system (1.0.2), such that L 1 (χ β j + P β j ) + πβ j = 0 in Rd div(χ β j ) = 0 in Rd (3.1.5) = 0 and χ β j = 0. π β j Y Y Note that χ β j H 1 (Y ) + π β j L 2 (Y ) C, (3.1.6) where C depends only on d and µ. Let  = (âαβ ij ), where â αβ ij = a per (χ β j + P β j, χα i + P α i ) (3.1.7) The homogenized system for the Stokes system (1.0.2) is given by { L0 (u 0 ) + p 0 = F, div(u 0 ) = g, where L 0 = div(â ) is a second-order elliptic operator with constant coefficients. The constant matrix  is called the homogenized matrix or effective matrix, and satisfies the following two properties. Remark The homogenized matrix  satisfies the ellipticity condition µ ξ 2 â αβ ij ξα i ξ β j µ 1 ξ 2 (3.1.8) for any ξ = (ξ α i ) R d d, where µ 1 depends only on d and µ. The upper bound is a consequence of the estimate χ β j L 2 (Y ) C(d, µ), while the lower bound follows from â αβ ij ξα i ξ β j = a per((χ β j + P β j )ξβ j, (χα i + Pi α )ξi α ) µ (χ α i + Pi α )ξi α 2 Y µ ξ 2. 12

21 Remark Let χ = (χ β j ) denote the matrix of correctors for the system (1.0.2) with A replaced by its adjoint A. Note that by definition, χ β j V per (Y ) and a per(χ β j, φ) = a per(p β j, φ) where a per(ψ, φ) = a per (φ, ψ). It follows that â αβ ij = a per (χ β j + P β j, χα i + P α i ) = a per (χ β j + P β j, P α i ) = a per (χ β j + P β j, χ α i = a per(χ α i + Pi α ) = a per(χ α i + Pi α, χ β j + P β j ) + Pi α, P β j ) = a per(χ α i + Pi α, χ β j + P β j ). (3.1.9) This, in particular, shows that (Â) = Â. Now we are ready to give the following homogenization theorem of Stokes systems with Dirichlet boundary conditions. It shows that the limiting solutions are actually solutions of Stokes system associated with the homogenized operator L 0 with the same Dirichlet boundary condition. Theorem Suppose that A(y) satisfies ellipticity (1.0.3) and periodicity (1.0.4) conditions. Let be a bounded Lipschitz domain. Let (u ε, p ε ) H 1 (; R d ) L 2 () be a weak solution of the following Dirichlet problem of Stokes system L ε (u ε ) + p ε = F in, div(u ε ) = g in, (3.1.10) u ε = h on, where F H 1 (; R d ), g L 2 () and h H 1/2 ( ; R d ) satisfying the Dirichlet compatibility condition (3.1.2). Assume that p ε = 0, then as ε 0, u ε u 0 strongly in L 2 (; R d ), u ε u 0 weakly in H 1 (; R d ), p ε p 0 weakly in L 2 (), A(x/ε) u ε Â u 0 weakly in L 2 (; R d d ). Moreover, p 0 = 0 and (u 0, p 0 ) is the weak solution of the homogenized problem L 0 (u 0 ) + p 0 = F in, div(u 0 ) = g in, (3.1.11) u 0 = h on. Proof. This homogenization theorem of Stokes systems is more or less proved in [10], using Tartar s oscillating testing function method. We therefore omit the details. 13

22 3.2 Asymptotic Expansions In this section we will apply the multi-scale method to the study of Stokes system. As we mentioned earlier, two scales describe the model: the variable x is the macroscopic one, while x/ε describe the microscopic one. Indeed, this suggests looking for a formal asymptotic expansion of solution (u ε, p ε ) in the form: ( u ε (x) = u 0 x, x ) ( + εu 1 x, x ) ( + ε 2 u 2 x, x ) +, ( ε ε ε p ε (x) = p 0 x, x ) ( + εp 1 x, x ) +, ε ε (3.2.1) where the functions u j (x, y), p j (x, y) are defined on Y and 1-periodic in y, for any x. Note that if φ ε (x) = φ(x, y) with y = x/ε, then φ ε (x) = 1 φ (x, x ) + φ (x, x ). x j ε y j ε x j ε Now if we consider the divergence-free Stokes system { Lε (u ε ) + p ε = F, div(u ε ) = 0, (3.2.2) which may now be rewritten as the following [ ] ( (ε 2 L 0 + ε 1 L 1 + ε 0 L 2 )u ε x, x ) + [ ] ( (ε 1 y + x )p ε x, x ) = F, ε ε [ ] ( (ε 1 div y + div x )u ε x, x ) = 0, ε (3.2.3) where the operators L 0, L 1, L 2 are defined by [ L 0 (φ(x, y)) ] ( ) α = a αβ ij y (y) φβ, i y j [ L 1 (φ(x, y)) ] ( ) α = a αβ ij x (y) φβ i y j [ L 2 (φ(x, y)) ] ( ) α = a αβ ij x (y) φβ. i x j y i ( ) a αβ ij (y) φβ, x j We identify the coefficients of the powers ε 2, ε 1, ε 0. This gives the following systems to be solved. As of order O(ε 2 ), we have Of order O(ε 1 ), we have L 0 (u 0 ) = 0, (3.2.4) { L 0 (u 1 ) + y p 0 = L 1 (u 0 ), div y (u 0 ) = 0. (3.2.5) 14

23 And of order O(ε), we obtain { L 0 (u 2 ) + y p 1 = F L 1 (u 1 ) L 2 (u 0 ) x p 0, div y (u 1 ) = div x (u 0 ). (3.2.6) Using the fact that u 0 (x, y) is 1-periodic in y, we may derive from (3.2.4) that u 0 (x, y) is independent of y, i.e., u 0 (x, y) = u 0 (x). (3.2.7) Then (3.2.5) reduces to L 0 (u 1 ) + y p 0 = L 1 (u 0 ). The second condition in (3.2.6) implies [div y (u 1 ) + div x (u 0 )] dy = 0, and since the above integral equals Y div(u 0 ), one has Y div(u 0 ) = 0. (3.2.8) Then the second condition in (3.2.6) is equivalent to div y (u 1 ) = 0; i.e., finally for (u 1, p 0 ), we need to solve the following system (note that L 1 = L 0 ), L 0 (u 1 ) + y p 1 = L 0 (P β j ) uβ 0 x j div y (u 1 ) = 0 (3.2.9) Recalling that the correctors (χ β j, πβ j ) are solution to the cell problem (3.1.5), then the general solution of (3.2.9) is, and u 1 (x, y) = χ β j (y) uβ 0 x j (x) + ũ 1 (x), (3.2.10) p 0 (x, y) = π β j (y) uβ 0 x j (x) + p 0 (x), (3.2.11) where ũ 1 (x) and p 0 (x) are independent of y. We now use the equations (3.2.10) and (3.2.11) in the first equation in (3.2.6) to obtain ( L 0 (u 2 ) ) [ α p 1 + = F α (x) + a αβ ij (y) + aαγ ik y (y) χγβ j α y k π β j (y) 2 u β 0 x α x j p 0 x α ] { } 2 u β 0 + a αβ ij x i x j y (y) ũβ 1 i x j 15

24 The above equation can be solved in (u 2, p 1 ) if we integrate both sides in y over Y, [ ] F α (x) = a αβ ij (y) + aαγ ik (y) χγβ j 2 u β 0 dy + p 0, Y y k x i x j x α where we have used the fact that Y πβ j dy = 0. The above equation is nothing else than where L 0 = div(â ) and  = (âαβ ij [ â αβ ij = Y L 0 (u 0 ) + p 0 = F, ) defined the same as in (3.1.7) that ] a αβ ij (y) + aαγ ik (y) χγβ x k So far by this multi-scale method, if we denote p 0 (x) by p 0 (x) for simplicity, we have formally shown that the homogenization problem of Stokes system (3.2.2) is exactly { L0 (u 0 ) + p 0 = F, (3.2.12) div(u 0 ) = Compactness Theorem We now prove a compactness theorem for a sequence of Stokes systems with coefficient matrices satisfying the same conditions and should be regarded as a compactness property of the Stokes systems with periodic coefficients. Its proof follows the Tartar s oscillating testing function method found in [10] for the proof of Theorem 3.1.5, and also uses the following observation. Proposition Suppose that {φ k } be a sequence of 1-periodic functions with φ k L 2 (Y ) C and ε k 0, then φ k (x/ε k ) φ k 0 weakly in L 2 (), (3.3.1) as k. Proof. Let u k Hper(Y 2 ) such that u k = φ k φ k, in Y. Let U k = u k. Then div(u k ) = φ k Y φ k and U k L 2 (Y ) φ k φ k L 2 (Y ) C. Y Y 16 Y j dy.

25 Then for any ϕ C0(), 1 [ ] φk (x/ε k ) φ k ϕ(x) = εk U k (x/ε k ) ϕ(x) 0, (3.3.2) Y as ε k 0, since U k (x/ε k ) L 2 () C U k L 2 (Y ) C, where we have used the periodicity of U k. We may now conclude that φ k (x/ε k ) φ k 0 weakly in L 2 (), Y as similarly φ k (x/ε) L 2 () C φ k L 2 (Y ) C The proof is now complete. We are now ready to prove our interior compactness theorem of Stokes systems. Theorem Let {A k (y)} be a sequence of 1-periodic matrices satisfies the ellipticity condition (1.0.3) (with the same µ). Let (u k, p k ) H 1 (; R d ) L 2 () be a weak solution of { div(a k (x/ε k ) u k ) + p k = F k, div(u k ) = g k in, where ε k 0, F k H 1 (; R d ) and g k L 2 (). We further assume that as k, F k F 0 strongly in H 1 (; R d ), g k g 0 strongly in L 2 (), u k u 0 weakly in H 1 (; R d ), p k p 0 weakly in L 2 (), Â k A 0, where Âk is the coefficient matrix of the homogenized system for the Stokes system with coefficient matrix A k (x/ε). Then, A k (x/ε k ) u k A 0 u 0 weakly in L 2 (; R d d ), and (u 0, p 0 ) is a weak solution of { div(a 0 u 0 ) + p 0 = F 0, in. (3.3.3) div(u 0 ) = g 0 17

26 Proof. Let A k = (a kαβ ij ) and (ξ k ) α i = a kαβ ij ( x ε k ) u β k x j Note that (ξ k ) α i L 2 () C. It suffices to show that if {ξ k } is a subsequence of {ξ k } and {ξ k } converges weakly to ξ 0 in L 2 (; R d d ), then ξ 0 = A 0 u 0. This would imply that (u 0, p 0 ) is a weak solution of (3.3.3) in. It also implies that the whole sequence {ξ k } converges weakly to A 0 u 0 in L 2 (; R d d ). Without loss of generality we may assume that ξ k ξ 0 weakly in L 2 (; R d d ). Note that ξ k, φ = F k, φ + p k, div(φ) (3.3.4) for all φ H0(; 1 R d ). Fix 1 j, d d and ψ C0(). 1 Let ( ) φ k (x) = P β j (x) + ε kχ k β j (x/ε k ) ψ(x), where χ k β j (and π k β j used in the following) are the correctors for the Stokes systems with coefficient matrix (A k ) (x/ε), introduced in Remark A computation shows that ξ k, φ k = A k (x/ε k ) u k, ( P β j (x) + ε kχ k β j (x/ε k ) ) ψ Since ( div + A k (x/ε k ) u k, ( P β j (x) + ε kχ k β j (x/ε k ) ) ψ = ψ u k, (A k ) (x/ε k ) ( P β j (x) + ε kχ k β j (x/ε k ) ) + A k (x/ε k ) u k, ( P β j (x) + ε kχ k β j (x/ε k ) ) ψ = (ψu k ), (A k ) (x/ε k ) ( P β j (x) + ε kχ k β j (x/ε k ) ) ( ψ)u k, (A k ) (x/ε k ) ( P β j (x) + ε kχ k β j (x/ε k ) ) + ξ k, ( P β j (x) + ε kχ k β j (x/ε k ) ) ψ. (A k ) (x/ε k ) [ P β j (x) + ε kχ k β j (x/ε k ) ]) [ ] = π k β j (x/ε k ) in R d, (3.3.5) it follows that the first term in the right hand side of (3.3.5) equals π k β j (x/ε), div(ψu k ) = π k β j (x/ε) π k β j, div(ψu k ). Using the fact that Y div(ψu k ) = ψ u k + ψg k ψ u 0 + ψg 0 strongly in L 2 () 18

27 and by Proposition 3.3.1, π k β j (x/ε) π k β j 0 weakly in L 2 (), Y we see that first term in the right hand side of (3.3.5) goes to zero. In view of the estimate ε k χ k β j (x/ε k ) L 2 () Cε k χ k β j L 2 (Y ) C ε k, it is easy to see that for the third term in the right hand side of (3.3.5) goes to ξ 0, P β j ψ. To handle the second term in the right hand side of (3.3.5), we note that again by Proposition 3.3.1, P α i (A k ) (x/ε k ) ( P β j (x) + ε kχ k β j (x/ε k ) ) converges weakly in L 2 () to Pi α (A k ) (y) ( P β j + χk β lim k Y j (y) ) dy = lim â kβα ji = a 0βα ji, k where Âk = (â kαβ ij ), A 0 = (a 0αβ ij ), and we have used the definition of matrices of effective coefficients as well as the assumption that Âk A 0. This, together with the fact that u k u 0 strongly in L 2 (; R d ), shows that the second term in the right hand side of (3.3.5) goes to a 0βα ji where we have used integration by parts. k, ξ k, φ k ξ 0, P β j ψ u α 0 = a 0βα ji ψ uα 0, x i x i To summarize, we have proved that as ψ + a0βα ji ψ uα 0 x i. (3.3.6) Finally, since φ k P β j ψ weakly in H1 0(; R d ) and F k F 0 strongly in H 1 (; R d ), we have F k, φ k F 0, P β j ψ. Also, since div(χβ j ) = 0 in Rd, p k, div(φ k ) = p k, div(p β j ψ) + p k, ε k χ k β j (x/ε) ψ p 0, div(p β j ψ). Thus, the right hand side of (3.3.4) converges to F 0, P β j ψ + p 0, div(p β j ψ) = ξ 0, (P β j ψ) = ξ 0, P β j ψ + ξ 0, ψ P β j, where the first equality follows by taking the limit in (3.3.4) with φ = P β j ψ. In view of (3.3.6) we obtain a 0βα ji ψ uα 0 = ξ 0, ψ P β j x. i Since ψ C0() 1 is arbitrary, this gives (ξ 0 ) β j = a0βα u α 0 ji x i, i.e., ξ 0 = A 0 u 0. The proof is complete. 19

28 3.4 Homogenization for Neumann problems The homogenization theory can be extended to Neumann problem of Stokes systems, following by an analogous argument as in Section 3.1 for Dirichlet boundary value problems. Here we state the main results. The following theorem gives us the existence and uniqueness of weak solution for Neumann problem of Stokes systems. Theorem Let be a bounded Lipschitz domain in R d. Suppose A(y) satisfies the ellipticity condition (1.0.3). Let F H 1 (; R d ), g L 2 () and f H 1/2 ( ; R d ) satisfy the following Neumann compatibility condition F + h = 0. (3.4.1) Then there exist a unique (u ε, p ε ) H 1 (; R d ) L 2 () ( unique in the sense of up to constants), such that (u ε, p ε ) is a weak solution of (1.0.2) and uε ν ε p ε n = h on, where uε ν ε is defined in (2.1.1) and n is the outward unit normal. Moreover, u ε H 1 () + p ε p ε L 2 () C { F H 1 () + g L 2 () + h H ( )}, (3.4.2) 1/2 where C depends only on d, µ, and. Proof. The existence and uniqueness of weak solutions for Neumann problem of Stokes system can be proved again by applying the Lax-Milgram Theorem. skip the details here. Remark If is C 1,1 and A is a constant matrix, the weak solution (u, p), given by Theorem 3.4.1, is in H 2 (; R d ) H 1 (), provided that F L 2 (; R d ), g H 1 () and h H 1/2 ( ; R d ). Moreover, u H 2 () + p L 2 () C { F L 2 () + g H 1 () + h H 1/2 ( ) }, (3.4.3) where C depends only on d, µ, and (see e.g. [24]). The following homogenization theorem of Stokes systems with Neumann boundary condition also shows that the limiting solutions are solutions of Stokes system associated with effective coefficients with the same Neumann boundary condition. Theorem Suppose A(y) satisfies ellipticity condition (1.0.3) and periodicity condition (1.0.4). Let be a bounded Lipschitiz domain. Let (u ε, p ε ) H 1 (; R d ) We 20

29 L 2 () to be a weak solution of the following Neumann problem of Stokes system L ε (u ε ) + p ε = F in, div(u ε ) = g in, (3.4.4) u ε p ε n = h on, ν ε where F H 1 (; R d ), g L 2 () and h H 1/2 ( ; R d ) satisfying the Neumann compatibility condition (3.4.1). Assume that u ε = p ε = 0, then as ε 0, u ε u 0 strongly in L 2 (; R d ), u ε u 0 weakly in H 1 (; R d ), Moreover, problem p ε p 0 weakly in L 2 (), A(x/ε) u ε Â u 0 weakly in L 2 (; R d d ). u 0 = p 0 = 0 and (u 0, p 0 ) is the weak solution of the homogenized L 0 (u 0 ) + p 0 = F in, div(u 0 ) = g in, u 0 ν 0 p 0 n = h on. (3.4.5) Proof. The proof will use the same approach as in the proof of Theorem Let ξ ε = A(x/ε) u ε. Note that ξ ε L 2 () C. We say (u ε, p ε ) is a weak solution of Neumann problem (3.4.4), if ξ ε, φ = F, φ + p ε, div(φ) + h, φ H 1/2 ( ;R d ) H 1/2 ( ;R d ), (3.4.6) for any φ H 1 (; R d ). By Theorem 3.4.1, we can extract a subsequence, still denoted by {u ε }, {p ε } and {ξ ε } such that u ε u 0 weakly in H 1 (; R d ); p ε p 0 weakly in L 2 (); ξ ε ξ 0 weakly in L 2 (; R d d ). Similarly, as in (3.3.5) of the proof of Theorem 3.3.2, we choose ( ) φ(x) = P β j (x) + εχ β j (x/ε) ψ(x), where 1 j, d d and ψ(x) C 1 0() are fixed. Similarly, ξ ε, φ = (ψu ε ), A (x/ε) ( P β j ( ψ)u ε, A (x/ε) ( P β j + ξ ε, ( P β j (x) + εχ β j (x/ε)) (x) + εχ β j (x/ε)) ψ. (x) + εχ β j (x/ε)) (3.4.7) 21

30 Since ( div A (x/ε) [ P β j (x) + εχ β j (x/ε)]) = π β j (x/ε) in R d, it follows that the first term in the right hand side of (3.4.7) equals π β j (x/ε), div(ψu ε ) = π β j (x/ε) π β j, div(ψu ε ). Using the fact that Y div(ψu ε ) = ψ u ε + ψg ψ u 0 + ψg strongly in L 2 () and by Proposition 3.3.1, π β j (x/ε) π β j 0 weakly in L 2 (), Y we see that first term in the right hand side of (3.4.7) goes to zero. In view of the estimate εχ β j (x/ε) L 2 () Cε χ β j L 2 (Y ) C ε, it is easy to see that for the third term in the right hand side of (3.4.7) goes to ξ 0, P β j ψ. To handle the second term in the right hand side of (3.4.7), we note that again by Proposition 3.3.1, Y P α i A (x/ε) ( P β j (x) + εχ β j (x/ε)) converges weakly in L 2 () to Pi α A (y) ( P β j + χ β j (y)) dy = â βα ji, where we have used the definition of matrices of effective coefficients. This, together with the fact that u ε u 0 strongly in L 2 (; R d ), shows that the second term in the right hand side of (3.4.7) goes to â βα ji ψ u α 0 = â βα ji ψ uα 0, x i x i where we have used integration by parts. To summarize, we have proved that, ξ ε, φ ξ 0, P β j ψ + âβα ji ψ uα 0. (3.4.8) x i 22

31 Finally, since φ P β j ψ weakly in H1 0(; R d ), we have F, φ F, P β j ψ. Also, since div(χ β j ) = 0 in Rd, p ε, div(φ) = p ε, div(p β j ψ) + p ε, εχ β j (x/ε) ψ p 0, div(p β j ψ). Thus, the right hand side of (3.4.6) converges to F, P β j ψ + p 0, div(p β j ψ) = ξ 0, (P β j ψ) = ξ 0, P β j ψ + ξ 0, ψ P β j, where the first equality follows by taking the limit in (3.4.6) with φ = P β j ψ. In view of (3.4.8) we obtain â βα ji ψ uα 0 = ξ 0, ψ P β j x. i Since ψ C0() 1 is arbitrary, this gives (ξ 0 ) β j = âβα u α 0 ji x i, i.e. we have prove that Taking limit in (3.4.6), we see that ξ 0 = Â u 0. Â u 0, φ = F, φ + p 0, div(φ) + h, φ H 1/2 ( ;R d ) H 1/2 ( ;R d ), this implies that (u 0, p 0 ) is the unique solution of Neumann problem L 0 (u 0 ) + p 0 = F in, div(u 0 ) = g in, u 0 ν 0 p 0 n = h on. satisfying u 0 = p 0 = 0. As a result we conclude that the whole sequence u ε u 0 weakly in H 1 (; R d ) and p ε p 0 weakly in L 2 (). Copyright c Shu Gu,

32 Chapter 4 Convergence Rates of Dirichlet Problems in Homogenization of Stokes Systems In this chapter we study the convergence rates in L 2 and H 1 of Dirichlet problems for Stokes systems with rapidly oscillating periodic coefficient, without any regularity assumptions on the coefficients. 4.1 Introduction More precisely, we consider the following Dirichlet problem for Stokes systems L ε (u ε ) + p ε = F in, div u ε = g in, u ε = h on, with the Dirichlet compatibility condition g h n = 0. By homogenization theorem (Theorem 3.1.5), we have shown that and p ε u ε u 0 weakly in H 1 (; R d ), p ε p 0 p 0 weakly in L 2 (), where (u 0, p 0 ) H 1 (; R d ) L 2 () is the weak solution of the homogenized problem L 0 (u 0 ) + p 0 = F in, div(u 0 ) = g in, u 0 = h on. The main purpose of this chapter is to investigate the rate of convergence of u ε u 0 L 2 () as ε 0, which is stated in the following theorem. Theorem Let be a bounded C 1,1 domain. Suppose that A satisfies the ellipticity condition (1.0.3) and periodicity condition (1.0.4). Given g H 1 () and f H 3/2 ( ; R d ) satisfying the Dirichlet compatibility condition (3.1.2), for 24

33 F L 2 (; R d ), let (u ε, p ε ), (u 0, p 0 ) be weak solutions of Dirichlet problems (3.1.10), (3.1.11), respectively. Then u ε u 0 L 2 () Cε u 0 H 2 (), (4.1.1) where the constant C depends only on d, µ, and. Theorem gives the optimal O(ε) convergence rate for the inverses of the Stokes operators in L 2 operator norm. Indeed, let T ε : F L 2 σ() u ε, where L 2 σ() = { F L 2 (; R d ) : div(f ) = 0 in }, and u ε denotes the solution of (3.1.10) with F L 2 σ(; R d ) and g = 0, f = 0. Then it follows from (4.1.1) and the estimate u 0 H 2 () C F L 2 () that where T 0 : F L 2 σ() u 0. T ε T 0 L 2 σ () L 2 σ () Cε, In this chapter we also obtain O( ε) rates for a two-scale expansion of (u ε, p ε ) in H 1 L 2. Let (χ, π) denote the correctors associated with A, defined by (3.1.5), and S ε the Steklov smoothing operator defined by (2.2.1). Theorem Let be a bounded C 1,1 domain. Suppose that A satisfies ellipticity (1.0.3) and periodicity (1.0.4) conditions. Let (u ε, p ε ) and (u 0, p 0 ) be the same as in Theorem Then u ε u 0 εχ ε S ε ( ũ 0 ) H 1 () C ε u 0 H 2 (), (4.1.2) where χ ε (x) = χ(x/ε) and ũ 0 is the extension of u 0 defined as in (4.2.1). Moreover, if p ε = p 0 = 0, then { } p ε p 0 π ε S ε ( ũ 0 ) π ε S ε ( ũ 0 ) L 2 () C ε u 0 H 2 (), (4.1.3) where π ε (x) = π(x/ε). The constants C in (4.1.2) and (4.1.3) depend only on d, µ, and. For the known results, as we mentioned earlier, consider the Dirichlet problem for the scalar elliptic equation L ε (u ε ) = F in a Lipschitz domain with u ε = f on. It is well known that u ε u 0 L 2 () Cε { 2 u 0 L 2 () + u 0 L ( )}. (4.1.4) 25

34 To see (4.1.4), one considers the difference between u ε and its first order approximation u 0 + εχ ε u 0 and let v ε = u ε u 0 εχ ε u 0. (4.1.5) To correct the boundary data, one further introduces a function w ε, where w ε is the solution to the Dirichlet problem: L ε (w ε ) = 0 in and w ε = εχ ε u 0 on. Using energy estimates, one may show that v ε w ε H 1 0 () Cε 2 u 0 L 2 (). The estimate (4.1.4) follows from this and the estimate w ε L () Cε u 0 L ( ), which is obtained by the maximum principle (see e.g. [31]). More recently, Griso [26, 27] was able to establish the much sharper estimate (4.1.1), using the method of periodic unfolding. We mention that in the case of scalar elliptic equations with bounded measurable coefficients, one may also prove (4.1.1) by using the so-called Dirichlet corrector. In fact, it was shown in [33] that u ε u 0 { Φ ε x } u 0 H 1 0 () Cε u 0 H 2 (), (4.1.6) where Φ ε (x) is the solution of L ε (Φ ε ) = 0 in with Φ ε = x on. In the case of elliptic systems, the estimates (4.1.6) and thus (4.1.1) continue to hold under the additional assumption that A is Hölder continuous. continuous and symmetric, it was proved in [32] that Moreover, if A is Hölder v ε H 1/2 () Cε u 0 H 2 (). (4.1.7) The approaches used in [32, 33] rely on the uniform regularity estimates established in [3, 37] and do not apply to operators with bounded measurable coefficients. Recently, by using the Steklov smoothing operator, T. A. Suslina [55, 56] was able to establish the O(ε) estimate (4.1.1) in L 2 for a boarder class of elliptic operators, which, in particular, contains the elliptic systems L ε in divergence form with coefficients satisfying the ellipticity condition a αβ ij ξα i ξ β j µ ξ 2 for any ξ = ( ) ξi α R m d. Since the correctors χ may not be bounded in the case of non-smooth coefficients, the idea is to consider the two-scale expansion v ε = u ε u 0 εχ ε S ε ( ũ 0 ), (4.1.8) where S ε is a smoothing operator at scale ε defined in (2.2.1) and ũ 0 an extension of u 0 to R d (also see [45,46,59] and their references on the use of S ε in homogenization). This reduces the problem to the control of the L 2 norm of w ε, where w ε is the solution 26

35 to the Dirichlet problem: L ε (w ε ) = 0 in and w ε = εχ ε S ε (ũ 0 ) on. Next, one considers h ε = w ε εχ ε θ ε S ε ( ũ 0 ), where θ ε is a cutoff function supported in an ε neighborhood of. Note that h ε = 0 on and L ε (h ε ) is supported in an ε neighborhood of. This allows one to approximate h ε in the L 2 norm by h 0, using an O( ε) estimate in H 1 and a duality argument, where L 0 (h 0 ) = L ε (h ε ) in and h 0 = 0 on. Finally, one estimates the L 2 norm of h 0 by another duality argument. In this chapter we extend the approach of Suslina to the case of Stokes systems, which do not fit the standard framework of second-order elliptic systems in divergence form. As expected in the study of Stokes or Navies-Stokes systems, the main difficulty is caused by the pressure term p ε. By carefully analyzing the systems for the correctors (χ, π) as well as their dual (φ αβ kij, qβ ij ), we are able to establish the O( ε) error estimates, given in Theorem 4.1.2, for the two-scale expansions of (u ε, p ε ) in H 1 L 2. This allows us to use the idea of boundary cutoff and duality argument in a manner similar to that in [55]. 4.2 Convergence rates for u ε in H 1 From now on we will assume that is a bounded domain with boundary of class C 1,1, F L 2 (; R d ), g H 1 (), and h H 3/2 ( ; R d ). We fix a linear continuous extension operator E : H 2 (; R d ) H 2 (R d ; R d ), and let ũ 0 = E u 0, (4.2.1) so that ũ 0 = u 0 in and ũ 0 H 2 (R d ) C u 0 H 2 (), (4.2.2) where C depends on. We introduce a first order approximation of u ε, v ε = u 0 + εχ ε S ε ( ũ 0 ). Let (w ε, τ ε ) H 1 (; R d ) L 2 () be a weak solution of L ε (w ε ) + τ ε = 0 in, div(w ε ) = ε div ( ) χ ε S ε ũ 0 in, w ε = εχ ε S ε ( ũ 0 ) on. (4.2.3) 27

36 We will use w ε to approximate the difference between u ε and its first order approximation v ε. To this end, for 1 i, j, α, β d, we let Note that b αβ ij b αβ ij (y) = aαβ ij (y) + aαγ ik (y) ( ) χ γβ j â αβ ij y. (4.2.4) k is 1-periodic. By the definition of χ and and, for each 1 α, β, j d, y i ( b αβ ij (y)) = Y y i ( a αβ ij (y)) + y i = y i ( a αβ ij (y)) y i = y α (π β j ). b αβ ij (y) dy = 0. ( a αγ j Â, bαβ ij ) ik (y) χγβ y k ( ) γβ a αγ j ik (y) P y k L 2 (Y ) satisfies + y α (π β j ) Lemma There exist Φ αβ kij H1 per(y ) and q β ij H1 per(y ) such that Moreover, (4.2.5) b αβ ij = (Φ αβ kij y ) + (q β ij ) and Φαβ kij k y = Φαβ ikj. (4.2.6) α where C depends only on d and µ. Φ αβ kij L 2 (Y ) + q β ij L 2 (Y ) C, (4.2.7) Proof. Fix 1 i, j, β d. There exist f β αβ ij = (fij ) H2 per(y ; R d ) and q β ij H1 per(y ) satisfying the following Stokes system, f β ij + qβ ij = bβ ij in Y, div(f β ij ) = 0 in Y, (4.2.8) f β ij dy = 0, where b β ij = (bαβ ij ). We now define Y Φ αβ kij (y) = (f αβ ij y k ) y i (f αβ kj ). 28