Corrector estimates for the homogenization of a locally-periodic medium with areas of low and high diffusivity
11 Corrector estimates for the homogenizationof a locally-periodic medium with areas oflow and high diffusivity
A. MUNTEAN , and T. L. VAN NOORDEN CASA - Centre for Analysis, Scientific computing and Applications,Department of Mathematics and Computer Science,Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands Institute for Complex Molecular Systems (ICMS), Technische Universiteit Eindhoven, P.O. Box 513,5600 MB Eindhoven, The Netherlands Chair of Applied Mathematics 1, Department of Mathematics, University of Erlangen-N¨urnberg,Martensstraße 3, 91058 Erlangen, Germany
We prove an upper bound for the convergence rate of the homogenization limit (cid:15) → (cid:15) is a suitable scale parameter. On this way, wejustify the formal homogenization asymptotics obtained by us earlier by proving an upperbound for the convergence rate (a corrector estimate). The main ingredients of the proofof the corrector estimate include integral estimates for rapidly oscillating functions withprescribed average, properties of the macroscopic reconstruction operators, energy boundsand extra two-scale regularity estimates. The whole procedure essentially relies on a goodunderstanding of the analysis of the limit two-scale problem. Keywords : Corrector estimates, transmission condition, homogenization, micro-macrotransport, reaction-diffusion system in heterogeneous materials
MSC 2010 : 35B27; 35K57; 76S05
We study the averaging of a system of reaction-diffusion equations with linear trans-mission condition posed in a class of highly heterogeneous media including areas of lowand high diffusivity. Our aim is twofold: on one hand, we wish to justify rigorously theformal asymptotics expansions performed in [30], while on the other hand we wish tounderstand the error caused by replacing a heterogeneous solution by an approximation(averaged) estimate. To this end, we prove an upper bound for the convergence rate ofthe limit procedure (cid:15) →
0, where (cid:15) is a suitable scale parameter (see section 2.1 forthe definition of (cid:15) ). The materials science scenario we have in view is motivated by avery practical problem: the sulfate corrosion of concrete. We refer the reader to [3] for anice and detailed description of the physico-chemical scenario and to [17] for the formal(two-scale) averaging of locally-periodic distributions of unsaturated pores attacked bysulphuric acid. The reference [18] contains the rigorous proof of the limiting procedure (cid:15) → a r X i v : . [ m a t h - ph ] A p r Adrian Muntean and Tycho van Noorden main working tools involve the two-scale convergence concept in the sense of Nguetsengand Allaire combined with a periodic unfolding of the oscillatory boundary. Here, weuse an energy-type method.The framework that we tackle in this context is essentially a deterministic one. Weassume that a distribution of the locally-periodic array of microstructures is known apriori . We refer the reader to [6] for related discussion of continuum mechanical descrip-tions of balance laws in continua with microstructure. At the mathematical level, wesucceed to combine successfully the philosophy of getting correctors as explained in theanalysis by Chechkin and Piatnitski [7] with the intimate two-scale structure of our sys-tem; see also [8, 9] for related settings where similar averaging strategies are used. Formethodological hints on how to get corrector estimates, we refer the reader to the analy-sis shown in [14] for the case of a reaction-diffusion phase field-like system posed in fixeddomains. For a higher level of discussion, see the monograph [10] which is a nicely writ-ten introductory text on homogenization methods and applications. It is worth notingthat the periodic homogenization of linear transmission problems is a well-understoodsubject (cf. [1, 15, 19, 28], e.g.), however much less is known if one steps away from theperiodic setting even if one stays within the deterministic case. If the microstructures aredistributed in a suitable random fashion, then concepts like random fields (see [4] or [5]and references cited therein; [21] and intimate connection with the stochastic geometryof the perforations) can turn to be helpful to getting averaged equations (eventually alsocapturing new memory terms), but corrector estimates seem to be very hard to get; cf.,e.g., [2]. Disordered media for which the stationarity and/or ergodicity assumptions onthe random measure do not hold or situations where at a given time scale length scalesare non-separated are typical situations that cannot be averaged with existing techniques(so it makes no sense to search here for correctors).This paper is organized as follows: In Section 2 we introduce the necessary notation, themicroscale and the two-scale limit model. Section 3 contains the main result of our paper– Theorem 3.1. In Section 4 we introduce the technical assumptions needed to obtain thecorrectors. At this point, we also collect the known well-posedness and regularity resultsfor both the microscopic model and the two-scale limit model. The rest of the paperconsists of the proof of the convergence rate and is the subject of Section 5. In this section we introduce our notation, the microscale model and the two-scale limitproblem. We start off with the definition of the locally periodic heterogeneous medium. For an introduction to two-scale convergence, see [23]. Here, we deviate from the purely periodic setting. It is worth noting also that the assump-tion of statistically homogeneity of distribution of microstructures, which is a crucial restrictionof stochastic homogenization, does not cover all possible configurations of locally-periodic ge-ometries. Also, as we formulate the working framework, we cannot treat neither randomly placedmicrostructures nor stochastic distributions of microgeometries. Consequently, the exact con-nection between these two averaging techniques is hard to make precise. orrector estimates of a locally-periodic scenario ! Figure 1. Schematic representation of a locally-periodic heterogeneous medium. The cen-ters of the gray circles are on a grid with width (cid:15) . These circles represent the areas oflow diffusivity and their radii may vary.We consider a heterogenous medium consisting of areas of high and low diffusivity.The medium is in the present paper represented by a two dimensional domain. For thedefinition of the locally-periodic medium with inclusions, we follow in main lines [7]. Wedenote the two dimensional bounded domain by Ω ⊂ R , with boundary Γ. Denote J (cid:15) := { j ∈ Z | dist( (cid:15)j, Γ) ≥ (cid:15) √ } , U := { y ∈ R | − / ≤ y i ≤ / i = 1 , } ) . A convenient way to parameterize the interface Γ (cid:15) between the high and low diffusivityareas, is to use a level set function, which we denote by S (cid:15) ( x ): x ∈ Γ (cid:15) ⇔ S (cid:15) ( x ) = 0 , Since we allow the size and shape of the perforations to vary with the macroscopic variable x , we use the following characterization of S (cid:15) : S (cid:15) ( x ) := S ( x, x/(cid:15) ) , (2.1)where S : Ω × U → R is 1-periodic in its second variable, and where S is independent of (cid:15) . We assume that S ( x, < const. < S ( x, y ) | y ∈ ∂U > const. > x ∈ Ω, sothat the areas of low diffusivity in each unit cell do not touch each other. We set Q (cid:15)j := { x ∈ (cid:15) ( U + j ) | S ( x, x/(cid:15) ) < } , Adrian Muntean and Tycho van Noorden and introduce the area of low diffusivity Ω (cid:15)l as follows:Ω (cid:15)l := (cid:91) j ∈ J (cid:15) Q (cid:15)j , and the area of high diffusivity Ω (cid:15)h as:Ω (cid:15)h := Ω \ Ω (cid:15)l . We call a medium of which the geometry is specified with a level set function of the typethat is given in (2.1) a locally periodic medium [7]. In Fig. 1 a schematic picture is givenof how such a medium might look like for a given (cid:15) >
0. Note that by construction thearea of high diffusivity Ω (cid:15)h is connected and the area of low diffusivity Ω (cid:15)l is disconnected.The interface between high and low diffusivity areas Γ (cid:15) is now given by Γ (cid:15) = ∂ Ω (cid:15)l andthe boundary of Ω (cid:15)h is given by ∂ Ω (cid:15)h = Γ (cid:15) ∪ Γ.Furthermore, we use the notation Ω (cid:15) := Ω \ (cid:83) j ∈ J (cid:15) ( (cid:15) ( U + j )), and we introduce for lateruse the smooth cut-off function χ (cid:15) ( x ) that satisfies 0 ≤ χ (cid:15) ( x ) ≤ χ (cid:15) ( x ) = 0 if x ∈ Ω (cid:15) and χ (cid:15) ( x ) = 1 if dist( x, Ω (cid:15) ) ≥ dist(Γ (cid:15) , Ω (cid:15) ). Moreover, (cid:15) |∇ χ (cid:15) | ≤ C and (cid:15) | ∆ χ (cid:15) | ≤ C , with C independent of (cid:15) , and also (cid:107) − χ (cid:15) (cid:107) L (Ω) ≤ (cid:15) / C, (cid:107)∇ χ (cid:15) (cid:107) L (Ω) ≤ (cid:15) − / C, (2.2) (cid:107) ∆ χ (cid:15) (cid:107) L (Ω) ≤ (cid:15) − / C, with C again independent of (cid:15) (see e.g. [7, 14]).In addition, we need to expand the normal ν (cid:15) to Γ (cid:15) in a power series in (cid:15) . This can bedone in terms of the level set function S (cid:15) , which we assume to be sufficiently regular sothat all the following computations make sense (see assumption (B1) in section 4): ν (cid:15) = ∇ S (cid:15) ( x ) |∇ S (cid:15) ( x ) | = ∇ S ( x, x/(cid:15) ) |∇ S ( x, x/(cid:15) ) | = ∇ x S + (cid:15) ∇ y S |∇ x S + (cid:15) ∇ y S | at x ∈ Γ (cid:15) . (2.3)First we expand |∇ S (cid:15) | . Using the Taylor series of the square-root function, we obtain |∇ S (cid:15) | = 1 (cid:15) |∇ y S | + O ( (cid:15) ) . (2.4)In the same fashion, we get ν (cid:15) = ν + (cid:15)ν + O ( (cid:15) ) , where ν := ∇ y S |∇ y S | and ν := ∇ x S |∇ y S | − ( ∇ x S · ∇ y S ) |∇ y S | ∇ y S |∇ y S | . (2.5)If we introduce the normalized tangential vector τ , with τ ⊥ ν , we can rewrite ν as ν = τ τ · ( ∇ x S ) |∇ y S | . (2.6) orrector estimates of a locally-periodic scenario ν (cid:15) = (cid:15) − ( ν (cid:15) − ν ) = ν + O ( (cid:15) ) . We focus on the following microscopic model (cid:110) ∂ t u (cid:15) = ∇ · ( D h ∇ u (cid:15) − q (cid:15) u (cid:15) ) in Ω (cid:15)h , (2.7) (cid:110) ∂ t v (cid:15) = (cid:15) ∇ · ( D l ∇ v (cid:15) ) in Ω (cid:15)l , (2.8) (cid:40) ν (cid:15) · ( D h ∇ u (cid:15) ) = (cid:15) ν (cid:15) · ( D l ∇ v (cid:15) ) u (cid:15) = v (cid:15) on Γ (cid:15) , (2.9) (cid:110) u (cid:15) ( x, t ) = u b ( x, t ) on Γ , (2.10) (cid:40) u (cid:15) ( x,
0) = u I(cid:15) ( x ) in Ω (cid:15)h ,v (cid:15) ( x,
0) = v I(cid:15) ( x ) in Ω (cid:15)l . (2.11)Here we denoted the tracer concentration in the high diffusivity area by u (cid:15) , the con-centration in the low diffusivity area by v (cid:15) , and the velocity of the fluid phase by q (cid:15) .Furthermore, D h denotes the diffusion coefficient in the high diffusivity region, D l thediffusion coefficient in the low diffusivity regions, ν (cid:15) denotes the unit normal to theboundary Γ (cid:15) ( t ), where u b denotes the Dirichlet boundary data for the concentration u (cid:15) , and where u I(cid:15) and v I(cid:15) are the initial values for the active concentrations u (cid:15) and v (cid:15) ,respectively.In order to formulate the upscaled equations and obtain closed-formulae for the effec-tive transport coefficients, we use the notations B ( x ) := { y ∈ U : S ( x, y ) < } , (2.12)and Y ( x ) := U − B ( x ) , (2.13)and we define the following x -dependent cell problem: ∆ y M j ( x, y ) = 0 for all x ∈ Ω , y ∈ Y ( x ) ,ν · ∇ y M j ( x, y ) = − ν · e j for all x ∈ Ω , y ∈ ∂B ( x ) ,M j ( x, y ) y -periodic , (2.14)for j ∈ { , } . To ensure the uniqueness of weak solutions to this cell problem, suitableconditions on the spatial averages of the cell functions need to be added.The solution to this cell problems allows us to write the results of the formal homog-enization procedure in the form of the following distributed-microstructure (two-scale) See, e.g. [11, 22], for the role of cell problems in formulating averaged equations.
Adrian Muntean and Tycho van Noorden model ∂ t v ( x, y, t ) = D l ∆ y v ( x, y, t ) for y ∈ B ( x ) , x ∈ Ω ,∂ t (cid:16) θ ( x ) u + (cid:82) | y | 0) = u I ( x ) for x ∈ Ω ,v ( x, y, 0) = v I ( x, y ) for y ∈ B ( x ) , x ∈ Ω . (2.17)where the porosity θ ( x ) of the medium is given by θ ( x ) := | Y ( x ) | = 1 − | B ( x ) | , and where the effective diffusivity tensor D ( x ) is defined by D ( x ) := D h (cid:90) Y ( x ) ( I + ∇ y M ( x, y )) dy, with M = ( M , M ). Recall that ( u (cid:15) , v (cid:15) ) is the solution vector for the micro problem and ( u , v ) is the solutionvector for the two-scale problem. We introduce now the macroscopic reconstructions u (cid:15) , v (cid:15) , u (cid:15) , which are defined as follows u (cid:15) ( x, t ) := u ( x, t ) for all x ∈ Ω (cid:15)h , (3.1) v (cid:15) ( x, t ) := v ( x, x/(cid:15), t ) for all x ∈ Ω (cid:15)l , (3.2) u (cid:15) ( x, t ) := u (cid:15) ( x, t ) + (cid:15)M ( x, x/(cid:15) ) ∇ u (cid:15) ( x, t ) for all x ∈ Ω (cid:15)h , (3.3)In the same spirit, we introduce the reconstructed flow velocity q (cid:15) ( x ) = q ( x ) for all x ∈ Ω (cid:15)h and the corresponding reconstructions u (cid:15) I , v (cid:15) I for the macroscopic initial data u I and v I , respectively.The main result of our paper is stated in the next Theorem. The applicability of thisresult confines to our working assumptions (A1), (A2), (B1), (B2), and (B3) that weintroduce in Section 4. Theorem 3.1 Assume (A1), (A2), (B1), (B2), and (B3). Then the following conver-gence rate holds || u (cid:15) − u (cid:15) || L ∞ ( I,L (Ω (cid:15)h )) + || v (cid:15) − v (cid:15) || L ∞ ( I,L (Ω (cid:15)l )) + || u (cid:15) − u (cid:15) || L ∞ ( I,H (Ω (cid:15)h )) + (cid:15) || v (cid:15) − v (cid:15) || L ∞ ( I,H (Ω (cid:15)l )) ≤ c √ (cid:15), (3.4) where I = (0 , T ] , and where the constant c is independent of (cid:15) . We borrowed this terminology from [14]. Note however that the concept of reconstructionoperators appears in various other frameworks like for the heterogeneous multiscale method [13]. orrector estimates of a locally-periodic scenario Functional setting As space of test functions for the microscopic problem, we take the Sobolev space H (Ω (cid:15) ; ∂ Ω) := (cid:8) ϕ ∈ H (Ω (cid:15) ) with ϕ = 0 at ∂ Ω (cid:9) . Since the formulation of the upscaled problem involves two distinct spatial variables x ∈ Ω and y ∈ B ( x ) (with B ( x ) ⊂ Ω), we need to introduce the following spaces: V := H (Ω) , (4.1) V := L (Ω; H ( B ( x ))) , (4.2) H := L θ (Ω) , (4.3) H := L (Ω; L ( B ( x ))) . (4.4)The spaces H and V make sense, for instance, as indicated in [26].4.1.2 Assumptions for the microscopic model Assumption (A1) : We assume the following restrictions on data and parameters: Take D h , D l ∈ ]0 , ∞ [, u I(cid:15) ∈ H (Ω (cid:15)h ), v I(cid:15) ∈ H (Ω (cid:15)l ), and u b ∈ H ( I ; H (Γ)). Furthermore, weassume || u I(cid:15) − u (cid:15) I || L ∞ (Ω (cid:15)h ) = O ( (cid:15) θ ) with θ ≥ || v I(cid:15) − v (cid:15) I || L ∞ (Ω (cid:15)l ) = O ( (cid:15) θ ) with θ ≥ Assumption (A2) : q (cid:15) ∈ H (Ω (cid:15)h ; R d ) ∩ L ∞ (Ω (cid:15)h ; R d ) with ∇ · q (cid:15) = 0 a.e. in Ω (cid:15)h . Addition-ally, we assume that || q (cid:15) − θ ¯ q || L (Ω (cid:15)h ; R d ) = O ( (cid:15) θ ) with θ ≥ . (4.5) Remark 4.1 If one wishes to replace q (cid:15) with the stationary Stokes or Navier-Stokesequations, then a few additional things have to be taken into account. One of the moststriking facts is that the exponent θ arising in (4.5) seems to be restricted to θ = . Consequently, this worsens essentially the convergence rate; see, for instance, [24](Theorem 1) for a discussion of homogenization of the periodic case.4.1.3 Assumptions for the two-scale model Assumption (B1) : The level set function S : Ω × U → R is 1-periodic in its secondvariable and is in C (Ω × U ). Adrian Muntean and Tycho van Noorden Assumption (B2) : We assume the following restrictions on data and parameters: θ, D ∈ L ∞ + (Ω) ∩ H (Ω) , ¯ q ∈ H (Ω; R d ) ∩ L ∞ (Ω; R d ) with ∇ · ¯ q = 0 a.e. in Ω ,u b ∈ L ∞ + (Ω × I ) ∩ H ( I ; H (Γ)) ,∂ t u b ≤ x, t ) ∈ Ω × I,u I ∈ L ∞ + (Ω) ∩ H ∩ H (Ω) ,v I ( x, · ) ∈ L ∞ + ( B ( x )) ∩ H for a.e. x ∈ Ω . Assumption (B3) : H := H (Ω) , (4.6) H := L (Ω; H ( B ( x ))) , (4.7) V := H θ (Ω) , (4.8) V := L (Ω; H ( B ( x ))) . (4.9) Remark 4.2 Following the lines of [26] and [29], Assumption (B1) implies in particularthat the measures | ∂B ( x ) | and | B ( x ) | are bounded away from zero (uniformly in x ).Consequently, the following direct Hilbert integrals (cf. [12] (part II, chapter 2), e.g.) L (Ω; H ( B ( x ))) := { u ∈ L (Ω; L ( B ( x ))) : ∇ y u ∈ L (Ω; L ( B ( x ))) } L (Ω; H ( ∂B ( x ))) := { u : Ω × ∂B ( x ) → R measurable such that (cid:90) Ω || u ( x ) || L ( ∂B ( x )) < ∞} are well-defined separable Hilbert spaces and, additionally, the distributed trace γ : L (Ω; H ( B ( x ))) → L (Ω , L ( ∂B ( x )))given by γu ( x, s ) := ( γ x U ( x ))( s ) , x ∈ Ω , s ∈ ∂B ( x ) , u ∈ L (Ω; H ( B ( x ))) (4.10)is a bounded linear operator. For each fixed x ∈ Ω, the map γ x , which is arising in (4.10),is the standard trace operator from H ( B ( x )) to L ( ∂B ( x )). We refer the reader to [25]for more details on the construction of these spaces and to [27] for the definitions of theirduals as well as for a less regular condition (compared to (B1)) allowing to define thesespaces in the context of a certain class of anisotropic Sobolev spaces.For convenience, we also introduce the evolution triple ( V , H , V ∗ ), where V := { ( φ, ψ ) ∈ V × V | φ ( x ) = ψ ( x, y ) for x ∈ Ω , y ∈ ∂B ( x ) } , (4.11) H := H × H , (4.12)4.1.4 Analysis of microscopic equations Definition 4.3 Assume (A1), (A2) and (B1). The pair ( u (cid:15) , v (cid:15) ), with u (cid:15) = U (cid:15) + u b andwhere ( U (cid:15) , v (cid:15) ) ∈ H (Ω (cid:15)h ; ∂ Ω) × H (Ω (cid:15)l ), is a weak solution of the problem ( P (cid:15) ) if the orrector estimates of a locally-periodic scenario (cid:90) Ω (cid:15)h ∂ t ( U (cid:15) + u b ) φ dx + (cid:90) Ω (cid:15)h ( D h ∇ ( U (cid:15) + u b ) − q (cid:15) ( U (cid:15) + u b )) · ∇ φ dx = − (cid:90) Γ (cid:15) ν (cid:15) · ( (cid:15) D l ∇ v (cid:15) ) φds, (4.13) (cid:90) Ω (cid:15)l ∂ t v (cid:15) ψ dydx + (cid:90) Ω (cid:15)l (cid:15) D l ∇ v (cid:15) · ∇ ψ dx = (cid:90) ∂ Γ (cid:15) ν (cid:15) · ( (cid:15) D l ∇ v (cid:15) ) ψ dsdx, (4.14)for all ( φ, ψ ) ∈ H (Ω (cid:15)h ; ∂ Ω) × H (Ω (cid:15)l ) and t ∈ I . Theorem 4.4 Assume (A1), (A2) and (B1). Problem ( P (cid:15) ) admits a unique global-in-time weak solution in the sense of Definition 4.3. Proof Since we deal here with a linear transmission problem, the proof of the Theoremcan be done with standard techniques (see [16], e.g.).4.1.5 Analysis of two-scale equations This section contains basic results concerning the well-posedness of the two-scale problem,which we reformulate here as:( P ) θ ( x ) ∂ t u − ∇ x · ( D ( x ) ∇ x u − qu ) = − (cid:82) ∂B ( x ) ν · ( D l ∇ y v ) dσ in Ω ,∂ t v − D l ∆ y v = 0 in B ( x ) ,u ( x, t ) = v ( x, y, t ) at ( x, y ) ∈ Ω × ∂B ( x ) ,u ( x, t ) = u b ( x, t ) at x ∈ ∂ Ω ,u ( x, 0) = u I ( x ) in Ω ,v ( x, y, 0) = v I ( x, y ) at ( x, y ) ∈ Ω × B ( x ) . Before starting to discuss the existence and uniqueness of weak solutions to problem( P ), we denote U := u − u b and notice that U = 0 at ∂ Ω. Definition 4.5 Assume (B1) and (B2). The pair ( u, v ), with u = U + u b where ( U, v ) ∈ V , is a weak solution of the problem ( P ) if the following identities hold (cid:90) Ω θ∂ t ( U + u b ) φ dx + (cid:90) Ω ( D ∇ x ( U + u b ) − q ( U + u b )) · ∇ x φ dx = − (cid:90) Ω (cid:90) ∂B ( x ) ν · ( D l ∇ y v ) φ dσdx, (4.15) (cid:90) Ω (cid:90) B ( x ) ∂ t vψ dydx + (cid:90) Ω (cid:90) B ( x ) D l ∇ y · ∇ y ψ dydx = (cid:90) Ω (cid:90) ∂B ( x ) ν · ( D l ∇ y v ) φ dσdx, (4.16)for all ( φ, ψ ) ∈ V and t ∈ I .0 Adrian Muntean and Tycho van Noorden Proposition 4.6 (Uniqueness) Assume (B1) and (B2). Problem ( P ) admits at most oneweak solution in the sense of Definition 4.5. Proof Since Problem ( P ) is linear, the uniqueness follows in the standard way andcan be done directly in the x -dependent function spaces: One takes two different weaksolutions to ( P ) satisfying the same initial data. Testing with their difference and usingthe Gronwall’s inequality conclude the proof. Theorem 4.7 Assume (B1) and (B2). Problem (P) admits at least a global-in-time weaksolution in the sense of Definition 4.5. Proof See the proof of Theorem 5.11 in [30].To get the correctors estimates stated in Theorem 3.1 we need more two-scale regularityfor the macroscopic reconstruction of the concentration field v . We state this fact in thefollowing result: Lemma 4.8 (Additional two-scale regularity) Assume (B1) and (B2). Then v (cid:15) ∈ L ( I ; H (Ω; H ( B ( x ))))) . (4.17) Proof The proof of this result is similar to the regularity lift proven in Claim 5.10 in [30].The main ingredients are fixing of the boundary and testing with difference quotients inthe weak formulation posed in fixed domains. We omit to show the details. Theorem 4.9 (On strong solutions) Assume (B1), (B2), and (B3). Problem (P) admitsone global-in-time strong solution. Proof Under the assumptions (B1) and (B2), Theorem 4.7 guarantees the existence ofglobal-in-time weak solutions. Relying on the assumption (B3), we can lift the regularityuntil getting a strong solution. A similar calculation is done in [16] (Theorem 5, pp.360–364). In particular, (B3) allows for a regularity lift such that ||∇ ∂ t u || L ( I × Ω) + ||∇ ∂ t v || L ( I × Ω × Y ) ≤ c. For our purpose, we only need ||∇ ∂ t u (cid:15) || L ( I × Ω (cid:15)h ) ≤ c, (4.18) || u (cid:15) || L ( I ; H (Ω (cid:15)h )) ≤ c (4.19)where u (cid:15) is the macroscopic reconstruction of u . Quite probably (4.19) could be relaxedto || u (cid:15) || L ( I ; H θ (Ω (cid:15)h )) for some θ > 0, but we don’t address here this issue. Since werather wish that the reader focusses on our strategy of getting the correctors, we omit toshow the proof details for this regularity result. orrector estimates of a locally-periodic scenario In this section, we give the proof of the main result of our paper, i.e. of Theorem 3.1. Theproof uses the auxiliary results stated in the lemmas below. They mainly concern integralestimates for rapidly oscillating functions with prescribed average; Related estimates canbe found, for instance, in [7] and Section 1.5 in [10]. Lemma 5.1 Assume the hypothesis of Theorem 3.1 to hold. Then (cid:90) Y ( x ) (cid:16) ∇ x · (( I + ∇ y M ) ∇ x u ) − θ ( x ) ∇ x · (cid:90) Y ( x ) ( I + ∇ y M ) ∇ x u dy (cid:17) dy = − (cid:90) ∂B ( x ) ν · ( I + ∇ y M ) ∇ x u dσ. Proof We compute (cid:90) Y ( x ) (cid:16) ∇ x · (( I + ∇ y M ) ∇ x u ) − θ ( x ) ∇ x · (cid:90) Y ( x ) ( I + ∇ y M ) ∇ x u dy (cid:17) dy = (cid:90) Y ( x ) ∇ x · (( I + ∇ y M ) ∇ x u dy − ∇ x · (cid:90) Y ( x ) ( I + ∇ y M ) ∇ x u dy. Reynolds’s transport theorem (see for instance [20]) gives ∇ x · (cid:90) Y ( x ) ( I + ∇ y M ) ∇ x u dy = (cid:90) Y ( x ) ∇ x · (( I + ∇ y M ) ∇ x u ) dy + (cid:90) ∂B ( x ) ∇ x S |∇ y S | ( I + ∇ y M ) ∇ x u dσ. By the boundary condition in (2.14), we have ν ( I + ∇ y M ) = ∇ y S |∇ y S | ( I + ∇ y M ) = 0 on ∂B ( x ) , so that we can write, using (2.5), ∇ x S |∇ y S | ( I + ∇ y M ) ∇ x u = (cid:32) ∇ x S |∇ y S | − ∇ x S · ∇ y S |∇ y S | ∇ y S |∇ y S | (cid:33) ( I + ∇ y M ) ∇ x u = ν · ( I + ∇ y M ) ∇ x u on ∂B ( x ). Combining the above expressions proves the conclusion of this lemma. Lemma 5.2 Assume the hypothesis of Theorem 3.1 to hold. Let Q ( x, y ) ∈ L (Ω; L ( B ( x )))and p ∈ L (Ω; L ( ∂B ( x ))). Furthermore, suppose that (cid:82) Y ( x ) Q ( x, y ) dy = (cid:82) ∂Y ( x ) p ( x, y ) dσ .Then the inequality (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Ω (cid:15)h Q ( x, x/(cid:15) ) φ ( x ) dx − (cid:15) (cid:90) Γ (cid:15) p ( x, x/(cid:15) ) φ ( x ) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C(cid:15) (cid:107) φ (cid:107) H (Ω (cid:15)h ) holds for every φ ∈ H (Ω (cid:15)h ; ∂ Ω). The constant C does not depend on the choice of (cid:15) .2 Adrian Muntean and Tycho van Noorden Proof The problem ∆ y Ψ( x, y ) = Q ( x, y ) in Y ( x ) ν · ∇ y Ψ = p ( x, y ) on ∂B ( x )has a 1-periodic in y solution that is unique up to an additive constant. We multiply thefirst equation above with φ and integrate over Ω (cid:15)h and use the thus obtained equality toget: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Ω (cid:15)h Q ( x, x/(cid:15) ) φ ( x ) dx − (cid:15) (cid:90) Γ (cid:15) p ( x, x/(cid:15) ) φ ( x ) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Ω (cid:15)h ∆ y Ψ( x, y ) | y = x(cid:15) φ ( x ) dx − (cid:15) (cid:90) Γ (cid:15) p ( x, x/(cid:15) ) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:15) (cid:90) Ω (cid:15)h (cid:16) ∇ x [ ∇ y Ψ( x, y ) | y = x(cid:15) ] − ∇ x ∇ y Ψ( x, y ) | y = x(cid:15) (cid:17) φ ( x ) dx − (cid:15) (cid:90) Γ (cid:15) p ( x, x/(cid:15) ) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:15) (cid:90) Γ (cid:15) ( ν + (cid:15)ν (cid:15) ) · ∇ y Ψ( a, y ) | y = x(cid:15) φ ( x ) ds − (cid:15) (cid:90) Ω (cid:15)h ∇ y Ψ | y = x(cid:15) ∇ x φ ( x ) dx − (cid:15) (cid:90) Ω (cid:15)h ∇ x ∇ y Ψ( x, y ) | y = x(cid:15) φ ( x ) dx − (cid:15) (cid:90) Γ (cid:15) p ( x, x/(cid:15) ) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Γ (cid:15) ν (cid:15) · ∇ y Ψ( a, y ) | y = x(cid:15) φ ( x ) ds (cid:12)(cid:12)(cid:12)(cid:12) + (cid:15) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Ω (cid:15)h ∇ y Ψ | y = x(cid:15) ∇ x φ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:15) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Ω (cid:15)h ∇ x ∇ y Ψ( x, y ) | y = x(cid:15) φ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15)C (cid:107) φ (cid:107) H (Ω (cid:15)h ) The lemma is now proved.The last auxiliary lemma is a special case of Lemma 4 in [7], and therefore we willstate it here without proof. Lemma 5.3 Let Π (cid:15) be a subset of { x ∈ Ω | dist( x, ∂ Ω) ≤ √ (cid:15) } . Then the followinginequality (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Π (cid:15) ∇ x u φ dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C(cid:15) / (cid:107) φ (cid:107) H (Ω (cid:15)h ) holds for all φ ∈ H (Ω (cid:15)h ; ∂ Ω). The constant C does not depend on (cid:15) . Proof (of Theorem 3.1) We define z (cid:15) ( x, t ) := u ( x, t ) + (cid:15)χ (cid:15) ( x ) M ( x, x/(cid:15) ) ∇ u ( x, t ) − u (cid:15) ( x, t ) ,w (cid:15) ( x, t ) := v ( x, x/(cid:15), t ) − v (cid:15) ( x, t ) . By, e.g., Theorem 4 in [23], the functions M ( x, x/(cid:15) ) and v ( x, x/(cid:15), t ) are well-definedfunctions in H (Ω (cid:15)h ) and L ( I ; H (Ω (cid:15)l )), respectively, and furthermore, we know by con- orrector estimates of a locally-periodic scenario C > (cid:107) z (cid:15) (cid:107) L ( I ; H (Ω (cid:15)h )) ≤ C, (5.1) (cid:107) w (cid:15) (cid:107) L ( I ; H (Ω (cid:15)l )) ≤ (cid:15)C, (5.2)where the constant C is independent of the choice of (cid:15) . In the following we will use thenotation u ( x, y, t ) = M ( x, y ) ∇ u ( x, t ). We compute:∆ z (cid:15) ( x, t ) =∆ u ( x, t ) + (cid:15)χ (cid:15) ∆ x u ( x, y, t ) | y = x(cid:15) + 2 χ (cid:15) ∇ x · ∇ y u ( x, y, t ) | y = x(cid:15) + 1 (cid:15) χ (cid:15) ∆ y u ( x, y, t ) | y = x(cid:15) + (cid:15) ∆ χ (cid:15) u ( x, y, t ) | y = x(cid:15) − ∆ u (cid:15) ( x, t )+ 2 (cid:15) ∇ χ (cid:15) · ∇ x u ( x, y, t ) | y = x(cid:15) + 2 ∇ χ (cid:15) · ∇ y u ( x, y, t ) | y = x(cid:15) ∆ x w (cid:15) ( x, x(cid:15) , t ) =∆ x v ( x, t ) + (cid:15) − ∇ x · ∇ y v ( x, y, t ) | y = x(cid:15) + (cid:15) − ∇ y · ∇ x v ( x, y, t ) | y = x(cid:15) + 1 (cid:15) ∆ y v ( x, y, t ) | y = x(cid:15) − ∆ x v (cid:15) ( x, y, t ) | y = x(cid:15) . We use that θ ( x ) ∂ t u − ∇ x · ( D ( x ) ∇ x u − ¯ qu ) = − (cid:90) ∂B ( x ) ν · ( D l ∇ y v ) dσ in Ωand ∆ y u ( x, y ) = 0 and ∂ t u (cid:15) = ∇ · ( D h ∇ u (cid:15) − q (cid:15) u (cid:15) ) to obtain D h ∆ x z (cid:15) − ∂ t z (cid:15) = D h ∆ x u + (cid:15)χ (cid:15) D h ∆ x u + 2 χ (cid:15) D h ∇ x · ∇ y u + (cid:15)D h ∆ χ (cid:15) u + 2 (cid:15)D h ∇ χ (cid:15) · ∇ x u + 2 D h ∇ χ (cid:15) · ∇ y u − θ (cid:16) ∇ x · ( D ( x ) ∇ x u − ¯ qu ) − (cid:90) ∂B ( x ) ν · ( D l ∇ y v ) dσ (cid:17) − (cid:15)χ (cid:15) ∂ t u − ∇ · ( q (cid:15) u (cid:15) ) ,(cid:15) D l ∆ x w (cid:15) − ∂ t w (cid:15) = (cid:15) D l ∆ x v + (cid:15)D l ( ∇ x · ∇ y v + ∇ y · ∇ x v ) . On Γ (cid:15) we have ν (cid:15) · ∇ z (cid:15) = − ν (cid:15) · ∇ u (cid:15) + ν (cid:15) · ∇ x u + (cid:15)ν (cid:15) · ∇ x u + ν (cid:15) · ∇ y u = − (cid:15) D l D h ν (cid:15) · ∇ v (cid:15) + ν (cid:15) · ∇ x u + (cid:15)ν (cid:15) · ∇ x u + ν (cid:15) · ∇ y u ,ν (cid:15) · ∇ w (cid:15) = − ν (cid:15) · ∇ v (cid:15) + ν (cid:15) · ∇ x v + (cid:15) − ν (cid:15) · ∇ y v . Adrian Muntean and Tycho van Noorden Now, we multiply with φ and integrate by parts to get (cid:90) Ω (cid:15)h ∂ t z (cid:15) φ dx + (cid:90) Ω (cid:15)h ( D h ∇ z (cid:15) · ∇ φ + q (cid:15) · ∇ z (cid:15) φ ) dx = (cid:15) (cid:90) Ω (cid:15)h χ (cid:15) ∂ t u φ dx − (cid:90) Ω (cid:15)h D h ∆ x u φ dx − (cid:15) (cid:90) Ω (cid:15)h D h χ (cid:15) ∆ x u φ dx − (cid:90) Ω (cid:15)h D h χ (cid:15) ∇ x · ∇ y u φ dx + (cid:90) Ω (cid:15)h θ (cid:16) ∇ x · ( D ( x ) ∇ x u ) − (cid:90) ∂B ( x ) ν · ( D l ∇ y v ) dσ (cid:17) φ dx (5.3)+ (cid:15) (cid:90) Γ (cid:15) D l ν (cid:15) · ∇ v (cid:15) φ ds − D h (cid:90) Γ (cid:15) ν (cid:15) · ∇ x u φ ds − (cid:15)D h (cid:90) Γ (cid:15) ν (cid:15) · ∇ x u φ ds − D h (cid:90) Γ (cid:15) ν (cid:15) · ∇ y u φ ds − (cid:90) Ω (cid:15)h ( (cid:15)D h ∆ χ (cid:15) u + 2 (cid:15)D h ∇ χ (cid:15) · ∇ x u + 2 D h ∇ χ (cid:15) · ∇ y u ) φ dx − (cid:90) Ω (cid:15)h ( 1 θ ¯ q − q (cid:15) ) · ∇ u φ dx − (cid:15) (cid:90) Ω (cid:15)h χ (cid:15) u q (cid:15) · ∇ φ dx, (5.4)and (cid:90) Ω (cid:15)l ∂ t w (cid:15) ψ dx + (cid:15) (cid:90) Ω (cid:15)l D l ∇ w (cid:15) · ∇ ψ dx = − (cid:15) (cid:90) Ω (cid:15)l D l ∆ x v ψ dx − (cid:15) (cid:90) Ω (cid:15)l D l ( ∇ x · ∇ y v + ∇ y · ∇ x v ) ψ dx − (cid:15) (cid:90) Γ (cid:15) D l ν (cid:15) · ∇ v (cid:15) ψ ds + (cid:15) (cid:90) Γ (cid:15) D l ν (cid:15) · ∇ x v ψ ds + (cid:15) (cid:90) Γ (cid:15) D l ν (cid:15) · ∇ y v ψ ds. (5.5)We take into account the identity (cid:0) ∇ y · ∇ x u ( x, y ) (cid:1) | y = x/(cid:15) = (cid:15) ∇ x · ( ∇ x u ( x, x/(cid:15) )) − (cid:15) (cid:0) ∆ x u ( x, y ) (cid:1) | y = x/(cid:15) , which gives (cid:15)D h (cid:90) Γ (cid:15) ν (cid:15) · ∇ x u | y = x/(cid:15) z (cid:15) ds = (cid:15)D h (cid:90) Ω (cid:15)h ∇ x u | y = x/(cid:15) · ∇ ( χ (cid:15) z (cid:15) ) + χ (cid:15) z (cid:15) ∇ x · ( ∇ x u | y = x/(cid:15) ) dx = (cid:15)D h (cid:90) Ω (cid:15)h ∇ x u · ∇ ( χ (cid:15) z (cid:15) ) + χ (cid:15) z (cid:15) ∆ x u dx + D h (cid:90) Ω (cid:15)h χ (cid:15) z (cid:15) ∇ y · ∇ x u dx, and also the boundary condition ν · ∇ y u = − ν · ∇ x u for y ∈ ∂B ( x ), which holdsalso on Γ (cid:15) , we add the two equations (5.3) and (5.5), substitute φ = z (cid:15) and ψ = w (cid:15) and orrector estimates of a locally-periodic scenario ∂ t (cid:107) z (cid:15) (cid:107) L (Ω (cid:15)h ) + 12 ∂ t (cid:107) w (cid:15) (cid:107) L (Ω (cid:15)l ) + (cid:90) Ω (cid:15)h ( D h ∇ z (cid:15) · ∇ z (cid:15) + q (cid:15) · ∇ z (cid:15) ) dx + (cid:15) D l (cid:107)∇ w (cid:15) (cid:107) L (Ω (cid:15)l ) = − (cid:90) Ω (cid:15)h D h ∆ x u z (cid:15) dx − (cid:90) Ω (cid:15)h χ (cid:15) D h ( ∇ x · ∇ y u ) z (cid:15) dx + (cid:90) Ω (cid:15)h θ ∇ x · ( D ( x ) ∇ x u ) z (cid:15) dx − (cid:15)D h (cid:90) Γ (cid:15) ν (cid:15) · ( ∇ x u + ∇ y u ) z (cid:15) ds − (cid:90) Ω (cid:15)h ( (cid:90) ∂B ( x ) ν · ( D l ∇ y v ) dσ ) z (cid:15) dx + (cid:15) (cid:90) Γ (cid:15) D l ν (cid:15) · ∇ y v z (cid:15) ds + (cid:15) (cid:90) Ω (cid:15)h χ (cid:15) ∂ t u z (cid:15) dx − (cid:15) (cid:90) Ω (cid:15)h D h ∇ x u · ∇ ( χ (cid:15) z (cid:15) ) dx − (cid:15) (cid:90) Ω (cid:15)l D l ∆ x v w (cid:15) dx − (cid:15) (cid:90) Ω (cid:15)l D l ( ∇ x · ∇ y v + ∇ y · ∇ x v ) w (cid:15) dx + (cid:15) (cid:90) Γ (cid:15) D l ν (cid:15) · ∇ x v w (cid:15) ds + (cid:15) (cid:90) Γ (cid:15) D l ν (cid:15) · ( (cid:15) ∇ v (cid:15) − ∇ y v ) u ds − (cid:90) Ω (cid:15)h ( (cid:15)D h ∆ χ (cid:15) u + 2 (cid:15)D h ∇ χ (cid:15) · ∇ x u + 2 D h ∇ χ (cid:15) · ∇ y u ) z (cid:15) dx − (cid:90) Ω (cid:15)h ( 1 θ ¯ q − q (cid:15) ) · ∇ u z (cid:15) dx − (cid:15) (cid:90) Ω (cid:15)h χ (cid:15) u q (cid:15) · ∇ z (cid:15) dx, where we have also used that φ − ψ = z (cid:15) − w (cid:15) = (cid:15)u on Γ (cid:15) and φ = (cid:15)u on ∂ Ω.We know that there exist β > γ ≥ β (cid:107) z (cid:15) (cid:107) H (Ω (cid:15)h ) ≤ (cid:90) Ω (cid:15)h ( D h ∇ z (cid:15) · ∇ z (cid:15) + q (cid:15) · ∇ z (cid:15) ) dx + γ (cid:107) z (cid:15) (cid:107) L (Ω (cid:15)h ) , and we use this to estimate12 ∂ t (cid:107) z (cid:15) (cid:107) L (Ω (cid:15)h ) + 12 ∂ t (cid:107) w (cid:15) (cid:107) L (Ω (cid:15)l ) + β (cid:107) z (cid:15) (cid:107) H (Ω (cid:15)h ) + (cid:15) D l (cid:107)∇ w (cid:15) (cid:107) L (Ω (cid:15)l ) ≤ γ (cid:107) z (cid:15) (cid:107) L (Ω (cid:15)h ) + I + I + ... + I Adrian Muntean and Tycho van Noorden where I = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Ω (cid:15)h D h ∇ x · ( ∇ x u + ∇ y u ) z (cid:15) dx − (cid:90) Ω (cid:15)h θ ∇ x · ( D ( x ) ∇ x u ) z (cid:15) dx + (cid:15)D h (cid:90) Γ (cid:15) ν · ( ∇ x u + ∇ y u ) z (cid:15) ds (cid:12)(cid:12)(cid:12)(cid:12) ,I = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Ω (cid:15)h ( (cid:90) ∂B ( x ) ν · ( D l ∇ y v ) dσ ) z (cid:15) dx − (cid:15) (cid:90) Γ (cid:15) D l ν · ∇ y v z (cid:15) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,I = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Ω (cid:15)h (1 − χ (cid:15) ) D h ( ∇ x · ∇ y u ) z (cid:15) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,I = (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) (cid:90) Γ (cid:15) (cid:16) D h ( ν + O ( (cid:15) )) · ( ∇ x u − ∇ y u ) + D l ν (cid:15) · ∇ y v (cid:17) z (cid:15) ds (cid:12)(cid:12)(cid:12)(cid:12) ,I = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Ω (cid:15)h ( (cid:15)D h ∆ χ (cid:15) u + 2 (cid:15)D h ∇ χ (cid:15) · ∇ x u + 2 D h ∇ χ (cid:15) · ∇ y u ) z (cid:15) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,I = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:15) (cid:90) Ω (cid:15)h χ (cid:15) ∂ t u z (cid:15) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,I = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:15) (cid:90) Ω (cid:15)h D h ∇ x u · ∇ ( χ (cid:15) z (cid:15) ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,I = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:15) (cid:90) Ω (cid:15)l D l ∆ x v w (cid:15) dx + (cid:15) (cid:90) Ω (cid:15)l D l ( ∇ x · ∇ y v + ∇ y · ∇ x v ) w (cid:15) dx − (cid:15) (cid:90) Γ (cid:15) D l ν (cid:15) · ∇ x v w (cid:15) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,I = (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) (cid:90) Γ (cid:15) D l ν (cid:15) · ( (cid:15) ∇ v (cid:15) − ∇ y v ) u ds (cid:12)(cid:12)(cid:12)(cid:12) ,I = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Ω (cid:15)h ( 1 θ ¯ q − q (cid:15) ) · ∇ u z (cid:15) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,I = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:15) (cid:90) Ω (cid:15)h χ (cid:15) u q (cid:15) · ∇ z (cid:15) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . For I we use that u = M ∇ u and that D ( x ) := D h (cid:82) Y ( x ) ( I + ∇ y M ) dy . We set Q ( x, y ) = ∇ x · (( I + ∇ y M ) ∇ x u ) − θ ( x ) ∇ x · (cid:90) Y ( x ) ( I + ∇ y M ) ∇ x u dy,p ( x, y ) = − ν · ( I + ∇ y M ) , and use Lemma 5.2 to obtain I ≤ (cid:15)C (cid:107) z (cid:15) (cid:107) H (Ω (cid:15)h ) . Lemma 5.1 asserts that the conditionsof Lemma 5.2 are satisfied for these choices of Q and p .For I we also apply Lemma 5.2, this time for the choice Q ( x, y ) = 1 θ ( x ) (cid:90) ∂B ( x ) ν · ∇ y v dσ,p ( x, y ) = ν · ∇ y v , orrector estimates of a locally-periodic scenario I ≤ (cid:15)C (cid:107) z (cid:15) (cid:107) H (Ω (cid:15)h ) with C independent of (cid:15) . For I we have I ≤√ (cid:15)C (cid:107) z (cid:15) (cid:107) H (Ω (cid:15)h ) . Application of the regularity results in Lemma 4.8 and Theorem 4.9results in I + I ≤ C(cid:15) (cid:107) z (cid:15) (cid:107) H (Ω (cid:15)h ) . With the use of Lemma 5.3 and the properties of χ (cid:15) we estimate I + I ≤ √ (cid:15)C (cid:107) z (cid:15) (cid:107) H (Ω (cid:15)h ) . Another application of the regularity resultsgives I ≤ (cid:15)C (cid:107) w (cid:15) (cid:107) H (Ω (cid:15)l ) and I ≤ C(cid:15) (cid:107) u (cid:107) H (Ω (cid:15)h ) ( (cid:107) v (cid:15) (cid:107) H (Ω (cid:15)l ) + (cid:107) v (cid:107) H (Ω; H ( B ( x ))) ). For I and I we use the assumptions on q and q (cid:15) (as stated in (A2) and (B2)) to get I + I ≤ √ (cid:15)C (cid:107) z (cid:15) (cid:107) H (Ω (cid:15)h ) .Now we obtain12 ∂ t (cid:107) z (cid:15) (cid:107) L (Ω (cid:15)h ) + 12 ∂ t (cid:107) w (cid:15) (cid:107) L (Ω (cid:15)l ) + D h (cid:107)∇ z (cid:15) (cid:107) L (Ω (cid:15)h ) + (cid:15) D l (cid:107)∇ w (cid:15) (cid:107) L (Ω (cid:15)l ) ≤ (cid:107) z (cid:15) (cid:107) L (Ω (cid:15)h ) + (cid:15)C (cid:107) u (cid:107) H (Ω (cid:15)h ) ( (cid:107) v (cid:15) (cid:107) H (Ω (cid:15)l ) + (cid:107) v (cid:107) H (Ω; H ( B ( x ))) ) + √ (cid:15)C (cid:107) z (cid:15) (cid:107) H (Ω (cid:15)h ) + (cid:15)C (cid:107) w (cid:15) (cid:107) H (Ω (cid:15)l ) Using the energy bounds (5.1) and (5.2) together with a Gronwall-type argument leadto (cid:107) z (cid:15) (cid:107) L ∞ ( I,L (Ω (cid:15)h )) + (cid:107) w (cid:15) (cid:107) L ∞ ( I,L (Ω (cid:15)l )) + (cid:107) z (cid:15) (cid:107) L ( I,H (Ω (cid:15)h )) ++ (cid:15) (cid:107) w (cid:15) (cid:107) L ( I,H (Ω (cid:15)l )) ≤ (cid:15) ˜ C + √ (cid:15) ˜ C (cid:107) z (cid:15) (cid:107) L ( I,H (Ω (cid:15)h )) , (5.6)which, in particular, implies (cid:107) z (cid:15) (cid:107) L ( I,H (Ω (cid:15)h )) ≤ (cid:15) ˜ C + √ (cid:15) ˜ C (cid:107) z (cid:15) (cid:107) L ( I,H (Ω (cid:15)h )) . and thus (cid:107) z (cid:15) (cid:107) L ( I,H (Ω (cid:15)h )) ≤ √ (cid:15) 12 ( ˜ C + (cid:113) ˜ C + 4 ˜ C ) . Combining this with (5.6) gives the result (cid:107) z (cid:15) (cid:107) L ∞ ( I,L (Ω (cid:15)h )) + (cid:107) w (cid:15) (cid:107) L ∞ ( I,L (Ω (cid:15)l )) + (cid:107) z (cid:15) (cid:107) L ( I,H (Ω (cid:15)h )) + (cid:15) (cid:107) w (cid:15) (cid:107) L ( I,H (Ω (cid:15)l )) ≤ c √ (cid:15), where the constant c is independent of (cid:15) . The last step uses the evident estimate (cid:107) (cid:15)u (1 − χ (cid:15) ) (cid:107) H (Ω (cid:15)h ) ≤ C √ (cid:15) , and the theorem is proven. Acknowledgments We acknowledge fruitful discussions with Gregory Chechkin regarding homogenizationtechniques for non-periodic media. We also thank Eduard Marusic-Paloka for an interest-ing correspondence on the best corrector estimates (upper bounds on convergence rates)existing for the stationary Stokes and Navier-Stokes problems. References [1] L. Baffico and C. Conca , Homogenization of a transmission problem in solid mechanics ,1999, 233 (J. Math. Anal. Appl.), pp. 659–680.[2] G. Bal and W. 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