HOMOGENIZATION OF STRATIFIED THERMOVISCOPLASTIC MATERIALS. (µ ε (x, θ ε ) vε = f,

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1 QUARTERLY OF APPLIED MATHEMATICS VOLUME, NUMBER XXXX XXXX, PAGES S X(XX- HOMOGENIZATION OF STRATIFIED THERMOVISCOPLASTIC MATERIALS By NICOLAS CHARALAMBAKIS (Department of Civil Engineering, Aristotle University, GR 5424 Thessaloniki, Greece and FRANÇOIS MURAT (Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Boîte courrier 87, Paris Cedex 5, France Abstract. In the present paper we study the homogenization of the system of partial differential equations ρ ε (x vε (µ ε (x, θ ε vε = f, ( c ε (x, θ ε θε v = µε (x, θ ε ε 2, posed in a < x < b, < t < T, completed by boundary conditions on v ε and by initial conditions on v ε and θ ε. The unknowns are the velocity v ε and the temperature θ ε, while the coefficients ρ ε, µ ε and c ε are data which are assumed to satisfy < c µ ε (x, s c 2, < c 3 c ε (x, s c 4, < c 5 ρ ε (x c 6, c 7 µε s (x, s, cε (x, s c ε (x, s ω( s s. This sequence of one dimensional systems is a model for the homogenization of nonhomogeneous, stratified, thermoviscoplastic materials exhibiting thermal softening and temperature dependent rate of plastic work converted into heat. Under the above hypotheses we prove that this system is stable by homogenization. More precisely one can extract a subsequence ε for which the velocity v ε and the temperature θ ε converge to some homogenized velocity v and some homogenized temperature θ which solve a system similar to the system solved by v ε and θ ε, for coefficients ρ, Received November 2, 25 and, in revised form, January 3, Mathematics Subject Classification. Primary 74Q5, 74Q, 35B27; Secondary 74C, 74F5, 35Q72, 35M2, 35K55. Key words and phrases. Homogenization, thermoviscoplastic materials. address: charalam@civil.auth.gr address: murat@ann.jussieu.fr c XXXX Brown University

2 2 englishnicolas CHARALAMBAKIS AND FRANÇOIS MURAT µ and c which satisfy hypotheses similar to the hypotheses satisfied by ρ ε, µ ε and c ε. These homogenized coefficients ρ, µ and c are given by some explicit (even if sophisticated formulas. In particular, the homogenized heat coefficient c in general depends on the temperature even if the heterogeneous heat coefficients c ε do not depend on it. Résumé. Dans cet article, nous étudions l homogénéisation du système d équations aux dérivées partielles ρ ε (x vε (µ ε (x, θ ε vε = f, ( c ε (x, θ ε θε v = µε (x, θ ε ε 2, posé dans a < x < b, < t < T et complété par des conditions aux limites sur v ε et des conditions initiales sur v ε et θ ε. Les inconnues sont la vitesse v ε et la température θ ε, alors que les coefficients ρ ε, µ ε et c ε sont des données qui vérifient < c µ ε (x, s c 2, < c 3 c ε (x, s c 4, < c 5 ρ ε (x c 6, c 7 µε s (x, s, cε (x, s c ε (x, s ω( s s. Cette suite de problèmes unidimensionnels modélise l homogénéisation de matériaux thermo-viscoplastiques hétérogènes dont la résistance diminue avec la température, et dont le taux de travail plastique converti en chaleur dépend de la température. Sous les hypothèses ci-dessus, nous démontrons que ce système est stable par homogénéisation. Plus précisément, on peut extraire une sous-suite ε pour laquelle la vitesse v ε et la température θ ε convergent vers une vitesse homogénéisée v et une température homogénéisée θ qui sont solution d un système similaire à celui dont v ε et θ ε sont solution, pour des coefficients ρ, µ et c qui satisfont des hypothèses analogues à celles satisfaites par ρ ε, µ ε et c ε. Les coefficients homogénéisés ρ, µ et c sont donnés par des formules explicites (même si elles sont assez compliquées. En particulier le coefficient thermique homogénéisé c dépend en général de la température, même si les coefficients thermiques hétérogènes c ε n en dépendent pas. Ë ÒÓÝ º ËØ Ò Ö ÙØ Ñ Ð ØÓ Ñ α < x < b < t < T Ø Ò Ó¹ ÑÓ ÒÓÔÓ ØÓÙ Ù Ø Ñ ØÓ ØÛÒ ÓÖ ôò Ü ô ÛÒ Ñ Ñ Ö Ô Ö ô ÓÙ ρ ε (x vε (µ ε (x, θ ε vε = f, ( c ε (x, θ ε θε v = µε (x, θ ε ε 2, Ñ ÙÒÓÖ ÙÒ Ø ÙÒ ÖØ v ε ÖÕ ÙÒ Ø ÙÒ ÖØ v ε θ ε º Ç ÒÛ Ø ÙÒ ÖØ Ò Ø Õ Ø Ø v ε ÖÑÓ Ö θ ε Òô Ó

3 englishhomogenization OF STRATIFIED THERMOVISCOPLASTIC MATERIALS 3 ÙÒØ Ð Ø ρ ε µ ε c ε Ò ÓÑ Ò ÙÒ ÖØ ÒÓÔÓ Ó Ò < c µ ε (x, s c 2, < c 3 c ε (x, s c 4, < c 5 ρ ε (x c 6, c 7 µε s (x, s, cε (x, s c ε (x, s ω( s s. À ÓÐÓÙ ÙØ ÑÓÒÓ Ø ØÛÒ Ù Ø Ñ ØÛÒ Ò Ò ÑÓÒØ ÐÓ Ø ÙÑÔ Ö ÓÖ ÒÓÑÓ Ó ÒôÒ ØÖÛÑ Ø ÑÓÖ ÛÒ ÖÑÓ ÜÛÔÐ Ø ôò ÙÐ ôò ÔÓÙ Õ Ö Ø ÖÞÓÒØ Ô ÖÑÓÑ Ð ØÓÔÓ ÔÐ Ø Õ Ü ÖØôÑ Ò Ô Ø ÖÑÓ Ö º Å Ø ÒÛØ ÖÛ ÔÖÓÙÔÓ ÔÓ Ò ÓÙÑ Ø ØÓ Ø Ñ ÙØ Ò Ù Ø Û ÔÖÓ Ø Ò ÓÑÓ ÒÓÔÓ º È Ó Ù Ö Ñ Ò Ø ÑÔÓÖÓ Ñ Ò Ü ÓÙÑ Ñ ÙÔÓ ÓÐÓÙ ε Ø Ò ÓÔÓ Ø Õ Ø Ø v ε ÖÑÓ Ö θ ε Ù ÐÒÓÙÒ ÔÖÓ ÔÓ ÓÑÓ ÒÓÔÓ ¹ Ñ Ò Ø Õ Ø Ø v ÖÑÓ Ö θ ÔÓÙ Ô Ð ÓÙÒ Ò Ø Ñ Ø ÑÓÖ Ñ ØÓ Ø Ñ ÔÓÙ Ô Ð ÓÙÒ Ó v ε θ ε ÙÒØ Ð Ø ρ µ c ÔÓÙ ÒÓÔÓ Ó Ò ÙÔÓ ÑÓ Ñ Ø ÙÔÓ ÔÓÙ ÒÓÔÓ Ó Ò Ó ρ ε µ ε c ε º ÙØÓ Ó ÓÑÓ ÒÓÔÓ Ñ ÒÓ ÙÒ¹ Ø Ð Ø ρ µ c ÓÒØ Ô ÜÓ Ò ÔÓÐ ÔÐÓ µ Ö º Ø Ö Ó ÓÑÓ ÒÓÔÓ Ñ ÒÓ ÖÑ ÙÒØ Ð Ø c Ü ÖØ Ø Ô Ø ÖÑÓ Ö Ñ Ò Ó ÒÓÑÓ Ó Ò ÙÒØ Ð Ø c ε Ò Ü ÖØôÒØ Ô ÙØ Òº Contents. Introduction 4 2. Setting of the problem and homogenization result Hypotheses Existence, uniqueness and continuity with respect to the data result Regularity results 2.4. Homogenization result 3. Definition of the homogenized viscosity function µ and of the homogenized heat coefficient c Transformation of the problem Definition of the homogenized functions µ and c Proofs of Propositions 3.3 and Proof of the regularity Theorem Proof of the homogenization Theorem First case: the case where fε is bounded in L 2 (, T; H ( Second case: the case where f ε is compact in L 2 (Q An example showing that the homogenized heat coefficient c depends in general on the temperature 38 References 42

4 4 englishnicolas CHARALAMBAKIS AND FRANÇOIS MURAT. Introduction. In the present work we consider the homogenization of nonhomogeneous, stratified, thermoviscoplastic materials exhibiting thermal softening which are subjected to simple shearing. In mathematical terms we consider the system of partial differential equations (posed in the given one dimensional space interval (a, b, a < b, and in the given time interval (, T, T > ρ ε (x vε = σε + f(t, x in (, T (a, b, (. c ε (x, θ ε θε = σε vε in (, T (a, b, (.2 σ ε = µ ε (x, θ ε vε in (, T (a, b, (.3 where the unknowns are the velocity v ε, the temperature θ ε and the stress σ ε, while the data are the density ρ ε (x, the inertial forces f(t, x, the heat coefficient c ε (x, s and the viscosity function µ ε (x, s. The system (. (.3 has to be completed by initial conditions on v ε and θ ε, namely v ε (, x = v ε (x, θ ε (, x = θ ε (x in (a, b, (.4 where v ε(x and θε (x are given, and by boundary conditions which impose the shearing, namely either v ε (t, a = va ε (t, vε (t, b = vb ε (t in (, T, (.5 where va ε(t and vε b (t are given, when the boundary velocities are imposed (Dirichlet boundary conditions, or σ ε (t, a = σa ε (t, σε (t, b = σb ε (t in (, T, (.6 where σa ε(t and σε b (t are given, when the boundary stresses are imposed (Neumann boundary conditions, or finally v ε (t, a = va ε (t, σε (t, b = σb ε (t in (, T (.7 where va ε(t and σε b (t are given, when the velocity is imposed in x = a and the stress is imposed in x = b (mixed boundary conditions. The main features of the data are first the fact that ρ ε, c ε and µ ε are bounded from above and from below by strictly positive constants, second the fact that the functions c ε are uniformly (in x and ε continuous with respect to s, and that the functions µ ε are uniformly (in x and ε Lipschitz continuous and non increasing with respect to s, and third the fact that all the functions ρ ε, c ε and µ ε depend on x in a measurable (and not necessarily continuous way. The latest property means that the materials under consideration are heterogeneous. Homogenization consists in studying the limit of the problems (. (.3 when ε, which represents the typical size of heterogeneities, tends to zero. Ten years ago, a new generation of heterogeneous materials, called functionally graded materials, appeared in the mechanical literature (see e.g. [], [2], [3], [4] and []. These materials, characterized by high resistance to loading and/or to temperature increase, have been studied intensively, and numerical homogenization formulas have been proposed under the assumption of smoothly varying fields, which means that the different components of these materials are supposed to be perfectly bonded by thermomechanical processing in order to exhibit continuously changing properties. In contrast, our analysis does not assume any continuity of the coefficients with respect to the variable x.

5 englishhomogenization OF STRATIFIED THERMOVISCOPLASTIC MATERIALS 5 The material described by (. (.3 is stratified in the sense that it is made of layers perpendicular to a given direction of space, here the x direction, while the shearing excerced by the boundary conditions is perpendicular to that direction, which allows one to reduce to one dimensional (in space partial differential equations. Homogenization then consists of considering a (very large number of (very fine layers (of tickness, say, ε made of different materials, and of describing the overall behaviour of the velocity v ε, temperature θ ε and stress σ ε for fixed external forces. In mathematical terms, this overall behaviour is expressed by the weak limits v, θ and σ of v ε, θ ε and σ ε, as ε tends to zero. We prove in the present paper that v, θ and σ satisfy a system of equations of the type (. (.3, corresponding to coefficients ρ (x, c (x, s and µ (x, s. This is the main result of the present paper. It means that system (. (.3 is stable under homogenization. In this process, the homogenized density ρ is obtained as the weak star limit of ρ ε, but such is not the case neither for the homogenized heat coefficient c (x, s nor for the homogenized viscosity function µ (x, s, which have to be defined through a much more sophisticated process (see Remark 3.5, where this process is summarized. In particular, a rather strange phenomenon occurs as far as c ε and c are concerned. Even if in problem (. (.3 the heat coefficients c ε (x, s do not depend on s, i.e. even if c ε (x, s = c ε (x, the homogenized heat coefficient c (x, s does depend in general on s (see Theorem 6., where explicit formula are given in the case of a material made of layers of some given homogeneous phases. This mathematical result is in accordance with recent experiments and theoretical mechanical studies based on temperature-dependent fraction of plastic work converted into heating (see [], [2] and [3]. Indeed the heat coefficient c is given by c = β ρh, where h is the specific heat coefficient and β the rate of plastic work converted into heating, a quantity which is related to the rearrangement of crystals during deformation. It is in general assumed in the literature that β =.9. In particular, β is constant with respect to the temperature θ. If ρ and h are assumed to be constant with respect to θ, which is a realistic assumption, then c does not depend on θ. Since our result shows that c depends on θ for homogenized materials, the hypothesis β =.9 is unrealistic. As experimental measurement of β for different temperatures is very difficult, estimating c may be very useful for understanding the overall behavior of stratified materials under high strain rates. Homogenization is now a well established mathematical theory, at least as elliptic or parabolic partial differential equations are concerned. We will only refer to the method of Tartar [5], and also, for general references, to the books of Sanchez-Palencia [4] and Bensoussan, Lions and Papanicolaou [6], even if those two books are concerned with the special case of periodic coefficients, an hypothesis which is not assumed in the present work. Let us explicitly observe that in the one dimensional (in space case, the homogenized coefficients can be in general expressed by formula involving the weak limits of the heterogeneous coefficients, or more exactly of some (nonlinear functions of them. Such is the case in the present paper.

6 6 englishnicolas CHARALAMBAKIS AND FRANÇOIS MURAT However, from the bibliographical standpoint, to the best of our knowledge, there is no paper studying the homogenization of thermoviscoplastic materials, even if the existence of solutions for (. (.3 has been often studied in the literature. As far as existence of a solution is concerned, we just quote the pionnering work of Dafermos and Hsiao [9], the works of Tzavaras [6] and [7], the recent paper [5] which is concerned with the numerical analysis of the problem, and our papers [7] and [8] (the second one presents existence and uniqueness results of the weak solution of (. (.3, as well as its approximation by finite elements. The plan of the paper is as follows. In Section 2, we give the precise hypotheses under which we study the problem (Subsection 2., then we recall a result of existence, uniqueness and continuity with respect to the data (Subsection 2.2 and finally we state regularity results (Subsection 2.3. Subsection 2.4 is devoted to the statement of the homogenization result (Theorem 2.5, which is the main result of the present paper. The rest of the paper is mainly concerned with the proof of this Theorem. We begin in Section 3 by defining the homogenized viscosity function µ and the homogenized heat coefficient c (Subsection 3.2, see in particular Remark 3.5. Before of that, we motivate these definitions by a change of unknown functions, which transforms the problem in an equivalent, but simpler one (Subsection 3.. The proofs concerned with the definitions of the homogenized viscosity function µ and heat coefficient c are given in Subsection 3.3. We then pass in Section 4 to the proof of the regularity (in time result stated in Subsection 2.3, and then in Section 5 to the proof the homogenization result. Finally Section 6 is devoted to the study of the model example in which an heterogeneous material is made of fine layers (of size ε of some homogeneous given phases. In this case we are able to give explicit formulas (Theorem 6. for the homogenized density ρ, the homogenized heat coefficient c and the homogenized viscosity function µ. These explicit formula show that in general the homogenized heat coefficient c depend on the temperature, even if the heat coefficients of the various phases do not depend on it. 2. Setting of the problem and homogenization result. 2.. Hypotheses. In the whole of the present paper, we consider a, b and T in R with a < b and T >, and we set (a, b =, (, T (a, b = Q. We also consider a sequence of strictly positive numbers ε which tend to zero. We consider a sequence of Carathéodory functions µ ε : R R (the viscosity functions which are decreasing in s and uniformly (in x and ε Lipschitz continuous in s, i.e. x µ ε (x, s is measurable s R, (2. c 7 (s s µ ε (x, s µ ε (x, s a.e. x, s, s R, s > s,

7 englishhomogenization OF STRATIFIED THERMOVISCOPLASTIC MATERIALS 7 where c 7 > is a given constant. We also assume that c µ ε (x, s c 2 a.e. x, s R, (2.2 where < c c 2 < + are given constants. We also consider a sequence of Carathéodory functions c ε : R R (the heat coefficients which are uniformly (in x and ε continuous in s, i.e. x c ε (x, s is measurable s R, (2.3 c ε (x, s c ε (x, s ω( s s a.e. x, s, s R, where ω is a modulus of continuity, i.e. a non decreasing continuous function ω : R + R + with ω( =. We also assume that c 3 c ε (x, s c 4 a.e. x, s R, (2.4 where < c 3 c 4 < + are given constants. We finally consider a sequence of L ( functions ρ ε (the densities. We assume that c 5 ρ ε (x c 6 a.e. x, (2.5 where < c 5 c 6 < + are given constants, and that ρ ε ρ in L ( weak star. (2.6 On the other hand, we consider sequences of external forces f ε, of Dirichlet boundary conditions va ε and vb ε, and of initial conditions vε and θ, ε and their limits f, va, vb, v and θ, which satisfy for a given constant K > f ε L 2 (Q, f L 2 (Q, f ε L 2 (Q K, (2.7 f ε f in L 2 (Q weak, va ε H (, T, vb ε H (, T, va H (, T, vb H (, T, va ε H (,T K, vb ε H (,T K, (2.8 va ε v a in H (, T weak, vb ε v b in H (, T weak, v ε H (, v H (, v ε H ( K, (2.9 v ε v in H ( weak, v ε (a = v ε a(, v ε (b = v ε b(, (2. θ ε L (, θ L (, γ L (, θ ε L ( K, θ ε θ in D ( weak star, θ ε γ in D ( weak star. (2.

8 8 englishnicolas CHARALAMBAKIS AND FRANÇOIS MURAT Observe that (2. are compatibility conditions between the boundary values and the initial value of v ε. Observe also introducing γ in hypothesis (2. is not necessary when θ ε, which is the physical case, since one has γ = θ in this case. Assuming that γ L ( will be used in (and only in the proof of Proposition 3.3 below Existence, uniqueness and continuity with respect to the data result. Theorems 2. and 3. of our paper [8] prove the following result of existence, uniqueness and local Lipschitz continuity with respect to the data for the case of Dirichlet boundary conditions (.5 (see also [7] for another proof of the existence result. Theorem 2. (Existence, uniqueness and local Lipschitz continuity for Dirichlet boundary conditions. Assume that hypotheses (2. (2. hold true. Then there exists a unique couple (v ε, θ ε which satisfies v ε L (, T; H ( H (, T; L 2 (, (2.2 θ ε W, (, T; L (, (2.3 ρ ε (x vε (µ ε (x, θ ε vε = f ε in D (Q, (2.4 ( c ε (x, θ ε θε v = µε (x, θ ε ε 2 in D (Q, (2.5 v ε (t, a = va ε (t, vε (t, b = vb ε (t a.e. t (, T, (2.6 Moreover the stress σ ε defined by satisfies v ε (, x = v ε (x a.e. x, (2.7 θ ε (, x = θ ε (x a.e. x. (2.8 σ ε = µ ε (x, θ ε vε and the following a priori estimates hold true (2.9 σ ε L (, T; L 2 ( L 2 (, T; H (, (2.2 v ε L (,T;H ( + vε L 2 (,T;L 2 ( C, (2.2 θ ε L (,T;L ( + θε L (,T;L ( C, (2.22 σ ε L (,T;L 2 ( + σ ε L 2 (,T;H ( C, (2.23 where C denotes a constant which depends only on c, c 2, c 3, c 4, c 5, c 6 and K.

9 englishhomogenization OF STRATIFIED THERMOVISCOPLASTIC MATERIALS 9 Finally, if (v ε, θ ε and (ˆv ε, ˆθ ε are the (unique solutions of (2. (2. for the data (f ε, va, ε vb ε, vε, θ ε and ( ˆf ε, ˆv a, ε ˆv b ε, ˆvε, ˆθ, ε and if these data satisfy (2.7 (2. for the same constant K, their difference satisfies v ε ˆv ε L (;T;L 2 ( + v ε ˆv ε L2 (,T;H ( + θ ε ˆθ ε L (,T;L ( C ( f ε ˆf ε L 2 (Q + va ε ˆvε a H (,T + vb ε ˆvε b H (,T + ( v ε ˆvε L 2 ( + θ ε ˆθ ε L (, where C denotes a constant which depends only on c, c 2, c 3, c 4, c 5, c 6, c 7, (b a and K. Observe that every term in equations (2.4 and (2.5 has a meaning in the sense of distributions, since µ ε (x, θ ε belongs to L (Q. Similarly the boundary conditions (2.6 have a meaning, since v ε belongs to L (, T; H ( and since H ( C (. Finally the initial conditions (2.7 and (2.8 have a meaning since v ε belongs to H (, T; L 2 ( C ([, T]; L 2 ( and since θ ε belongs to W, (, T; L ( C ([, T]; L (. The regularity (2.2 of the stress σ ε is very specific to the fact that the problem is one dimensional. This regularity immediately follows, for the first statement, from the definition (2.9 of σ ε and from (2.2, (2.2, and for the second statement, from equation (2.4, which reads as σε = ρε vε fε, and from (2.5, (2.2 and (2.7. Remark 2.2 (Neumann and mixed boundary conditions. As pointed out in [8], a result similar to the result of Theorem 2. still holds true when the Dirichlet boundary conditions (.5 on v ε are replaced either by the Neumann boundary conditions (.6 on σ ε or by the mixed boundary conditions (.7 on v ε and σ ε. In those cases, we consider sequences of boundary conditions σa ε and σε b which satisfy σa ε L 2 (, T, σb ε L2 (, T, σa L 2 (, T, σb L2 (, T, σa ε L 2 (,T K, σb ε L 2 (,T K, (2.25 σa ε σ a in L 2 (, T weak, σb ε σ a in L 2 (, T weak, when Neumann boundary conditions are concerned, and va ε H (, T, σb ε L2 (, T, va H (, T, σb L2 (, T, va ε H (,T K, σb ε L 2 (,T K, (2.26 va ε v a in H (, T weak, σb ε σ a in L 2 (, T weak, with the compatibility condition v ε (a = vε a (, (2.27 when mixed boundary conditions are concerned.

10 englishnicolas CHARALAMBAKIS AND FRANÇOIS MURAT Under these hypotheses, results of existence, uniqueness and local Lipschitz continuity with respect to the data similar to Theorem 2. continue to hold true in the case of Neumann or mixed boundary conditions Regularity results. In this Subsection, we state two regularity results on σ ε and v ε which will play a crucial role in the proof of the homogenization result. When = (a, b R, an interpolation result (see Lemma 3. of [8] asserts that L (, T; L 2 ( L 2 (, T; H ( L 6 (Q (2.28 (this result is specific to the one dimensional case. Combined with (2.2 and (2.23, this immediately implies the following regularity result on σ ε. Proposition 2.3 (L 6 Regularity of σ. Assume that the hypotheses of Theorem 2. hold true. Then the stress σ ε defined by (2.9 satisfies σ ε L 6 (Q, σ ε L 6 (Q CC, (2.29 where C is the constant which appears in the a priori estimate (2.23 and where C is a constant which depends only on (b a. v ε On the other hand, as stated by the following Theorem, is bounded in L loc (, T; L2 loc ( L2 loc ((, T; H fε loc (, when, further to the above hypotheses, is bounded in L 2 (, T; H (. Theorem 2.4 (Regularity of v. Assume that the hypotheses of Theorem 2. hold true and that f ε L 2 (, T; H (, (2.3 fε L 2 (,T;H ( K. For δ > sufficiently small, denote by δ the interval Then the unique solution (v ε, θ ε of (2.2 (2.8 satisfies with δ = (a + δ, b δ. (2.3 v ε L (δ, T δ; L 2 ( δ L 2 (δ, T δ; H ( δ, (2.32 vε L (δ,t δ;l 2 ( δ + vε L 2 (δ,t δ;h ( δ Cδ, (2.33 where Cδ denotes a constant which depends only on c, c 2, c 3, c 4, c 5, c 6, c 7, b a, K, K and δ.

11 englishhomogenization OF STRATIFIED THERMOVISCOPLASTIC MATERIALS Theorem 2.4 is similar to Theorem 3.2 of [8], except the fact that the regularity is now local in time (i.e. in (δ, T δ and not in (, T as well as in space (i.e. in δ and not in. This Theorem will be proved in Section 4 by a rather classical proof. Similar regularity results hold true as far as Neumann and mixed conditions are concerned Homogenization result. The main result of the present paper is the following. Theorem 2.5 (Homogenization. Consider sequences of viscosity functions µ ε, heat coefficients c ε, densities ρ ε, external forces f ε, boundary conditions va ε and vε b and initial conditions v ε and θε, which satisfy (2. (2.. Further to (2.7 assume that either (2.3, i.e. f ε is bounded in L 2 (, T; H (, (2.34 or that f ε f in L 2 (Q strongly, (2.35 holds true. Then there exist a subsequence ε, an homogenized viscosity function µ and an homogenized heat coefficient c which satisfy (2. (with a constant c 7 possibly different of c 7, (2.2, (2.3 (with a modulus of continuity ω possibly different of ω and (2.4 (with constants c 3 and c 4 possibly different of c 3 and c 4, such that the unique solution (v ε, θ ε of (2.2 (2.8 and the stress σ ε defined by (2.9 satisfy v ε v in L (, T; H ( weak star, (2.36 v ε v in L 2 (, T; L 2 ( weak, (2.37 θ ε θ in D (Q weak star, (2.38 { σ ε σ in L (, T; L 2 ( weak star and in L 2 (, T; H ( L 6 (2.39 (Q weak, where (v, θ is the unique solution of (2.2 (2.8 for the viscosity function µ, the heat coefficient c, the density ρ, the external force f, the boundary conditions v a and v b and the initial conditions v and θ, and where the stress σ is defined by Theorem 2.5 will be proved in Section 5. σ = µ (x, θ v. (2.4 Remark 2.6 (Summary of Theorem 2.5. Theorem 2.5 asserts that there exist a subsequence ε, an homogenized viscosity function µ (x, s and an homogenized heat conduction coefficient c (x, s. Actually the subsequence ε and the homogenized functions µ and c only depend (and do depend on the sequences µ ε, c ε and θ ε, but do not depend on the other data ρ ε, f ε, v ε a, v ε b and vε.

12 2 englishnicolas CHARALAMBAKIS AND FRANÇOIS MURAT The functions µ and c are explicitly constructed through a rather complicated process, which involves weak limits (for the subsequence ε and for every s fixed of some functions of µ ε (x, s, c ε (x, s and θ ε. This process will be described in Section 3 below, where the functions µ and c as well as the subsequence ε will be defined (see Propositions 3.3 and 3.4, and also Remark 3.5 where this process is summarized. Remark 2.7 (Stability by homogenization of c ε (x, s. Theorem 2.5 proves that the system (2.2 (2.8 is stable by homogenization, or in other terms is stable for sequences of viscosity functions µ ε, heat coefficients c ε and densities ρ ε which satisfy (2. (2.5. Note however that the subclass where c ε only depends on x, i.e. the class where c ε (x, s = c ε (x with c ε (x L (, (2.4 c 3 c ε (x c 4 a.e. x, is not stable by homogenization. Indeed we will exhibit in Section 6 below a sequence of heat coefficients c ε (x satisfying (2.4 for which the homogenized heat coefficient c (in the sense of Theorem 2.5 is a heat coefficient c (x, s which does depend on s. Remark 2.8 (Oscillation and non oscillation of the unknowns. Let us examine more in detail the behaviour of the quantities v ε, vε, vε, σε, θ ε and θε. We will assume in this Remark that hypothesis (2.34 holds true. In view of (2.2, the velocities v ε are bounded in H (Q, and therefore do not oscillate. On the other hand, in view of (2.32 and of Aubin s compactness Lemma, the accelerations vε are relatively compact in L 2 loc (Q, and therefore do not oscillate. Finally in view of (2.23, the stresses σ ε are bounded in L 2 (, T; H (. Also, since σ ε = µε (x, θ ε 2 v ε + µε s (x, θε θε v ε = = µ ε (x, θ ε 2 v ε + µε s (x, θε (µ ε (x, θ ε 2 c ε (x, θ ε (σε 3, the stress rates σε are bounded in L 2 loc (Q + L2 (Q L 2 loc (Q in view of (2.32 and (2.29. Thus the stresses σ ε are bounded in Hloc (Q, and therefore do not oscillate. On the other hand, in view of (2.22, the temperatures θ ε are bounded in W, (, T; L (, and therefore do not oscillate in time. Similarly the strain rates vε, which are given by v ε = µ ε (x, θ ε σε, (2.42 do not oscillate in time since the stresses σ ε do not oscillate and since the temperatures θ ε (and therefore the coefficients µ ε (x, θ ε do not oscillate in time. Finally the temperature increases θε, which are given by θ ε = µ ε (x, θ ε c ε (x, θ ε (σε 2, (2.43 do not oscillate in time for the same reason.

13 englishhomogenization OF STRATIFIED THERMOVISCOPLASTIC MATERIALS 3 In contrast, we claim that the temperatures θ ε do oscillate in space, which implies, in view of (2.42 and (2.43, that the strain rates vε and the temperature increases θ ε also do oscillate in space. Indeed in the proofs below (see Sections 3, 4 and 5 (and already in the proofs in [7] and [8], very important quantities, herein named the transformed temperatures τ ε (see (3.9 below, naturally appear. These transformed temperatures τ ε are proved to be bounded in W, (Q (see (3.34 below, and therefore do not oscillate. Such is not the case for the temperatures θ ε, since the formula θ ε (t, x = N ε (x, τ ε (t, x (see (3.2 below implies that the temperatures θ ε do oscillate in space. Remark 2.9 (Estimates on the constants. Estimates on the constants c 7, c 3, c 4 and on the modulus of continuity ω in terms of c, c 2, c 3, c 4, c 7 and ω are given in (3.49, (3.5 and (3.5. In particular, when the functions c ε (x, s are uniformly (in x and ε Lipschitz continuous in s, i.e. when ω( s = c 8 s, then also c is uniformly (in x Lipschitz continuous in s with a constant c 8 possibly different of c 8 (see Remark 3.7. Remark 2. (Non equivalence of hypotheses (2.34 and (2.35. Hypotheses (2.7 and (2.34 do not imply (2.35, and conversely hypothesis (2.35 not imply (2.34. Therefore neither of the two hypotheses (2.34 and (2.35 reduces to the other one, even if they are very similar in spirit. Remark 2. (Neumann and mixed boundary conditions. Homogenization results similar to Theorem 2.5 hold true as far as Neumann and mixed boundary conditions are concerned. We leave their statements and proofs to the reader. Let us emphasize that the homogenized viscosity function µ and heat coefficient c are independent of the type of the boundary conditions and are identical in these cases to the ones which appear in Theorem Definition of the homogenized viscosity function µ and of the homogenized heat coefficient c. 3.. Transformation of the problem. As in [7] and [8], we will perform in this Subsection a change of unknown functions and we will write the system (2.2 (2.8 in an equivalent but simpler form. Although this transformation is not needed in order to define the functions µ and c, it motivates these definitions, and plays an important role in the proof of Theorem 2.5. In this Subsection, we will also give a regularity result on a very important quantity, the transformed temperature. We first make a translation which reduces the problem to homogeneous boundary conditions. (Such a translation has not to be done when Neumann boundary conditions

14 4 englishnicolas CHARALAMBAKIS AND FRANÇOIS MURAT are concerned, and it has to be conveniently modified when mixed boundary conditions are concerned. We define v ε (t, x = (x avε b (t + (b xvε a(t, (3. b a u ε (x = vε (x vε (, x, (3.2 u ε (t, x = v ε (t, x v ε (t, x. (3.3 Note that u ε H ( (3.4 when v ε, vε a and vb ε satisfy hypotheses (2.8, (2.9 and (2.. Then (v ε, θ ε is a solution of (2.2 (2.8 if and only if (u ε, θ ε is a solution of ρ ε (x uε ( ( u µ ε (x, θ ε ε + vε = f ε ρ ε (x vε in D (Q, (3.5 ( c ε (x, θ ε θε u = µε (x, θ ε ε 2 + vε in D (Q, (3.6 u ε (t, a =, u ε (t, b = a.e. t (, T, (3.7 u ε (, x = u ε (x a.e. x, (3.8 θ ε (, x = θ ε (x a.e. x. (3.9 We then perform a change of unknown function by replacing θ(t, x by a new unknown τ(t, x. We define the functions M ε : R R by M ε (x, s = s Since in view of (2.2 and (2.4 one has θ ε (x c ε (x, s µ ε (x, s ds a.e. x, s R. (3. < c 3 c c ε (x, s µ ε (x, s c 4 c 2 < +, the functions s M ε (x, s are Carathéodory functions which are uniformly (in x and ε Lipschitz continuous in s and which satisfy c 3 c Mε s (x, s c 4c 2 a.e. x, s R. (3. Moreover the functions s Mε s (x, s = cε (x, sµ ε (x, s are Carathéodory functions which are uniformly (in x and ε continuous in s, since in view of (2. (2.4 one has M ε Mε (x, s s s (x, s c 7c 4 s s + c 2 ω( s s a.e. x, s, s R. (3.2 Finally for almost every x, the function s M ε (x, s is one-to-one from R onto R and satisfies M ε (x, θ ε (x = a.e. x. (3.3 We define N ε (x, r as the reciprocal function of M ε (x, s, i.e. M ε (x, s = r N ε (x, r = s a.e. x, s, r R. (3.4

15 englishhomogenization OF STRATIFIED THERMOVISCOPLASTIC MATERIALS 5 The function r N ε (x, r is a one-to-one function from R onto R and satisfies Since M ε (x, N ε (x, r = r, the chain rule yields N ε (x, = θ ε (x a.e. x. (3.5 M ε s (x, Nε (x, r Nε (x, r =. (3.6 r Thus the functions r N ε (x, r are Carathéodory functions which are uniformly (in x and ε Lipschitz continuous in s and which satisfy Nε (x, r a.e. x, s R. (3.7 c 4 c 2 r c 3 c Moreover, the functions s Nε (x, r are Carathéodory functions which are uniformly r (in x and ε continuous in s, since in view of (3.6, (3. and (3.2, one has N ε Nε (x, r r r (x, ε M r = s (x, Nε (x, r Mε s (x, Nε (x, r M ε s (x, Nε (x, r Mε s (x, Nε (x, r ( (c 3 c 2 c 7 c 4 N ε (x, r N ε (x, r + c 2 ω( N ε (x, r N ε (x, r, which using (3.7 implies N ε Nε (x, r r r (x, r c 7c 4 c 3 r r + c ( 2 r r 3 c3 c 2 ω 3 c2 c 3 c a.e. x, r, r R. We now define the transformed temperature τ ε by (3.8 τ ε (t, x = M ε (x, θ ε (t, x. (3.9 In view of the definition (3.4 of the reciprocal function N ε, definition (3.9 is equivalent to θ ε (t, x = N ε (x, τ ε (t, x. (3.2 The reason for the definition of the transformed temperature τ ε is that, multiplying equation (3.6 by µ ε (x, θ ε (which is bounded from above and from below by strictly positive constants, we see that (3.6 is equivalent to τ ε = µε (x, θ ε ( u ε + vε while the initial condition (3.9 is equivalent to 2 τ ε (, x = a.e. x. We finally define the function λ ε : R R by in D (Q, λ ε (x, r = µ ε (x, N ε (x, r a.e. x, r R. (3.2 In view of (2., (2.2, and (3.7 the functions λ ε are Carathéodory functions which are Lipschitz continuous in r and which satisfy

16 6 englishnicolas CHARALAMBAKIS AND FRANÇOIS MURAT Then we have c λ ε (x, r c 2 a.e. x, r R, (3.22 c 7 λε (x, r a.e. x, r R. (3.23 c 3 c r µ ε (x, θ ε (t, x = λ ε (x, τ ε (t, x, (3.24 ( σ ε (t, x = µ ε (x, θ ε (t, x vε u (t, x = λε (x, τ ε ε vε (t, x (t, x + (t, x. (3.25 It is now straightforward to prove the following equivalence result. Proposition 3. (Equivalence. Assume that hypotheses (2. (2. hold true, and define v ε, u ε, Mε, N ε and λ ε by (3., (3.2, (3., (3.4 and (3.2. Defining u ε and τ ε by (3.3 and (3.9, the couple (v ε, θ ε satisfies (2.2 (2.8 if and only if the couple (u ε, τ ε satisfies u ε L (, T; H ( H (, T; L 2 (, (3.26 τ ε W, (, T; L (, (3.27 ρ ε (x uε ( ( u λ ε (x, τ ε ε + vε = f ε ρ ε (x vε in D (Q, (3.28 τ ε ( = u λε (x, τ ε ε + vε 2 in D (Q, (3.29 u ε (, x = u ε (x a.e. x, (3.3 τ ε (, x = a.e. x. (3.3 Moreover the stress σ ε defined by (2.9 is equivalently defined by (3.25, and satisfies σ ε L (, T; L 2 ( L 2 (, T; H (. (3.32 Finally the norms of v ε and θ ε in the spaces L (, T; H ( H (, T; L 2 ( and W, (, T; L ( and the norms of u ε and τ ε in the spaces L (, T; H ( H (, T; L 2 ( and W, (, T; L ( are equivalent, with constants in the equivalences which depend only on c, c 2, c 3, c 4, (b a and K. Note that the initial conditions (3.3 and (3.3 have a meaning since H (, T; L 2 ( C ([, T]; L 2 ( and since W, (, T; L ( C ([, T]; L (. Actually, the transformed temperature τ ε does not only belong to W, (, T; L ( as stated in (3.27, but also enjoys regularity properties which are not enjoyed by the temperature θ ε. Indeed we have the following result of regularity of τ ε (which goes back to Proposition 3. of [7], see also Proposition 3. of [8].

17 englishhomogenization OF STRATIFIED THERMOVISCOPLASTIC MATERIALS 7 Proposition 3.2 (Regularity of the transformed temperature. Assume that hypotheses (2. (2. hold true. Then the transformed temperature τ ε defined by (3.9 satisfies τ ε W, (, T; L ( W, (, T; H ( H (, T; W, (, (3.33 and the following a priori estimates hold true τ ε W, (,T;L ( + τ ε W, (,T;H ( + τ ε H (,T;W, ( F, (3.34 where F depends only on c, c 2, c 3, c 4, c 5, c 6 and K. The proof of Proposition 3.2 just consists in writing (see (3.25, (3.29 and (3.3 τ ε = σε 2, τ ε (, x =, which using (2.2 and (2.23 implies that τ ε W, (, T; L ( and the corresponding a priori estimate, but which also implies that ( τ ε = 2σ ε σε, τ ε (, x =, which using again (2.2 and (2.23 implies the other assertions of Proposition Definition of the homogenized functions µ and c. The goal of this Subsection is to state results which define the functions µ and c which appear in Theorem 2.5. These results will be proved in Subsection 3.3. For convenience the statements are divided into two Propositions. Proposition 3.3 (Definition of the subsequence ε and of the functions N and λ. There exist a subsequence ε and two Carathéodory functions N and λ : R R, with r N (x, r one-to-one and strictly increasing from R onto R for almost every x, which satisfy for almost every x and for every r, r R N (x, = θ (x, (3.35 N (x, r, (3.36 c 4 c 2 r c 3 c N N (x, r r r (x, r c 7c 4 c 3 r r + c ( 2 r r 3 c3 c 2 ω, ( c2 c 3 c c λ (x, r c 2, (3.38 c 7c 2 2 c 3 c 3 λ (x, r, (3.39 r such that for the functions N ε and λ ε defined by (3.4, (3. and (3.2, one has for every r R N ε (, r N (, r in D ( weak star, (3.4 λ ε (, r λ (, r in L ( weak star. (3.4

18 8 englishnicolas CHARALAMBAKIS AND FRANÇOIS MURAT From N and λ defined in Proposition 3.3, we define µ and c in the following way. Proposition 3.4 (Definition of the homogenized functions µ and c. Define first M (x, s as the reciprocal function of N (x, r, i.e. M (x, s = r N (x, r = s a.e. x, s, r R. (3.42 Then M : R R is a Carathéodory function, with s M (x, s one-to-one and strictly increasing from R onto R for almost every x, which satisfies for almost every x and for every s, s R M (x, θ (x =, (3.43 c 3 c M s (x, s c 4c 2, (3.44 M M (x, s s s (x, s c 7c 4 ( 4 c3 2 c 3 s s + c2 4 c3 2 c4 c 2 3 c3 c 2 ω s s. ( c2 c 3 c Define then µ and c : R R by µ (x, s = λ (x, M (x, s, (3.46 c (x, s = M s (x, s µ (x, s, (3.47 for almost every x and for every s R. Then µ and c are Carathéodory functions which satisfy for almost every x and for every s, s R, s s c µ (x, s c 2, (3.48 c 7c 4 c 3 2 c 3 c 3 (s s µ (x, s µ (x, s, (3.49 c 3 c c (x, s c 4c 2, (3.5 c 2 c ( c (x, s c (x, s c7 c 4 ] 4c 3 2 c 3 + c 7c 2 4c c4 c 3 c 5 s s + c2 4c 3 ( 2 c4 c 2 c 2 ω s s. (3.5 3 c3 c 3 c For these functions µ and c and for the subsequence ε defined in Proposition 3.3, Theorem 2.5 holds true. Remark 3.5 (Summary of the definitions of the homogenized functions µ and c. Let us summarize the way in which the homogenized functions µ and c and of the subsequence ε are defined. From the functions µ ε (x, s and c ε (x, s and from the initial datum θ ε (x, we define the function M ε (x, s by (3., i.e. by M ε s (x, s = cε (x, sµ ε (x, s, M ε (x, θ ε (x =, and the function r N ε (x, r as the reciprocal function of s M ε (x, s (see (3.4. We then set (see (3.2 λ ε (x, r = µ ε (x, N ε (x, r.

19 englishhomogenization OF STRATIFIED THERMOVISCOPLASTIC MATERIALS 9 The subsequence ε and the functions N (x, r and λ (x, r are then choosen such that for every fixed r R N ε (x, r N (x, r λ ε (x, r λ (x, r in D ( weak star, in L ( weak star. Finally, the function s M (x, r is defined as the reciprocal function of r N (x, r (see (3.42. We then set µ (x, s = λ (x, M (x, s, (see (3.46, and finally define c by c (x, s = M s (x, s µ (x, s. We will give in Section 6 explicit formulas for µ and c (see formulas (6.2 and (6.5 and Theorem 6. in the case of a material made of layers of given homogeneous phases characterized by densities ρ i (x = ρ i, viscosity functions µ i (x, s = µ i (s and heat coefficients c i (x, s = c i (s. Remark 3.6 (Dependance of the functions µ and c on µ ε, c ε and θ ε. Note that the subsequence ε, the homogenized viscosity function µ and the homogenized heat coefficient c depend on (and depend only on the sequences µ ε (x, s, c ε (x, s and θ ε (x. Let us emphasize the µ and c do depend on the sequence of initial temperatures θ ε. Remark 3.7 (Case where the functions c ε are Lipschitz continuous in s. When the functions c ε (x, s are uniformly (in x and ε Lipschitz continuous in s, i.e. when or in other terms when ω( s = c 8 s, c ε (x, s c ε (x, s c 8 s s a.e. x, s, s R, then formula (3.5 implies that c is also uniformly (in x Lipschitz continuous in s with a constant c 8 possibly different of c Proofs of Propositions 3.3 and 3.4. We begin with a classical Lemma which is a basic tool in the proofs of Propositions 3.3 and 3.4. Lemma 3.8 (Extracting a weak converging subsequence for Carathéodory functions. Let E be an open bounded set, E R m, m, and let F ε : E R k R, k, be a sequence of Carathéodory functions which satisfy, for almost every x E and for every r, r R k F ε (x, r F ε (x, r ω( r r, (3.52 F ε (x, G ε (x, (3.53

20 2 englishnicolas CHARALAMBAKIS AND FRANÇOIS MURAT where ω is a modulus of continuity and where G ε is a sequence of functions in L (E such that G ε bounded in L (E, G L (E, (3.54 G ε G in D (E weak star. Then there exist a subsequence ε and a Carathéodory function F : E R k R which satisfies (3.52 and F (x, G (x a.e. x E, (3.55 such that for every fixed r R k Moreover, if in (3.52 one has F ε (, r F (, r in D (E weak star. (3.56 ω( r = C r (3.57 for some constant C >, i.e. if F ε is uniformly (in x and ε Lipschitz continuous in r, and if z ε is a sequence such that then z ε z in L (E k strong, (3.58 F ε (, z ε F (, z in D (E weak star. (3.59 Proof of Lemma 3.8. First step: Definition of the subsequence ε and of the function F. The estimate F ε (x, r G ε (x + ω( r a.e. x, r Rk, (3.6 which follows from (3.52 and (3.53, implies that F ε (, r is bounded in L (E for every fixed r R k. Therefore one can extract a (diagonal subsequence ε such that for every q Q k F ε (, q F q in D (E weak star, (3.6 where F q belongs to D (E. For r =, one deduces from (3.53, (3.54 and (3.6 that G F G which implies that F L ( and satisfies in D (E, F (x G (x a.e. x E. (3.62 Moreover, for every ϕ Cc (E one has (see (3.52 (F ε (x, q F ε (x, q ϕ(xdx ϕ L (E ω( q q. Passing to the limit along the subsequence ε thanks to (3.6 implies that for every ϕ C c (E and for every q, q Q k < F q F q, ϕ > ϕ L (E ω( q q,

21 englishhomogenization OF STRATIFIED THERMOVISCOPLASTIC MATERIALS 2 which implies that F q F q belongs to L (E with Since F L (E, this implies that F q F q L (E ω( q q q, q Q k. (3.63 F q L (, F q F q L (E E ω( q q q, q Q k. (3.64 Therefore F q n is a Cauchy sequence in L ( if q n is a Cauchy sequence in R k. For every r R k, we define the function F r as the limit in L (E of any sequence F q n, where q n Q k is any sequence which tends to r in R k. This definition is licit since F r depends only on r and not on the sequence q n, as it is easily seen by considering two sequences q n and q n which tend to the same r. Moreover inequality (3.63 and the definition of F r implies that F r F r L (E ω( r r r, r R k. (3.65 We now define the function F : E R k R by F (x, r = F r (x a.e. x E, r R k. It immediately follows from this definition and from (3.65 and (3.62 that F is a Carathéodory function which satisfies (3.52 and (3.55. Second step: Proof of (3.56. Let us now prove that (3.56 holds true for every r R k with the above defined function F r and the same subsequence ε for every r R k. Let r R k be fixed. In view of (3.6 we can extract from the subsequence ε a further subsequence, say ε, which depends on r, such that F ε (, r F r in D (E weak star, where F r belongs to D (E. From (3.52 applied to r and r = q Q k, we deduce that for every ϕ C c ( and every q Qk one has < F r F q, ϕ > ϕ L (E ω( r q. Using a sequence q n Q k which converges to r and the definition of Fr, this implies that Fr = Fr. Since the limit of the subsequence F ε (, r is uniquely determined independently of the subsequence ε, the whole sequence F ε (, r converges to Fr, which is nothing but F (, r, and (3.56 is proved. Third step: Proof of (3.59. It remains to prove that (3.59 holds true when one assumes (3.57 and (3.58. Since z L (E k, for every δ > there exists a step function ζ (i.e. a function of the form ζ(x = i c i χ i (x a.e. x E, where the sum is finite, where the c i belong to R k, and where the χ i are characteristic functions of disjoints measurable sets such that χ i (x = in E, such that i z ζ L (E k δ.

22 22 englishnicolas CHARALAMBAKIS AND FRANÇOIS MURAT From the formula F ε (x, ζ(x = i F ε (x, c i χ i (x, and from (3.56 one deduces that F ε (, ζ F (, ζ in D (E weak star. On the other hand, one deduces from (3.52 and (3.57 that F ε (, z ε F ε (, ζ L (E C z ε ζ L (E k. Similarly, since F satisfies (3.52, one has F (, ζ F (, z L (E C z ζ L (E k C δ. The convergence (3.59 then follows from the previous results, from the convergence z ε ζ L (E k z ζ L (E k δ and from the formula F ε (, z ε F (, z = F ε (, z ε F ε (, ζ + F ε (, ζ F (, ζ + F (, ζ F (, z. Lemma 3.8 is proved. Proof of Proposition 3.3. Since N ε satisfies (3.7 and (3.5 one has, for almost every x and for every r, r R N ε (x, r N ε (x, r r r, c 3 c N ε (x, = θ ε(x, (3.66 N ε (x, r c 3 c r + θ ε (x. Therefore one can apply Lemma 3.8 to the sequence N ε (here we use the fact that γ L ( in hypothesis (2.. Similarly, in view of (3.8 and (3.7, one has, for almost every x and for every r, r R N ε Nε (x, r r r (x, r c 7c 4 Nε (x, r. c 4 c 2 r c 3 c c 3 3 c3 r r + c 2 c 2 ω 3 c2 Therefore one can apply Lemma 3.8 to the sequence Nε r. ( r r c 3 c, (3.67

23 englishhomogenization OF STRATIFIED THERMOVISCOPLASTIC MATERIALS 23 This implies that there exists a subsequence ε and two Carathéodory functions N and D, which satisfy for almost every x and for every r, r R N (x, r N (x, r r r, c 3 c N (x, = θ (x, D (x, r D (x, r c 7c 4 c 3 r r + c 2 3 c3 c 2 ω 3 c2 ( r r c 3 c, such that for every r R D (x, r, c 4 c 2 c 3 c N ε (, r N (, r in D ( weak star, N ε r (, r D (, r in L ( weak star. From the mean value theorem and from (3.67, we have for some η = η(x, r, h with < η < Nε (x, r + h N ε (x, r Nε (x, r h r = N ε Nε (x, r + ηh h (x, r h r r ( c7 c 4 h c 3 h + c ( 2 h 3 c3 c 2 ω. 3 c2 c c 3 Passing to the weak star limit in D ( along the subsequence ε implies that namely that ( N (x, r + h N (x, r D c7 c 4 (x, r h h c 3 3 c3 h + c 2 c 2 3 c2 ( h ω c c 3, D (x, r = N (x, r a.e. x, r R. This proves the assertions of Proposition 3.3 as far as the functions N ε are concerned. Similarly, in view of (3.22 and of (3.23, the Carathéodory functions λ ε (x, r satisfy c 2 λ ε (x, r c a.e. x, r R, λ ε (x, r λ ε (x, r c 7 c 2 (r r a.e. x, r, r R, r > r. c 3 c Therefore one can apply Lemma 3.8 to the sequence F ε =. This implies that there λε exists a further subsequence, still denoted by ε, and a Carathéodory function, that we denote by, such that for every r R λ λ ε (, r λ (, r in L ( weak star,

24 24 englishnicolas CHARALAMBAKIS AND FRANÇOIS MURAT where the function λ satisfies c 2 λ (x, r c a.e. x, r R,, λ (x, r λ (x, r c 7 c 2 (r r a.e. x, r, r R, r > r, c 3 c which implies that c λ (x, r c 2 a.e. x, r R, λ (x, r λ (x, r c 7c 2 2 c 3 c 3 (r r a.e. x, r, r R, r > r. This proves the assertions of Proposition 3.3 as far as the functions λ ε are concerned. Proposition 3.3 is proved. Proof of Proposition 3.4. Since for almost every x, the function r N (x, r is one-to-one and strictly increasing from R into R, since N (x, = θ (x and since N (x, r, c 4 c 2 r c 3 c its reciprocal function s M (x, s defined by (3.42 is one-to-one and strictly increasing from R onto R and satisfies (3.43 and (3.44, and the function M : R R is a Carathéodory function. Since N (x, M (x, s = s, the chain rule yields N r (x, M (x, s M (x, s =, s which implies, using (3.36 and (3.37 M M (x, s s s (x, s = N r (x, M (x, s N r (x, M (x, s N r (x, M (x, s N r (x, M (x, s ( (c 4 c 2 2 c7 c 4 c 3 M (x, s M (x, s + c 2 3 c3 c 2 ω 3 c2 ( M (x, s M (x, s c 3 c, which using (3.44 implies (3.45. Defining now µ by (3.46, properties (3.48 and (3.49 immediately follows from (3.38, (3.39 and (3.44. Similarly defining c by (3.47, property (3.5 immediately follows from (3.44 and (3.48, and property (3.5 immediately follows from c (x, s c (x, s µ (x, s M M (x, s s and from (3.48, (3.45, (3.44 and (3.49. Proposition 3.4 is proved. s (x, s + M s (x, s µ (x, s µ (x, s µ (x, sµ (x, s,

25 englishhomogenization OF STRATIFIED THERMOVISCOPLASTIC MATERIALS Proof of the regularity Theorem 2.4. The idea of the proof is to differentiate equation (2.4 with respect to time; setting vε = wε, and since (µε (x, θ ε = µε s (x, θε θε = µε s (x, θε µε (x, θ ε ( v ε 2 c ε (x, θ ε = = µε s (x, θε c ε (x, θ ε µ ε (x, θ ε (σε 2, one formally obtains ρ ε (x wε (µ(x, θ ε wε = fε + ( µ ε s (x, θε c ε (x, θ ε (µ ε (x, θ ε 2 (σε 3 The right-hand side of this equation is bounded in L 2 (, T; H ( in view of hypothesis (2.3 on fε, of the a priori estimate (2.29 on σε and of the estimate µ ε s (x, s c ε (x, s(µ ε (x, s 2 c 7 c 3 c 2. Using w ε as test functions in this equation formally implies that w ε = vε is bounded in L (, T; L 2 ( L 2 (, T; H (, or more exactly in L loc (, T; L2 loc ( L 2 loc (, T; H vε loc (, since the boundary conditions on only belong to (and are bounded in L 2 (, T, and since the initial conditions are even worse. The above computation is formal, but we will make it rigourous below by writing the equation on the differential quotients h v ε in place of vε, and by using as test functions in place of h v ε, the function ϕ 2 (tψ 2 (x h v ε, where ϕ and ψ are cut-off functions. Note that the proof holds true in any dimension, whenever the a priori estimate (2.29 holds true.. Let us now pass to the correct proof of Theorem 2.4. Let (v ε, θ ε be the solution (2.2 (2.8. For h >, we denote by h v ε, h θ ε and h f ε the differential quotients h v ε (t, x = vε (t + h, x v ε (t, x, h h θ ε (t, x = θε (t + h, x θ ε (t, x, h h f ε (t, x = fε (t + h, x f ε (t, x. h

26 26 englishnicolas CHARALAMBAKIS AND FRANÇOIS MURAT Making the difference of equation (2.4 at time t + h and at time t, we have ρ ε h v ε (µ ε (x, θ ε (t + h, x h v ε = h f ε + ( γh ε v ε in D (Q, (4. where we have set γh ε (t, x = µε (x, θ ε (t + h, x µ ε (x, θ ε (t, x. (4.2 h We now fix δ > sufficiently small and define ϕ δ (t and ψ δ (x by ϕ δ (t = t δ ψ δ (x = x a δ if t δ, ϕ δ (t = if t δ, if a x a + δ, ψ δ (x = if a + δ x b δ. ψ δ (x = b x δ if b δ x b, The test function ϕ 2 δ (tψ2 δ (x h v ε (t, x belongs to L 2 (, T; H ( H (, T; L 2 ( and can be used as test function in (4.. This yields 2 d ρ ε (xϕ 2 δ dt (tψ2 δ (x h v ε (t, x 2 dx+ + = µ ε (x, θ ε (t + h, xϕ 2 δ (tψ2 δ (x h v ε 2 (t, x dx = ρ ε (xϕ δ (t dϕ δ dt (tψ2 δ(x h v ε (t, x 2 dx+ µ ε (x, θ ε (t + h, xϕ 2 δ (t2ψ δ(x dψ δ dx (x h v ε (t, x h v ε (t, xdx+ + h f ε (t, xϕ 2 δ(tψδ(x 2 h v ε (t, xdx γh(t, ε x vε (t, xϕ2 δ(tψδ(x 2 h v ε (t, xdx γh ε vε (t, x (t, xϕ2 δ (t2ψ δ(x dψ δ dx (x h v ε (t, xdx.

27 englishhomogenization OF STRATIFIED THERMOVISCOPLASTIC MATERIALS 27 Integration in time from to t, estimates (2.2 and (2.5 on µ ε and ρ ε, and Young s inequality yield c 5 2 t + c c 6 δ + c 4 + t + c 4 ϕ 2 δ (tψ2 δ (x h v ε (t, x 2 dx+ t t t ϕ 2 δ(t ψδ(x 2 h v ε 2 (t, x dxdt h v ε (t, x 2 dxdt + ϕ 2 δ(t ψδ(x 2 h v ε (t, x 2 dxdt + c 4c 2 2 δ 2 h f ε (t, xϕ 2 δ(t ψ 2 δ(x h v ε (t, xdxdt + ϕ 2 δ(t ψδ(x 2 h v ε 2 dxdt + t c t h v ε (t, x 2 dxdt + γh(t ε, x 2 v ε 2 (t, x dxdt + + t h v ε (t, x 2 dxdt + t γh ε δ δ (t, x 2 v ε 2 (t, x dxdt. (4.3 Let us now estimate the various terms which appear in the right-hand side of (4.3. We first estimate the fourth term, say IV ε, of the right-hand side of (4.3. Since by hypothesis (2.3, the functions fε are bounded in L 2 (, T; H (, there exist functions g ε L 2 (Q such that f ε = gε, gε L 2 (Q CK, (4.4 where the constant C only depends on (b a. Therefore h f ε (t, x = h = h t +h t t +h t f ε (s, xds = ( g ε (s, xds = h t +h t g ε (s, xds.