MEMORY EFFECT IN HOMOGENIZATION OF VISCOELASTIC KELVIN-VOIGT MODEL WITH TIME DEPENDENT COEFFICIENTS

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1 Mathematical Models and Methods in Applied Sciences c World Scientific Publishing Company MEMORY EFFECT IN HOMOGENIZATION OF VISCOELASTIC KELVIN-VOIGT MODEL WITH TIME DEPENDENT COEFFICIENTS ZOUHAIR ABDESSAMAD, ILYA KOSTIN, GRIGORY PANASENKO, Laboratoire de Mathématiques de l Université de Saint-Etienne Université Jean Monnet 23, rue du Docteur Paul Michelon 4223, Saint-Etienne, France zouhair.abdessamad@univ-st-etienne.fr, kostin@free.fr, Grigory.Panasenko@univ-st-etienne.fr VALERY SMYSHLYAEV Departement of Mathematical Sciences Bath BA2 7AY. United Kingdom vps@maths.bath.ac.uk Received Day Month Year Revised Day Month Year Communicated by xxxxxxxxxx The paper is devoted to the modelling of the procedure of formation of a composite material constituted of solid fibers and of a solidifying matrix. The solidification process for the matrix depends on the temperature and on the reticulation rate which are used for the modelling of the mechanical properties of the matrix. The mechanical properties are described by the viscoelastic media equation with rapidly oscillating periodic coefficients depending on the temperature and the reticulation rate. That is modelled as an initial boundary value problem with time-dependent elasticity and viscosity tensors to account for the solidification, and the termal forcing. First we prove the existence and uniqueness of the solution for the problem and obtain a priori estimates. Then we derive the homogenized problem, characterize its coefficients including explicit memory terms, and prove that it admits a unique solution. Finally, we prove error bounds for the asymptotic solution, and discuss the related regularity properties of the homogenized solution. Keywords: Composite materials; homogenization; viscoelastic media; memory effect. AMS Subject Classification: 65M12, 35K55; 35B27; 74A4 1. Introduction We consider a composite material made of solid fibers included in a resin solidifying matrix which becomes solid when it is heated up reaction of reticulation. The solidification process depends on the temperature and on the reticulation rate, satisfying the heat equation coupled to the kinetic equation for the reticulation rate. 1

2 2 Z. ABDESSAMAD, I. KOSTIN, G.PANASENKO, and V. P. SMYSHLYAEV The homogenization technique applied to this problem reduced it to the homogenized problem with constant coefficients. The estimates have been proved for the difference between the exact solution and solution of the homogenized problem. In particular, an estimate For the difference between the exact temperature T and the homogenized temperature T has been obtained 7, 8. Here the temperature calculated at the first stage and the reticulation rate are used to model the mechanical properties of the composite material which has a periodic geometry with a small period > corresponding to the distance between neighboring fibers. The mechanical properties are described by a viscoelastic media equation Kelvin-Voigt model with rapidly oscillating periodic coefficients depending on the temperature and the reticulation rate, that is, solution of the heat transfer problem, 7, 8. Specifically, the deformation of a viscoelastic medium under the thermal influence is described by the following problem: ρ x ü T B,T u T ij x j A,T u T ij x j = fx, t, x, t Ω, τ, u T x, t =, x, t Ω, τ, 1.1 u T x, = u T x, =, x Ω. Here u T x, t is the unknown displacement vector field, u T x, t = u T x, t,k 1 k n; u T and ü T denote the first and the second time derivatives of u T, respectively. The summation over the repeating subscripts is assumed. The volume density ρξ is a scalar function which is also 1-periodic. The smooth vector function fx, t is given, and describes forces due to the thermal effects. The linear elastic tensor A,T ij A,T ij = A kl ij and the viscosity tensor B,T ij 1 k,l n, B,T ij = B kl ij are matrix valued entities: 1 k,l n, i, j = 1,..., n, which are functions of T, the solution of?? and they depend periodically on x. This dependence means that the visco-elastic properties depend on the temperature. For example, for low temperatures the elastic tensor A is greater than B while for high tempertaures B is greater. So, this temperature depending Kelvin-Voigt model simulates the thermo-chemico-visco-elastic process for a composite material. In this model first the thermo-chemical problem 7, 8 should be solved, and then its solution is used in the visco-elastic equation 1.1. There is a high interest to the combined models of this type, see for example a recent paper 24 where a viscoplastic model is considered; it takes into account the non-linear hardening effect. Our model 1.1 takes into account the dependence of the visco-elastic properties on the temperature. Let us replace T by T the solution of the homogenized heat problem in the coefficients A ij, B ij of problem 1.1. Denote u T a solution of this problem. This new problem with the unknown u T is a particular case of the time dependent problem for the Kelvin-Voigt equation. Taking into account the regularity of T we will

3 Memory Effect in Homogenization of Viscoelastic Kelvin-Voigt Model with time dependent coefficients 3 further consider the following auxiliary visco-elasticity equation: x ρ ü B ij x, t, x u A ij x, t, x u = fx, t, 1.2 x j x j u = on Ω, 1.3 u t= = u t= =.1.4 Here the linear elastic tensor A ij x, t, ξ and the viscosity tensor B ij x, t, ξ are matrix-valued entities: A ij x, t, ξ = A kl ij x, t, ξ, B 1 k,l n ijx, t, ξ = B kl ij x, t, ξ, which are periodic with respect to ξ. The volume density ρξ 1 k,l n is a scalar function which is also periodic. When the point x belongs to a fiber, A kl ij x, x, t, Bkl ij x, x, t and ρ x are respectively the elasticity, the viscosity and the density coefficients of the fiber, otherwise their values describe the physical properties of the resin. The smooth vector function fx, t is given, and describes forces due to the thermal effects. Since the temperature and the reticulation rate change with x and t, so does f. Also, the elastic and viscous tensors are assumed to depend on the temperature and hence on x and t in order to reflect the partial solidification of the medium when the temperature decreases. A scalar case with time-independent coefficients was studied in 1 using the Laplace transform methods, with a fading long term memory effect observed. This effect for visco-elasticity with time-independent coefficients has been first discovered by E.Sanchez-Palencia in 4 Chapter 6. More detailed result involving the weak convergence of the displacement field of a body to the displacement field of a homogenized material with fading memory has been obtained in 26 ; moreover there was considered the thermodynamics term in the right hand side of the Kelvin-Voigt, although the coefficients were still time-independent. The aim of the present paper is to establish that the memory effect in fact takes place in a general situation of time-dependent vector problem. In this case the Laplace transform methods cease to be applicable, and we develop instead a version of the method of asymptotic expansions, supplemented by its rigorous justification, including establishing the error bounds. Mathematically, the memory effect is a particular example of more general non-local effects emerging as a result of homogenization. Various aspects of the latter have been extensively studied and documented in the literature, see e.g. 5, 12 2, 25. The present problem nevertheless, from the mathematical point of view, bears essential specifics. Addressing those requires developing certain nontrivial modifications of both the general theory of hyperbolic systems with variable coefficients and of the non-local homogenization theory, addressed in this paper. The main result of the paper is in establishing the structure of the homogenized equation corresponding to 1.2. Namely, we show that under appropriate technical assumptions it has the following form: ρ vx, t σ i x, t = fx, t, 1.5

4 4 Z. ABDESSAMAD, I. KOSTIN, G.PANASENKO, and V. P. SMYSHLYAEV where σ i x, t = B ij x, t v x, t + x Âijx, t v x, t j x j t + Ê ij x, t, t v x, t + x D ij x, t, t v x, t dt. 1.6 j x j Here ρ is the mean value of density ρ, and Âij, B ij, D ij and Êij are homogenized characteristics explicitly expressible in terms of solutions of appropriate unit cell problems, see 2.1 and 2.2. In particular, the memory or non-local terms are those containing the integro-differential operators with the kernels D ij and Êij, explicitly given by We will adopt following notational conventions throughout the paper. In 1.2 and henceforth the summation with respect to repeated indices is implied. A matrix followed by a vector implies a standard multiplication of the matrix by a vector, being a vector; Q = [, 1] n denotes the reference periodicity cell; C# Q stands for the subspace of infinitely smooth functions C R n whose elements are periodic with respect to Q, i.e. 1-periodic with respect to each of its n variables; H# 1 Q and L 2 # Q denote the closures of C # Q in the norms of the standard spaces H1 Q and L 2 Q, respectively. For any space X, we denote the space X n by X, and for any tensor M = Mij kl M n 2 n 2R the set of n2 n 2 real matrices, we denote by I M u, v Ω the bilinear form I M u, v Ω = Ω M kl ij u l x j v k dx u, v H 1 Ω. Similarly, we define I M u, v Q for all u, v H 1 #Q by the same formula with Ω replaced by Q. We start by analyzing the general problem Section 1. Namely, we use a version of the Faedo-Galerkin method see e.g. 6 to prove that for all >, equation 1.2 together with boundary condition 1.3 and general initial conditions specified below, admits a unique solution and we obtain a priori estimates for the solution. Then, in Section 2, we develop a modification of the traditional techniques of the method of asymptotic expansions see, e.g., 2 4 to derive homogenized equation and to characterize its coefficients. Afterwards we establish that the homogenized problem admits a unique solution v and study the convergence of the exact solution to the approximate solution as tends to zero, obtaining tight error bounds on the difference between them, see Theorem 3.3. Finally we study related regularity properties of the homogenized solutions Section 3. We derive sufficient conditions for the regularity for a specific case but under rather generic assumptions on the viscosity and elasticity coefficients, allowing them to be microscopically spatially discontinuous as e.g. in a matrix-inclusion composite.

5 Memory Effect in Homogenization of Viscoelastic Kelvin-Voigt Model with time dependent coefficients 5 2. Existence and uniqueness for original problem Let Ω be a bounded domain of R n n 2 with a Lipschitz boundary, τ be a positive number, and let Ω τ := Ω, τ. Let for all i, j = 1,..., n and all x Ω, ξ Q and t, τ the tensors A ij x, t, ξ and B ij x, t, ξ belong to M n,n R, being measurable functions of their arguments, periodic in ξ. We will set ξ = x/ for any positive and will regard A ij and B ij as functions on Ω τ, depending on as a parameter, assuming A ij x, t, ξ and B ij x, t, ξ have sufficient regularity for this to make sense, as will be specified further later in the paper. At the moment we assume that: H1- For all > and i, j = 1,..., n, both A ij and its time derivative Ȧij belong to L Ω τ ; M n,n R. Moreover, there exists a positive constant ν independent of, x and t, such that A ij L Ω τ ν 1 and Ȧij L Ω τ ν 1. H2- For all > the tensors B and Ḃ belong to L Ω τ ; M n2,n 2,symR, where M n 2,n 2,symR is the set of symmetric elasticity tensors such that Bij kl = Bil kj = Bji lk and such that Ḃij L Ω τ ν 1. Additionally, B is uniformly elliptic, i.e. for all symmetric matrices η = ηj l Rn n, for almost all x, t Ω τ and all > ν η k i η k i B kl ij x, t, x ηk i η l j ν 1 η k i η k i. H3- The Q-periodic function ρ belongs to L Q; R and is uniformly positive, i.e. there exists a constant ρ 1 such that 1 ρξ ρ 1, for all ξ Q. In assumptions H1 and H2 we employed the L Ω τ -norm of an n n matrix valued function, which can be defined e.g. as the L -norm on Ω τ for the matrix euclidian norm. We consider deformation of a viscoelastic medium with thermal effects, with rapidly oscillating properties in Ω τ, described by the initial boundary value problem: x ρ ü B ij x, t, x u A ij x, t, x u = fx, t, 2.1 x j x j u = on Ω, 2.2 u t= = ϕ, u t= = ψ.2.3 The following theorem establishes the existence, uniqueness and a priori estimates for a weak solution to the above initial boundary value problem: Theorem 2.1. Let f L 2, τ; H 1 Ω, ϕ H 1 Ω, ψ L 2 Ω and let assumptions H1-H3 hold. Then, for all >, problem admits a unique weak solution u in H 1, τ; H 1 Ω and there exists a constant C 1 depending only on ν, τ, ρ 1 and Ω such that u Ωτ u L,τ;H 1 Ω + u L2,τ;H 1 Ω + u L,τ;L 2 Ω

6 6 Z. ABDESSAMAD, I. KOSTIN, G.PANASENKO, and V. P. SMYSHLYAEV C 1 f L 2,τ;H 1 Ω + ϕ H 1 Ω + ψ L 2 Ω. 2.4 Proof: Let u x, t H 1, τ; H 1 Ω be a weak solution of , i.e. Ω u x, = ϕx, x Ω, 2.5 and, for any z H 1, τ; H 1 Ω such that zx, τ =, τ B ij x, t, x u + A ij x, t, x u z dx dt x j x j τ Ω x ρ Ω x ρ ψx zx, dx u x, t żx, t + fx, t zx, t dx dt =. 2.6 In order to prove the results of the theorem we use the Faedo-Galerkin method see e.g. 6, 7.1. We denote by w i i N an orthogonal basis of H 1 Ω, and for any fixed positive integer m we introduce an approximate problem which consists in finding a function u mt defined by: u mt = m d i mt w i x, t τ, i=1 where d i mt 1 i m satisfy the following system of ordinary differential equations: for j = 1,..., m ρ ü mt, w j Ω + I B u mt, w j Ω + I A u mt, w j Ω = ft, w j Ω, 2.7 d j m = ϕ m, w j Ω, d j m = ψ m, w j Ω, 2.8 with ϕ m and ψ m being respectively the orthogonal projections in H 1 Ω of ϕ and ψ on the finite space Spanw 1, w 2,..., w m, with, Ω denoting the standard inner product in L 2 Ω. Using H3 we conclude that the matrix ρw i, ρw j 1 i,j m is invertible, so the system admits a unique solution u m in H 2, τ; H 1 Ω. We derive next a priori estimates for u m and u m. To this end, we multiply 2.7 by d j mt and sum with respect to j = 1,..., m. Thus we obtain ρ ü mt Ω, u mt Ω + I B u mt, u mt Ω = ft, u mt Ω I A u mt, u mt Ω. Since u m H 2, τ; H 1 Ω, we can re-express the last identity in the following form: 1 d ρ u 2 dt mt 2 L 2 Ω + I B u m, u m Ω = f, u m Ω I A u m, u m Ω 2.9 We aim next at estimating all the terms in the right hand side of 2.9 via I B, Ω. We first use the fact that for u H 1 Ω, the assumption H2 and the standard

7 Memory Effect in Homogenization of Viscoelastic Kelvin-Voigt Model with time dependent coefficients 7 Korn inequality see e.g. 5, 1, 11 ensure the equivalence of I B u, u Ω to the H 1 norm, i.e. ν 2 u 2 H 1 Ω I Bu, u Ω n 2 ν 1 u 2 H 1 Ω. 2.1 Use H1-H2 and the Poincaré-Friedrichs inequality for the terms of the right hand side of 2.9 and apply 2.1. Thus we check that there exists a constant c 1, e.g. c 1 = 4 max{1 + CΩ 2 ν 1, 2 n 4 ν 4 } where CΩ > is the constant appearing in the Poincaré Friedrichs inequality, such that d ρ u m 2 L dt 2 Ω + I B u m, u m Ω c 1 I B u m, u m Ω + f 2 H 1 Ω We deduce from 2.11 that d ρ u dt mt 2 L 2 Ω c 1 I B u mt, u mt Ω + ft 2 H 1 Ω On the other hand, multiplying relations 2.7 by d j mt and summing them up for all j = 1,..., m, we obtain ρ ü mt, u mt Ω + I B u mt, u mt Ω + I A u mt, u mt Ω = ft, u mt Ω. By using the symmetric properties of B ij assumption H2, we can write the last identity in the following form: d ρ u mt, u mt Ω + 12 dt I Bu mt, u mt Ω = ρ u mt 2 L 2 Ω I Ḃ u mt, u mt Ω I A u mt, u mt Ω + ft, u mt Ω. In the same manner as previously, in order to estimate the right hand side of the last identity via I B, Ω, we use H1 and 2.1 to justify that there exists a constant c 2 e.g. c 2 = max{ 1 2, 3 n2 ν CΩ 2 ν 1 }, such that d dt ρ u mt, u mt Ω + 12 I Bu mt, u mt Ω I B u mt, u mt Ω + ft 2 H 1 Ω ρ u mt 2 L 2 Ω + c 2 Multiply then the last inequality by a real γ > and add it to 2.12 to arrive at ds γ t c 3 ρ u dt mt 2 L 2 Ω + I Bu mt, u mt + ft 2 H 1 Ω, 2.13 where e.g. c 3 = γ + c 1 + γ c 2 and S γ t := ρ u mt 2 L 2 Ω + γ 2 I Bu mt, u mt + γ ρ u mt, u mt Ω. ν Lemma 2.1. For < γ < 4 ρ 1 CΩ, there exists a constant c 2 4, depending only on ν and γ, such that, for t τ, the following inequality holds: ρ u mt 2 L 2 Ω + γ 2 I Bu mt, u mt Ω c 4 S γ + t. f 2 H 1 Ω dt 2.14

8 8 Z. ABDESSAMAD, I. KOSTIN, G.PANASENKO, and V. P. SMYSHLYAEV Proof: By using assumptions H1 and H3, the Poincaré-Friedrichs inequality and 2.1, it is easy to see that for < γ ν/ 4 ρ 1 CΩ 2 we have 1 2 ρ u mt 2 L 2 Ω + γ 4 I Bu mt, u mt γ ρ u mt, u mt Ω >. Thus we deduce that S γ t 1 2 ρ u mt 2 L 2 Ω + γ 4 I Bu mt, u mt Ω ν Further, we fix γ in ], 4 ρ 1 CΩ [ in inequality 2.13 and, using 2.15, find a constant c 5 = c 3 max{ 4 γ 2, 2} such that ds γ t c 5 S γ t + ft 2 H dt 1 Ω. Then, by the Gronwall s inequality e.g. 6, B2.j, we deduce that t S γ t e S c5 t γ + ft 2 H 1 Ω dt, t [, τ]. Finally, comparing the last inequality with 2.15, we obtain 2.14 with e.g. c 4 = 2 e c5 τ. Now we will use inequality 2.14 to establish a priori estimates for u m. Indeed, we estimate the term on the left of 2.14 from below by using 2.1 and thus we check that for all t [, τ] we have u m H 1 Ω + u m L2 Ω c 6 f L2,τ;H 1 Ω + ϕ H 1 Ω + ψ L2 Ω On the other hand, by integrating 2.11 with respect to t and by using 2.16, we verify the following estimate: u m 2 L 2,τ;H 1 Ωτ c 7 f 2 L 2,τ;H 1 Ω + ϕ 2 H 1 Ω + ψ 2 L 2 Ω From estimates we conclude that the sequences u m m N and u m m N are bounded in L, τ; H 1 Ω and L 2, τ; H 1 Ω L, τ; L 2 Ω, respectively. Therefore we can extract a subsequence u m m N such that, when m, we have: u m u L, τ; H 1 Ω and u m u L 2, τ; H 1 Ω L, τ; L 2 Ω, where denotes the weak-star convergence. Lastly, we integrate identity 2.7 over [, τ] integrating the first term once by parts in t and pass to the limit using the above convergence results. So we prove that u satisfy In the same way we pass to the limit in estimates and prove that u satisfy the estimate 2.4 of the Theorem. Thus we have proved that there exists a function u belonging to H 1, τ; H 1 Ω and satisfying the weak formulation of problem

9 Memory Effect in Homogenization of Viscoelastic Kelvin-Voigt Model with time dependent coefficients 9 In order to prove that the above solution is unique, it suffices to show that problem with f ϕ ψ has no non-trivial solutions. To verify this, we fix s τ and substitute s t u x, r dr when t s, vx, t = when s t τ into the identity 2.6, adopting in this case the form τ τ τ ρ u v dx dt + I B u, v Ω dt + I A u, v Ω dt =, u x, =. Ω After integration by parts in the first and the second terms, we obtain 1 2 ρu s 2 L 2 Ω + s I B u, u dt s IḂu, v Ω + IA u, v Ω dt 2.18 Now, by using 2.1, we estimate the right hand side of 2.18 via I B, Ω and obtain inequality 1 2 ρu s 2 L 2 Ω n2 s s ν 2 I B u, u Ω dt Finally, we choose τ = τ 1 small enough so that τ 1 < ν2 4n 2. The latter estimate implies that for any s in [, τ 1 ], we have u s L 2 Ω =. Finally, if τ 1 < τ, we apply repeatedly the same argument on the smaller intervals within [, τ] to deduce that u. 3. Asymptotic expansion of the solution In this section we consider the above model of a viscoelastic deformation of an - periodic composite material, treating as a small parameter. The functions ρξ, A ij x, t, ξ and B ij x, t, ξ are Q-periodic with respect to ξ e.g. with the unit periodicity cell Q. We assume that these functions satisfy hypotheses fully analogous to H1-H3, as clarified later. We describe the asymptotic behaviour of the solution of problem when is small. According to the traditional asymptotic expansion method, a formal asymptotic solution to the problem is sought in the following two-scale form: u x, t v x, t + N x, t, x + 2 u 2 x, t, x Here vx, t is the leading term, Nx, t, x is the corrector -term found from appropriate unit cell problem, and Nx, t, ξ, u 2 x, t, ξ are assumed to be Q-periodic in ξ. Near the boundary Ω the asymptotic expansion 3.1 is expected to contain a usual boundary layer cf. e.g. 2 Chapter 9.

10 1 Z. ABDESSAMAD, I. KOSTIN, G.PANASENKO, and V. P. SMYSHLYAEV Substituting the ansatz 3.1 into 2.1 and collecting formally the terms with equal powers of, we obtain a sequence of initial boundary value problems which will be given explicitly later on. A key role in the subsequent asymptotic constructions will be played by various versions of the following cell problem, which we first study here in maximal generality. For any x Ω, we seek ux,, H 1, τ; H such that B ij x, t, ξ u Here H is Hilbert space defined by equipped with the norm A ij x, t, ξ u = gx, t, ξ, 3.2 ux,, ξ = φx, ξ. 3.3 H := {u H 1 #Q; uξ ξ = }, 3.4 u H := ξ u L 2 Q n. Henceforth the angular brackets with subscript ξ denote the mean of the appropriate function with respect to ξ: uξ ξ := uξ dξ. 3.5 We will also make use of space H 1 # Q which is by definition a dual space of linear continuous functionals g on H 1 # Q, with their values upon action on u H1 # Q denoted g, u. For g H 1 # Q, g ξ is defined as g, 1, where 1 is the identical unity for all the components, with the latter definition being consistent with 3.5 for more regular g, e.g. g L 2 Q. Theorem 3.1. Assume that for all x Ω the functions φ and g belong respectively to H and H 1, τ; H 1 # Q. Let gx, t, satisfy g ξ =. Let also for any x Ω the straightforward modifications of assumptions H1-H2 hold, namely with x replaced by ξ, Ω by Q and Ω τ by Q τ := Q, τ. Then the problem admits for any x a unique solution ux,, in H 1, τ; H. Moreover, there exists a constant C 2 which depends only on ν and τ such that, for any x in Ω, Q u L,τ;H + u L 2,τ;H C 2 g L 2,τ;H 1 Q + φ H. 3.6 The proof of Theorem 2.1 is fully analogous to that of Theorem 1.1. Namely, for the existence we use the Galerkin approximations techniques exploiting the invertibility of the matrix analogous in the new variables to I B w i, w j, and the uniqueness follows immediately from a straightforward modification of the Gronwall s inequality. We do not reproduce the proof here. We next study regularity properties of the solutions to the problem Mathematically, the latter is subsequently required for constructing the higherorder terms in the asymptotic expansion and for eventually obtaining the error

11 Memory Effect in Homogenization of Viscoelastic Kelvin-Voigt Model with time dependent coefficients 11 bound. Physically, the dependence of the viscoelastic coefficients A ij and B ij on x and t results from the dependence of these coefficients on the temperature. Consequently, the regularity of the temperature in x and t studied e.g. in 7 determines the regularity of A ij and B ij on x and t, which allows us to study the regularity of the solution of problem With this aim we state and proof the following lemma. Lemma 3.1. Let ux,, H 1, τ; H be the unique solution of problem and assume that A ij, B ij C p Ω τ ; L Q, φ C p Ω; H and g C p Ω τ ; H 1 Q, for some p 1. Then we have u, u C p Ω τ ; H. Proof: The proof is by induction in p. Theorem 2.1 implies that for any x in Ω there exists a unique solution ux,, in H 1 [, τ]; H and it satisfies 3.6. It is easy to see first that ux,, is in fact in C[, τ]; H, cf. 6. For example, fixing x and introducing a small h, t+h ux, t + h, ux, t, H = ux, s, ds t H 1/2 t+h ux, s, Hds 2 h 1/2 ch 1/2 as h with appropriate constant c. t Let next the assumptions of the lemma hold for p = 1 and let us prove that then u, u C 1 Ω τ ; H. Initially we will use only the fact that φ CΩ; H and g CΩ τ ; H 1 Q. For any fixed value of x in Ω and for any vector h R n such that x + h in Ω, we have ux + h,, C[, τ]; H. We subtract equation 3.2 written down for x from the same equation with x replaced by x + h and integrate the result over Q [, τ]. Thus we verify that for any z H 1, τ; H 1 # Q, the function uh defined by u h t, ξ := ux + h, t, ξ ux, t, ξ, t, ξ [, τ] Q satisfies τ Q A ij u h z τ u h dξ dt + B ij z τ dξ dt = g h z dξ dt, Q Q u h t= = φ h,

12 12 Z. ABDESSAMAD, I. KOSTIN, G.PANASENKO, and V. P. SMYSHLYAEV where φ h x, ξ = φx + h, ξ φx, ξ, and g h x, t, ξ = gx + h, t, ξ + gx, t, ξ + From 3.6 we obtain A ij x + h, t, ξ A ij x, t, ξ u x + h, t, ξ B ij x + h, t, ξ B ij x, t, ξ u x + h, t, ξ u h L,τ;H C 2 ϕ h H + g h L2,τ;H 1 Q. The continuity of functions A ij, B ij, ϕ and g on Ω τ implies that the term on the right hand side of the last inequality tends to zero as h vanishes, and consequently, u h H converges to zero as h for any t in [, τ]. Then, using the fact that u C[, τ]; H for any x in Ω, we deduce that, for any x, t Ω τ, the following relation holds: ux + h, t + ζ, ξ ux, t, ξ H, when h, ζ, in R n+1. So, we have u CΩ τ ; H and now we will prove that u belongs to CΩ τ ; H. To this end, we fix x, t in Ω τ and write 3.2 in the form: B ij x, t, ξ u = Gx, t, ξ. Here Gx, t, ξ := A ij x, t, ξ u + gx, t, ξ. By using H1-H2 and the Korn inequality which is valid for any u in H 1 #Q we conclude that the H-norm and I B, Q are equivalent. Therefore there exists a constant c depending only on ν and the space dimension n, such that u H c G H 1 Q. Since the last inequality holds for all x, t Ω τ, we use the same techniques as above to prove that for small parameters h, ζ R n+1, such that x + h, t + ζ in Ω τ, we have ux + h, t + ζ, ξ ux, t, ξ H c Gx + h, t + ζ, ξ Gx, t, ξ H 1 Q. Taking into account that u, g, A ij and B ij are continuous on Ω τ one can show that G belongs to CΩ τ ; H 1 Q and so u CΩ τ ; H. Thus we have proved that for a given g CΩ τ ; H 1 Q and φ CΩ; H, the function u, solution of problem and its time derivative u both belong to CΩ τ ; H. Next formally differentiate equation 3.2 with respect to t and use the fact that functions A ij, B ij C 1 Ω τ ; L Q, φ C 1 Ω; H and g C 1 Ω τ ; H 1 Q to verify that u satisfies the same type of problem as with the right hand.

13 Memory Effect in Homogenization of Viscoelastic Kelvin-Voigt Model with time dependent coefficients 13 side belonging to CΩ τ ; H 1 Q and the initial condition u in CΩ; H. Therefore the above result implies that the second time derivative of u denoted ü is in fact in CΩ τ ; H. Now we will study the differentiability of u with respect to x. To this end, we fix a positive integer l n and differentiate formally equation 3.2 with respect to x l. As above, we check that there exists a function u such that u, u CΩ τ ; H, satisfying the following: τ Q B ij u with g = g x l + Bij x l u + A ij z τ dξ dt = g z dξ dt Q ux,, ξ = φ x, ξ x l u + Aij x l u. By using 3.6 and the same techniques as above, we check that for a small scalar parameter δ and for all x Ω, we have ũ u L,τ;H C 2 ũ u H + g δ L2,τ;H 1 Q, where ũx, t, ξ = ux+δ e l,t,ξ ux,t,ξ δ ; e l 1 l n is the canonical basis of R n and g δ x = Bij x + δ e l, t, ξ B ij x, t, ξ u x + δ e l, t, ξ gx, t, ξ δ + Aij x + δ e l, t, ξ A ij x, t, ξ u x + δ e l, t, ξ δ + gx + δ e l, t, ξ gx, t, ξ δ Then, taking into account that u and u belong to CΩ τ ; H and using the assumption of the lemma for p = 1, we prove that u x l exists and is equal to u. Thus we have u u x l, x l CΩ τ ; H and finish the proof of the assertion of Lemma 2.1 for p = 1. The remainder of the proof repeats the above argument with minor modifications. Namely, assume that the statement of Lemma 2.1 is valid for some positive integer p and suppose that φ C p+1 Ω; H, g C p+1 Ω τ ; H 1 Q and A ij, B ij C p+1 Ω τ ; L Q. Then the differentiation of 3.6 with respect to t implies that u satisfies 3.6 with the right hand side in C p Ω τ ; H 1 Q. Similarly, differentiating 3.6 with respect to x l, l = 1,..., n, we check that u x l is a solution of 3.2 with the right hand side belonging to C p Ω τ ; H 1 Q. Thus, the induction assumption implies that u, ü, u x l, u x l C p Ω τ ; H and we deduce the result of the Lemma for p + 1, completing the proof.

14 14 Z. ABDESSAMAD, I. KOSTIN, G.PANASENKO, and V. P. SMYSHLYAEV 3.1. The main unit cell problem Substituting the ansatz 3.1 into 2.1 and taking into account that v is independent of ξ, the identification of the terms corresponding to order 1 leads to the following equation: where B ij x, t, ξ 2 N x, t, ξ t F x, t, ξ := B ik x, t, ξ v x, t + x k A ij x, t, ξ N x, t, ξ = F x, t, ξ 3.7 Equation 3.7 has to be supplemented by the initial condition A ik x, t, ξ v x, t. 3.8 x k Nx,, ξ =, 3.9 to comply with the first initial condition in 2.3. This leads to a version of the main unit cell problem, typical for homogenization problems. By Theorem 2.1 its solution Nx, t, ξ, periodic in ξ, exists for any given vx, t such that v, t H 1, τ for any x Ω. The solution Nx, t, ξ exists and is unique up to a function depending only on x and t i.e. a constant with respect to ξ, c.f. Theorem 2.1. To select a unique solution, we require that N has zero mean value with respect to ξ. Henceforth, we apply this selection criterium whenever a boundary value problem with periodic conditions is stated. The following lemma establishes the structure of function N. Lemma 3.2. The following representation holds: Nx, t, ξ = t v x k x, t N B k x, t t, t, ξdt + t v x k x, t N A k x, t t, t, ξdt 3.1 Here Nk Ax, t, s, ξ and N k B x, t, s, ξ are periodic with respect to ξ solutions of the following initial boundary value problems: B ij x, s + t, ξ N k A! x, t, s, ξ A ij x, s + t, ξ N A k x, t, s, ξ! =, 3.11 B ij x, s + t, ξ N k B! x, t, s, ξ A ij x, s + t, ξ N B! k x, t, s, ξ =, 3.12 Nk A x,, s, ξ = gk A x, s, ξ, Nk B x,, s, ξ = gk B x, s, ξ In turn, gk Ax, s, ξ, gb k x, s, ξ solve the following cell problems: B ij x, s, ξ gk A x, s, ξ B ij x, s, ξ gk B x, s, ξ = A ik x, s, ξ, 3.14 = B ik x, s, ξ. 3.15

15 Memory Effect in Homogenization of Viscoelastic Kelvin-Voigt Model with time dependent coefficients 15 Proof: We check here the formal part of the proof of Lemma 2.2. The existence and uniqueness of the solutions of these boundary value problems will be established later. From 3.1, we obtain N t = t v x k x, t t N B k x, t t, t,, ξdt + + v x k x, tn B k x,, t, ξ + t v x k x, t t N A k x, t t, t, ξdt v x k x, tn A k x,, t, ξ Substitute 3.1 and 3.16 into 3.7. Using 3.11, 3.12 and 3.13 we conclude that the left hand side of 3.7 equals B ij x, t, ξ v gk B x, t B ij x, t, ξ v gk A x, t x k x k Finally, using 3.14 and 3.15, we conclude that 3.17 transforms into the right hand side of 3.7 given by 3.8. Notice that according to 3.15, for any fixed x and t, gk B are standard unit cell solutions corresponding to the elliptic operator with coefficients B ij x, t, ξ in homogenization for the classical elliptic operators with periodic coefficients cf. eg. 2,9, whereas gk A does not have such a direct standard analogue. Lemma 3.3. i Let the modification of assumptions H1-H2 stated in Theorem 2.1 hold. Then for any k = 1,..., n, equation 3.14 admits a unique solution such that for all x, t Ω τ, g A k Hn. Moreover, there exists a constant C 3 which depends only on ν and n such that g A k H n + ġ A k H n C 3, x, t Ω τ ii Let p N and A ij, B ij C p Ω τ ; L Q. Then, for any k = 1,..., n, Q g A k C p Ω τ ; H n Proof: We fix k and consider x and s as parameters, i.e. gk A depends only on ξ. i Let gk A ξ be a weak solution of 3.14, i.e. φ g A k φ B ij dξ = A ik dξ, φ H 1 ξ #Q n. 3.2 i Q Let us make use of the equivalence between I B, Q and the H-norm which we have mentioned above. We check that the term on the left hand side of 3.2 is a symmetric and positive definite bilinear form on H n, while the right hand side is a continuous linear form on H n. Thus, the Lax-Milgram lemma implies the existence and uniqueness of the solution gk A Hn of 3.2 and we obtain the following estimate: g A k H n 2 n ν In order to study the existence of the time derivative of gk A, we use the same techniques as in the proof of Lemma 2.1 to conclude that ġk A Hn exists and is

16 16 Z. ABDESSAMAD, I. KOSTIN, G.PANASENKO, and V. P. SMYSHLYAEV bounded in the H n -norm by a constant depending on only ν. Thus, using 3.21, we arrive at ii The proof of 3.19 is an induction over p and it is similar to that of Lemma 2.1. First we prove that the continuity of A ij on Ω τ implies the continuity of g A k in x and t. Second, we check that if A ij C p+1 Ω τ ; L Q, the function ga k x exists l in H n for all x = x 1,..., x n, t Ω τ, and satisfies 3.2 with a right hand side in C p Ω τ ; L Q. Thus we deduce that ga k x l C p Ω τ ; H n and obtain Remark 3.1. Since the functions gk A and gb k satisfy the equations of the same type, 3.14 and 3.15 respectively, the results of Lemma 2.1 are equally valid for the function gk B. To conclude, the solutions of the problem , N k A and, also exist and are as regular as the solutions of N B k 3.2. Derivation of homogenized equation. The third term of ansatz 3.1, u 2 x, t, ξ, solves the equation which results from equating the terms of order after substituting 3.1 into 2.1. This equation is of the form B ij x, t, ξ u2 A ij x, t, ξ u2 = Fx, t, ξ Here Fx, t, ξ = B ij x, t, ξ + A ij x, t, ξ N x j v + Ṅ x j + + v A ij x, t, ξ + N x j B ij x, t, ξ Ṅ x j + f ρ v The homogenized equation for v is obtained following a standard recipe as a necessary condition for the existence of u 2 x, t, ξ as a solution of problem with the initial condition u 2 x,, ξ = Indeed, cf. Theorem 2.1, in order for u 2 x, t, ξ satisfying 3.22 to exist, the function Fx, t, ξ should have zero mean value with respect to ξ over Q: Fx, t, ξ ξ = Substituting 3.23 into 3.25 and using 3.1 and 3.16, we obtain: ρ vx, t σ i x, t = fx, t, 3.26

17 Memory Effect in Homogenization of Viscoelastic Kelvin-Voigt Model with time dependent coefficients 17 where ρ = ρ ξ and σ i x, t = B ij x, t v x, t + x Âijx, t v x, t j x j t + Ê ij x, t, t v x, t + x D ij x, t, t v x, t dt j x j Importantly, the homogenized relations display the memory effect due to the integral terms in Expression 3.27 uses the notation  ij x, t = A ij x, t, ξ + B ik x, t, ξ ga j ξ k ξ, Bij = B ik x, t, ξδ kj I + gb j ξ k here B ij is the conventional homogenized tensor for B ij, cf. 2 and be ij x, t, t = bd ij x, t, t = * * A ik x, t, ξ N B j ξ k x, t t, t, ξ A ik x, t, ξ N A j ξ k x, t t, t, ξ + + ξ ξ + + * * B ik x, t, ξ 2 N B j ξ k t x, t t, t, ξ B ik x, t, ξ 2 N A j ξ k t x, t t, t, ξ The above memory terms are consistent with those derived for a particular case of scalar problems with time-independent coefficients in 1, which are known to be generally present, see 1 for some explicit examples. + + ξ ξ ξ, Existence and uniqueness of the homogenized solution In this section we study the existence and uniqueness of the homogenized solution v, which satisfies the following problem formally derived above: ρ vx, t σ i x, t = fx, t, v Ω,τ =, v t= = v t= =, 3.28 together with the constitutive relations 3.27 with memory. Theorem 3.2. Let f L 2, τ; H 1 Ω such that there exists a constant τ < τ, such that fx, t = for all t τ, and assume that H1 H3 hold. Then problem 3.28 admits a unique solution v H 1, τ; H 1 Ω. Proof: For any v in H 1, τ; H 1 Ω, we define the vectorial function h v by h v i x, t := t Ê ij x, t, t v x, t + x D ij x, t, t v x, t dt, j x j i = 1,..., n

18 18 Z. ABDESSAMAD, I. KOSTIN, G.PANASENKO, and V. P. SMYSHLYAEV and introduce the linear mapping L : H 1, τ; H 1 Ω H 1, τ; H 1 Ω by the relation Lv = ṽ, where ṽ satisfies the following problem: ρ ṽ B ij x, t ṽ  ij x, t ṽ = fx, t + hv i x, t3.29 x j x j ṽ = on Ω,3.3 ṽ t= =, ṽ t= =.3.31 We start the proof by showing that the homogenized coefficients ρ, Âij and B ij satisfy the same assumptions as those imposed on the coefficients ρ, A ij and B ij of the original problem. It is easy to check that ρ satisfies assumption H3. For the symmetry of Bij, first we use 3.2 to verify that = B mj is for all 1 i, j, m, s n. Second, we use the weak formulation of 3.15 see 3.2 and 5, p.151 to justify that 1 i, j, m, s n B ij ms = B qp lk Zps k g B j + ξ j I Z qm l g B i + ξ i I, 3.32 ξ where B ms ij Z ps k g j + ξ j I = 1 [ g ps ] j + ξ j δ ps + g ks j + ξ j δ ks. 2 ξ k ξ p Making use of the relation B qp lk = Bpq kl Let η = ηi m Rn n, such that ηi m from 3.32 the following inequality: ν η m i η m i in 3.32, we get B ms ij = B sm ji. = η i m. Taking H2 into account, we obtain B ms ij η m i η s j 2 ν n max 1 i n gb i 2 H n ηi m ηi m Thus, by using 3.21 in the last inequality, we prove that B ij are uniformly bounded and uniformly elliptic. Notice in passing that the above properties of symmetry, uniform ellipticity and boundedness for the homogenized tensor are known to hold for classical elliptic problems of linear elasticity, e.g. 2, according to which recipe ˆB is associated with B, as mentioned above. Similarly, it suffices to use 3.18 in the definitions of  ij and the time derivatives of  ij and B ij to prove that they are uniformly bounded on H n by a constant c depending only on ν. Thus we have proved that, under the assumptions of the Theorem, the homogenized coefficients satisfy assumptions H1 H3 although the constants of upper bounds may not be the same. Now we can apply Theorem 1.1 to problem for ṽ with a given v. In particular, if f, then ṽ H1,τ;H 1 Ω C 1 h v L2,τ;L 2 Ω.

19 Memory Effect in Homogenization of Viscoelastic Kelvin-Voigt Model with time dependent coefficients 19 By 3.18 the functions Nk A, N k B and their time derivatives N k A, N k B are uniformly bounded, and so the latter inequality implies that there exists a constant C 3 depending only on ν, ρ 1, τ and Ω this constant C 3 converges to zero when τ vanish, such that ṽ H1,τ;H 1 Ω C 3 v H1,τ;H 1 Ω Let τ = τ 1 > be small enough for C 3 to satisfy C 3 < 1, implying that for any given f the mapping L is contracting. Then the Banach fixed point theorem implies that L has a unique fixed point v = ṽ, which is the solution of problem 3.28 on Ω τ1. Since for t [, τ], we have vt H 1 Ω, we can then repeat the above argument to extend our solution to the time interval [τ 1, 2 τ 1 ], and so on. After a finite number of steps we construct a solution existing on the interval [, τ]. To prove uniqueness, it is enough to take into account that ṽ = v in 3.34 and deduce that for C 3 < 1, the zero function is the unique solution of problem 3.28 with f Justification of the asymptotics If v solves the homogenized problem 3.28, then, by its derivation, the solvability condition for the equation 3.22 for u 2 is satisfied. Hence there exists a solution u 2 x, t, ξ of Consider the following representation for the exact solution: u x, t = vx, t + N x, t, x + 2 u 2 x, t, x + r x, t Our aim is to obtain an estimate for the remainder r x, t for sufficiently small. In order to justify the asymptotic expansion of u we will impose additional regularity assumptions on the viscoelastic coefficients. Namely, we will assume that A ij and B ij are both smooth with respect to x and t, and periodic and piecewise smooth with respect to ξ. More precisely, we will assume that there exist disjoint periodic subdomains D m R n, m = 1,..., L, such that R n = L m=1d m and that each D m is in Hölder class C 1,β D m of periodic functions, with < β 1. We hence require the physical characteristics of the composite media to be smooth e.g. constant in ξ in each subdomain D m assumed itself having a sufficiently smooth boundary, but possibly discontinuous across their boundaries. Now by first assuming that vx, t, Nx, t, ξ and u 2 x, t, ξ are smooth with respect to x and t in Ω τ and piecewise smooth with respect to ξ, we will prove the convergence of u to v when the parameter tends to zero and establish the relevant error bounds. In the next section we will describe sufficient conditions which ensure the validity of these assumptions on vx, t, Nx, t, ξ and u 2 x, t, ξ.

20 2 Z. ABDESSAMAD, I. KOSTIN, G.PANASENKO, and V. P. SMYSHLYAEV Theorem 3.3. We assume that: H4- v, v C 3 Ω τ ; N, Ṅ C 2 Ω τ ; K; u 2, u 2 C 1 Ω τ ; K; and for m = 1,..., L, we have A ij, B ij C 2 Ω τ ; C 1,λ D m, with < λ < 1. Then there exists a constant C independent of such that u v + N L,τ;H 1 Ω + u v + Ṅ L 2,τ;H 1 Ω C 1/2, u v L,τ;L 2 Ω + u v L,τ;L 2 Ω C 1/2. Henceforth we denote for a fixed < ζ < 1 K := { u C 1,ζ D m, m = 1,..., L }. Proof: Let χ x be a differentiable function whose support belongs to the - neighbourhood of the boundary of Ω, such that χ Ω = 1, χ 1 and χ x j CΩ c, with a constant c independent of. Such a cut-off function χ exists, see e.g 2 and 9. Set ũ 2x, t = vx, t + 1 χ x N x, t, x + 2 u 2 x, t, x, noticing that ũ 2 thereby satisfies the boundary condition 2.2. By using H4 we notice that for all m = 1,..., L we have: A ij, B ij, N, Ṅ, u 2 and u 2 [ belong to C 1 Ω τ D m [, hence ] one can use the chain rule: wx, x/ = wx, ξ ]ξ=x/ + 1 wx, ξ, for any of the above ξ=x/ functions. Thus, we substitute r = u ũ 2 into the original equation 2.1. According to the derivation of the terms of expansion 3.35, the terms of the order 1 and will vanish. As a result, taking into account , 3.28 and the zero initial conditions for N and u 2, we obtain the following problem for r x, t: ρ r B ij x, t, x x r x r A ij x, t, = h 1 + h2 + h3, 3.36 i x j x j r = on Ω, r x, =, 3.37 r x, = χ x 1Ṅ x,, x + 2 χ 1 u 2 x,, x on Ω Here [ ] h 1 x, t := A ij x, t, ξ N x, t, ξ + B ij x, t, ξ Ṅ x, t, ξ x j x j [ + A ij x, t, ξ u2 x, t, ξ + B ij x, t, ξ ] u2 x, t, ξ + ρ x χ x 1 Nx, t, x + 2 ü 2 x, t, x, ξ= x ξ= x

21 Memory Effect in Homogenization of Viscoelastic Kelvin-Voigt Model with time dependent coefficients 21 and [ h 2 x, t := 2 A ij x, t, x u2 x, t, ξ + B ij x, t, x x j ] u2 x, t, ξ, x j ξ= x h 3 x, t := A ij x, t, x B ij x, t, x [ χ x N + 2 u 2 x j [ χ x x Ṅ + 2 u 2 j Now, using H4, we establish the following estimate: ] x, t, x x, t, x ]. h 1 L 2,τ;L 2 Ω + h 2 L 2,τ;L 2 Ω c, 3.39 with c denoting henceforth a constant whose precise value is insignificant and may change from line to line. Further, taking into account the above described properties of χ, we obtain cf. e.g. 2,9 h 3 L2,τ;L 2 Ω c 1/2, r H 1 Ω + r L2 Ω c. 3.4 Thus, applying Theorem 1.1 to problem and using estimates , we obtain the following inequality: and r L,τ;H 1 Ω + r L,τ;L 2 Ω + r L 2,τ;H 1 Ω c 1/2. Finally, we take into account that ũ 2 v + N L,τ;H 1 Ω + ũ 2 v + Ṅ L 2,τ;H 1 Ω c 1/2 ũ 2 v L,τ;L 2 Ω + ũ 2 v L,τ;L 2 Ω c to arrive at the claimed estimates. 4. Sufficient condition for regularity In this section we study the regularity properties of the terms of asymptotic expansion 3.1. In particular, we are interested in sufficient conditions for v to satisfy the assumptions of the Theorem 2.3 above. The appearance of a long memory integral term in the homogenized equation presents substantial technical complications for the study of the regularity of v, N and u 2. To simplify the matters, we prove here that the required regularity holds at least in a particular case. Namely, we consider the case when the elastic and viscous characteristics are proportional, i.e. there exists a constant κ > such that A ij = κ B ij, i, j = 1,..., n. 4.1

22 22 Z. ABDESSAMAD, I. KOSTIN, G.PANASENKO, and V. P. SMYSHLYAEV In this case 2.1 takes the form x ρ ü B ij x, t, x u + κ u = fx, t. x j Moreover, the main corrector of the asymptotic expansion 3.1 is in this case given by Nx, t, ξ = t e κ t t g k x, t, ξ x k v + κ v x, t dt, where, for any x in Ω, the function g k x,, H 1, τ; H n satisfies the equation B ij x, t, ξ g k = B ik x, t, ξ. 4.2 The latter can be verified by direct inspection of 3.7 when 4.1 is held. Combined with the expressions for the homogenized coefficients Âij, B ij, D ij and Êij, the latter formulae imply via : B ij = κ 1 g j  ij = B ij + B ik, Ê ij = ξ D ij =. k ξ Substituting these into , we conclude that the homogenized equation takes the form ρ vx, t B ij x, t v + κ v = fx, t, 4.3 x j with the memory term vanishing. Remind in passing that the memory term does not generally vanish 1. Lemma 4.1. Let m N such that m > n 2, and let Ω be an open bounded set of R n with C 2m+2 boundary. Assume that B ij C m+1 Ω τ ; L Q and f k L 2, τ; H 2m 2k Ω, k =,..., m Here f k is the kth time derivative of f, such that there exists a constant τ < τ, such that fx, t = for all t τ. Then v, v C m [ n 2 ] Ω τ. Proof: Let v be a solution of 4.3 supplemented by boundary and initial conditions as in By Theorem 2.2, v exists and is unique in H 1, τ; H 1 Ω. Throughout the proof we will use the following notation: ṽ := v + κ v and f := f + κ ρ v. Thus ṽ satisfies ρ ṽ ṽ B ij = x f, ṽ =, x Ω, ṽ t= =. 4.4 j The validity of the Lemma is known in the case when the coefficients B ij are independent of time see e.g. 6.

23 Memory Effect in Homogenization of Viscoelastic Kelvin-Voigt Model with time dependent coefficients 23 We will use the Faedo-Galerkin s method. Let B ij C m+1 Ω τ and f k L 2, τ; H 2m 2k Ω, k =,..., m. We will show that then ṽ k L 2, τ; H 2m+2 2k Ω, k =,..., m + 1, 4.5 and there exists a constant C depending only on m, τ, Ω and ν, such that m+1 k= ṽ k L2,τ;H 2m+2 2k Ω C m f k L2,τ;H Ω m 2k First notice that Lemma 2.3 implies that if B ij C m+1 Ω τ ; Q then the function g j solving 4.2 belongs to C m+1 Ω τ, thus we have B ij C m+1 Ω τ. i Let f L 2, τ; L 2 Ω and B ij C 1 Ω τ, and let us prove that 4.5 and 4.6 hold for m =. Let ṽ L 2, τ; H 1 Ω C[, τ]; L 2 Ω be a weak solution of 4.4, i.e. ρ ṽ, z Ω + I bb ṽ, z Ω = f, z Ω, z H 1 Ω in D, τ and ṽ =. 4.7 k= Hence v H 1, τ; H 1 Ω and then f L 2, τ; L 2 Ω. Thus, by using H1, we prove that for all integer p 1 there exists a unique function: p ṽ p t = d i pt w i x H 1, τ H 1 Ω, i=1 solving the approximate Galerkin s problem: for j = 1,..., p, ρ ṽ p t, w j Ω + I bb ṽ p t, w j Ω = ft, w j Ω in D, τ and ṽ p =. 4.8 Here w i i is an orthogonal basis of H 1 Ω. Multiplying 4.8 by d j p and summing them for j = 1,..., p, we obtain ρ ṽ p, ṽ p + I Ω B bṽ p, ṽ p Ω = f, ṽ p Then, by using 2.1 and the Gronwall s Lemma see proof of Theorem 1.1, we deduce that there exists a constant c which depends only on ν, τ, ρ 1 and Ω such that ṽ p L,τ;H 1 Ω + ṽ p L2,τ;L 2 Ω c f L2,τ;L 2 Ω. 4.9 The last estimate allows us to pass to the limit as p in a standard way and to check that ṽ L, τ; H 1 Ω, ṽ L 2, τ; L 2 Ω and satisfy 4.7. Thus, for almost all t [, τ] we have ft ṽt L 2 Ω and according to Thm 4, we obtain from 4.7 ṽ H2 Ω c f L2 Ω + ṽ L2 Ω + ṽ L2 Ω. Ω.

24 24 Z. ABDESSAMAD, I. KOSTIN, G.PANASENKO, and V. P. SMYSHLYAEV Finally we integrate the last inequality over [, τ] and employ 4.9 to obtain ṽ L2,τ;H 2 Ω + ṽ L2,τ;L 2 Ω C f L2,τ;L 2 Ω, which completes the proof of for m =. ii Assume that are valid for m N and let B ij C m+2 Ω τ, f k L 2, τ; H 2m+2 2k Ω, k =,..., m + 1. Differentiate 4.7 with respect to t and then check that ṽ satisfy 4.7 with the right hand side defined by: F = f + κ ρ v + ṽ Bij x j Relations of order m imply that for k =,..., m + 1, the function ṽ k belongs to L 2, τ; H 2m+2 2k Ω. Then, we deduce that F k belongs to L 2, τ; H 2m 2k Ω, k =,..., m. Further, from 4.4 at t =, we deduce that ṽ =. Thus by applying the induction assumption we get. ṽ k L 2, τ; H 2m+4 2k Ω, k = 1,..., m + 2. Taking into account 4.6, we obtain m+2 k=1 ṽ k L 2,τ;H 2m+4 2k Ω C m+1 k= f k L2,τ;H 2m+2 2k Ω. 4.1 According to , we have ṽ H 2m+4 Ω c f H 2m+2 Ω + ṽ H 2m+2 Ω + ṽ 2 L 2 Ω. Finally, we integrate the last inequality over [, τ] and use 4.1 to assert the following estimate: m+2 k= ṽ k L 2,τ;H 2m+4 2k Ω C m+1 k= f k L 2,τ;H 2m+2 2k Ω. Thus is proved. Finally, we will use these results to prove the lemma. Indeed, since equation 4.3 is accompanied by trivial initial conditions v = v =, we have ṽ =. It results from that under assumption of the lemma we have v, v H m+1, τ; H 2m+2 Ω, and therefore we deduce the assertion of the lemma by the Sobolev embedding. Remark 4.1. Lemma 3.1 gives a sufficient condition for the inclusions v, v C 3 Ω τ which are assumed in H4: relations 4.1 and inclusions A ij, B ij C [ n 2 ]+4 Ω τ ; L Q, and for all k =, 1, 2, 3, f k L 2, τ; H 6 2k Ω. Henceforth we assume that these inclusions are valid.

25 Memory Effect in Homogenization of Viscoelastic Kelvin-Voigt Model with time dependent coefficients 25 Lemma 4.2. Let D m R n, m = 1,..., L be disjoint periodic subdomains such that each D m is of class C 1,β with < β 1 and R n = L m=1d m. We also assume that B ij x, t, ξ are 1-periodic with respect to ξ, such that B ij C 2 Ω τ ; C,λ D m, < λ < 1, m = 1,..., L. Then N, Ṅ C 2 Ω τ ; K, u 2, u 2 C 1 Ω τ ; K. Proof: By taking account of 4.1 in 3.7 we check that for all x, t Ω τ the function N H 1 #Q satisfies the following equation B ij x, t, ξ Ṅ + κ N = B ij x, t, ξ v + κ v.4.11 x j Let D R n be a bounded subdomain such that Q D. By using the periodicity property of N and B ij, we can study 4.11 in D. By assumption, for all x, t in Ω τ we have B ij x, t, C,λ D m, m = 1,..., L. Then according to 21 see also 22 we conclude that for all x, t Ω τ, for any δ > and for all < λ 1 min{λ, }, multiplying Ṅ + κ N by e κt τ β 2 β+1 and integrating in τ, we have: N, Ṅ C 1,λ1 D m D δ, m = 1,..., L, where D δ = {ξ D; distξ, D > δ}. In particular, ξ N and ξ Ṅ are bounded. Let us recall that the earlier assumption B ij C [ n 2 ]+4 Ω τ ; L Q implies that N, Ṅ C 2 Ω τ ; H 1 #Q see Lemma 3.1. Thus we use the continuity of B ij on x and t with respect to C,λ D m -norm and that v, v C 1 Ω τ and we employ the same techniques as in the proof of Lemma 2.1 to prove that N,Ṅ CΩ τ ; C 1,λ1 D m D δ and that ξ N, ξ Ṅ CΩ τ ; L D δ. Similarly, we differentiate separately 4.11 with respect to x and t and we use that B ij C 1 Ω τ ; C,λ D m to check that there exists a constant < λ 2 < 1, such that N, Ṅ x, N x belong to CΩ τ ; C 1,λ2 D m D δ and that ξ N, ξ Ṅ x, ξ N x belong to CΩ τ ; L D δ, having denoted for brevity x N by N x. In the same way, by differentiating separately 4.11 two times with respect to t and x and using that B ij C 2 Ω τ ; C,λ D m we check that there exists a constant < λ 3 < 1, such that the third time derivative N 3 of N and N xx, Ṅ xx belong to CΩ τ ; C 1,λ3 D m D δ for m = 1,..., L, and that ξ N 3, ξ N xx, ξ Ṅ xx belong to CΩ τ ; L D δ. Thus we fix δ so that Q D δ. We deduce from the results above that N, Ṅ C 2 Ω τ ; C 1,γ D m for all m = 1,..., L, and that ξ N, ξ Ṅ C 2 Ω τ ; L D δ, here γ = min 1 i 3 λ i. Now we will study the problem of u 2 in D δ that will be denoted by D. To be