Two-Scale Convergence

Transcription

1 Two-Scale Convergence Emmanuel Frénod August 4, Contents Introduction. On Two-Scale Convergence first statements How Homogenization brought the concept A remark concerning periodicity A remark concerning weak-* convergence Two-Scale Convergence - Definition and esults 5. Definitions Link with Weak Convergence Injection Lemma Two-Scale Convergence criterion Strong Two-Scale Convergence criterion Application : Homogenization of linear Singularly Perturbed Hyperbolic Equations 5 3. Equation of interest and setting A priori estimate Weak Formulation with Oscillating Test Functions Order Homogenization - Constraint Order Homogenization - Equation for V Order Homogenization - Preparations: equation for U and u Order Homogenization - Strong Two-Scale convergence of U Order Homogenization - Function W Order Homogenization - A priori estimate and convergence Order Homogenization - Constraint Order Homogenization - Equation for V For the numerics Université Européenne de Bretagne, Lab-STICC (UM CNS 39, Université de Bretagne-Sud, Centre Yves Coppens, Campus de Tohannic, F-567, Vannes & Projet INIA Calvi, Université de Strasbourg, IMA, 7 rue ené Descartes, F-6784 Strasbourg Cedex, France

2 Introduction. On Two-Scale Convergence first statements The concept of Two-Scale Convergence was introduced in two papers of Nguetseng [, 3] in 989. Then in 99, Allaire [3] produced a synthetic and very readable proof of the result.. How Homogenization brought the concept The concept of Two-Scale Convergence emerged from questions of Periodic Homogenization. Homogenization is a Mathematical Theory, or more precisely, an Asymptotic Analysis Theory that originates from Material Engineering, or more precisely, from understanding the way Constitutive Equation of composite material can be gotten from the Constitutive Equation of each component of the concerned material and from their topological and geometrical distributions. Material shape Microstructure Figure.: Composite material has a macroscopic shape and a microstructure. The ratio between the size of the microstructure and the size of the material is. In order to make the purpose clear, we first consider the simplest - but rich enough - example I know. (The explanation that follows does not aim to be mathematically rigorous. It clearly appeals to intuition and to non-rigorous vocabulary. Imagine that we want to get the temperature field within a composite material which is in thermal equilibrium, knowing the temperature on its boundary. Symbolically, as represented in Figure. (in a bi-dimensional setting, the composite material has a macroscopic shape at a macroscopic size. Within it, heterogeneities are more or less periodically distributed with a periodicity - or a characteristic size - which is times smaller than its macroscopic size, where is a small parameter. This makes up what is usually called the microstructure of the composite material. Now, to achieve our goal, we contemplate the following Heat

3 Equation [ a (x, x u] = u within the material, given on the boundary of the material, (. that is supposed to describe how the temperature u is being splitted within the material from its distribution on the boundary. In this equation, a stands for the Thermal Diffusion Coefficient (it is the ratio Thermal Conduction over Calorific Capacity times material Density, and stand for the gradient and divergence operators. (If a unidimensional material is considered, x = x lives in, if a bi-dimensional material is considered, x = (x, y lives in and if a tridimensional material is considered, x = (x, y, z lives in 3. The fact that a depends on x and x/ needs to be understood in the following sense. Variable x is the dimensionless position, meaning that when used to describe the material at its macroscopic scale, the needed variations of variable x are of the order of. Beside this, the dependance of a in x/ models the variation of the Thermal Diffusion at the microstructure scale. To illustrate this ability of x/ dependance to describe variations at the microscopic scale, I show the graph of some kinds of functions in one and two dimensions. In Figure Figure.: Graph of sin(x + + cos(x for = / (left, /4 (center and /8 (right between π and π.., is drawn the graph, between π and π, of a (x, x/ = (/ sin(x + + cos(x/ for = /, /4 and /8. Those functions have a variation at the macroscopic scale, which is described by the piece (/ sin(x+ and variations at much smaller scales, which are the microscopic variations. In this example, the microscopic variations can be qualified of being high frequency periodic oscillations with small amplitude. They are described by the term cos(x/ which needs to be multiplied by (explaining the presence of superscript in a to insure amplitude of size of those high frequency periodic oscillations. The next figure (figure.3 shows the graph of a (x, x/ = a(x, x/ = (/ sin(x + + (/ cos(x/ for the same values of as previously. Here, the macroscopic scale variation is always given by the piece (/ sin(x + and the variations at a smaller scale, making up the microscopic variations, are given by (/ cos(x/. (This term is not multiplied by, bringing the uselessness of superscript in a. In this case the microscopic variations can be qualified of being high frequency periodic oscillations with large amplitude or periodic 3

4 Figure.3: Graph of sin(x + + cos(x for = / (left, /4 (center and /8 (right between π and π Figure.4: Graph of (sin(x + cos(x for = / (left, /4 (center and /8 (right between π and π. 4

5 Strong Oscillations. Figure.4 shows the ability of a function depending on x and x/ to describe situations with microscopic variations which are with modulated amplitude - with regions where they are Strong Oscillations and regions where they are not Strong. The drawn function is (/(sin(x + cos(x/. In this expression, the microscopic variations, which are high frequency periodic oscillations, are described by factor cos(x/ and the modulated amplitude is (/(sin(x +. Figure.5 shows function (/ cos(x + + (/(sin(x Figure.5: Graph of cos(x + + (sin(x + cos(x for = / (left, /4 (center and /8 (right between π and π. cos(x/ for always the same values of. Those functions possesses both macroscopic scale variation and modulated amplitude high frequency oscillations as microscopic variations. In every example above the microscopic scale variations are periodic. Yet, the x/ dependance may also describe microscopic scale variations that are not periodic. This is illustrated in figure.6 where function (/ sin(x/3 + + (/4 cos(x/ sin((π/4x/ is drawn. Despite the fact that, for visibility reasons, the chosen value of is not very small (/, figures.7 and.8 show bi-dimensional functions with periodic Strong Oscillations (figure.7 and modulated amplitude in one direction (figure.8. At the end of the day, functions writing a (x, x/ have the ability to describe a variety of coupling both macroscopic and microscopic variations which is wide enough. It is practicable to introduce a variable (say ξ, which may be ξ, (ξ, υ or (ξ, υ, ζ, depending on the dimension number which describes the variations at the microscopic scale. This consists in considering that a in fact depends on two variables: a (x, ξ and that the coefficient in equation (. is a (x, ξ = x. (. Applying this practicable trick to the examples involved in figures above, the following 5

6 ï ï8 ï6 ï4 ï ï8 ï6 ï4 ï ï8 ï6 ï4 ï ï ï x x πx sin( + + cos( sin( for = / (top, /4 (center and /8 (botom between 3π and 3π. Figure.6: Graph of y x y Figure.7: Graph of x + y + (sin( + + (sin( + (left and of x + y + (sin( + x (sin( + (right for = / on [, 3]. 6

7 5 5 3 Figure.8: Graph of x + y + sin(x(sin( y + + (sin(x + for = / on [, 3]. 3 formulas are gotten: a (x, ξ = sin(x + + cos(ξ, in the case of figure., a (x, ξ = a(x, ξ = sin(x + + cos(ξ, in the case of figure.3, a (x, ξ = a(x, ξ = (sin(x + cos(ξ, in the case of figure.4, a (x, ξ = a(x, ξ = cos(x + + (sin(x + cos(ξ, in the case of figure.5, a (x, ξ = a(x, ξ = sin(x cos(ξ sin(π 4 ξ, a (x, y, ξ, υ = a(x, y, ξ, υ = x + y + (sin(υ + + (sin(ξ +, and a (x, y, ξ, υ = a(x, y, ξ, υ = x + y + (sin(υ + (sin(ξ + in the case of figure.7, and, a (x, y, ξ, υ = a(x, y, ξ, υ = x + y + (sin(υ + (sin(ξ +, in the case of figure.8. (.3 If ξ a (x, ξ is periodic, the microscopic scale variations are qualified of high frequency periodic oscillations. emark. Two-Scale Convergence is essentially designed to be used in the context of high frequency periodic oscillations. 7

8 Going back to the question we imagine we are interested in, we need to implement a numerical method, for instance a Finite Difference Method or a Finite Element Method, in order to compute an approximated solution of Partial Differential Equation (.. Since, x is a dimensionless variable, the domain on which (. is set has a size which order of magnitude is. But, to get a reasonable result, we must choose a discretization step x which is such that x <<. Otherwise, the effect of the microstruture is not taken into account, and the resulting computation has nothing to do with reality. Hence, if is very small, meaning that the microstructure is much smaller than the macroscopic size, the computation can be very expensive and even not feasible. For instance, if we consider a tridimensional material, with = 3 then, with the constraint x <<, that we consider to be achieved with x =, the order of magnitude of the number of degrees of freedom needed for the computation is ( 3 3 =. This is quite expensive. Leading such a computation may not be completely unreasonable if we are interested in knowing the intimate distribution of the temperature at the microstructure scale, but in most of the situations the tiny variations at this scale have no interest. In those cases, it is highly unreasonable to lead such a heavy computation to get the result. Hence, we would like to have on hand an equation, which does not explicitly involve any microstructure, but which contains, or more precisely, which induces in its solution the average effect of the microstructure (which is described by the x/ dependance of a in.. Denoting symbolically this equation by Hu =, (.4 involving an operator H, with the constraint that u is close to u in some senses, we can expect to implement a numerical method to compute an approximated solution of (.4 - which is also an approximation of u - with a cost much smaller (because the constraint x << is useless than the one needed for the direct approximation of equation (.. Homogenization Theory gathers a collection of methods that allows us to build operators H satisfying the required constraint. The first Homogenization methods were set out by Engineers in the middle of the 97s and then formalize by Mechanical Scientists. They are based on Asymptotic Expansion. Applying them in the case of equation (. consists in writing u (x = U(x, x + U (x, x + U (x, x +..., (.5 with functions U(x, ξ, U (x, ξ, U (x, ξ,... periodic with respect to ξ, in injecting this expansion in equation (., in identifying and gathering terms in factor of,,,,,... and then in deducing a set of equations: H U =, (.6 H U = I(U, (.7 H U = I (U, U, (

9 Extracting information from those equations, we get well-posed equations for U, U, U,.... For a full understanding of the methods based on Asymptotic Expansion, I refer to the books by Sanchez-Palencia [7] and to the one by Bensoussan, Lions & Papanicolaou [5]. Then, if we want to get a rigorous mathematical justification of the process just described, we need to prove results like u (x U(x, x?, (.9 for a norm? to be determined, or in a weaker sense, (u (x U(x, x. (. If we want to get justification at higher orders, we need to prove convergence results like u (x U(x, x U (x, ξ, (. in some senses, ( ( (u (x U(x, x U (x, ξ U (x, ξ, (. in some senses, and so on. In the case of a Parabolic Partial Differential Equation like (., with Dirichlet boundary conditions, convergence results of the kind of (.9 can be brought out using the Maximum Principle and boundary estimates (see Bensoussan, Lions & Papanicolaou [5]. In other cases, the way is less straightforward, and appeals to Oscillating Test Functions used within a Weak Formulation of the Partial Differential Equation. Passing to the limit using Compensated Compactness like results (see Tartar [9] may give convergence results of the kind of (.9 or (.. This method is called "Energy Method" or "Oscillating Test Function Method" and was designed by Tartar [8] in collaboration with Murat [] (see also [5]. For mathematical justification, works of Engquist have also to be consulted, in particular [7]. The Weak Formulation with Oscillating Test Functions writes, in the case of equation (., [ a (x, x ] u (x ϕ(x, x dx =, (.3 Material which yields, using the Stokes Formula, a (x, x u (x [ϕ(x, x ] dx = Material Boundary Something (.4 9

10 or Material a (x, x u (x [ x ϕ(x, x + ξϕ(x, x ] dx = Something. (.5 Boundary In those integrals, u converges generally in a weak sense only and this is the same for the other involved functions: a (x, x, xϕ(x, x and ξϕ(x, x. It is well known that passing to the limit in a product of two weak converging sequences of functions is a nonstraightforward task. Hence, passing to the limit in (.3, (.4 or (.5 is not so easy and consequently involves relatively sophisticated analytical methods (like Compensated Compactness results. The situation just described is typical of the mathematical justification of homogenization results. Two-Scale Convergence offers an efficient framework to pass to the limit in such terms, in the case when oscillations are periodic. It is certainly possible to infer that Two-Scale Convergence emerges from those kind of questioning. Yet, as it will be illustrated by the example treated in section 3, Two-Scale Convergence is much more than a method to justify Asymptotic Expansion: it a constructive Homogenization Method very well adapted to Singularly Perturbed Hyperbolic Equations. Of course, we can try to provide an answer to question of the same type. For instance we can be interested in the time varying version of (.: u [ a (x, x u] = within the material, u (t,. given on the boundary of the material, u (, x given at initial time within the material, (.6 where t stands for a dimensionless time and where x has the same meaning as in equation (.. and We can also consider the following equations : z [a(t, t ], x zɛ = c(t, t, x, (.7 z [a(t, t ], x zɛ = c(t, t, x, (.8 which are relevant models for the short-term and long-term dynamics of dunes on a seabed of a coastal ocean where tide is strong. In equation (.7 and (.8, z = z (t, x, where t stands is the dimensionless time and x is the bi-dimensional dimensionless position variable, is the dimensionless seabed altitude at time t and in position x. Those equations were widely studied in Faye, Frénod & Seck [9, 8].

11 We can also pay attention to the following Vlasov equations : and f + v xf + (E(x, t + v B(x, t vf =, (.9 f + v x f + v x f + (E(x, t + v B(x, t vf =, (. which are models involved in Tokamak Plasma physic. In those equations x 3 stands for the dimensionless position, v 3 for the dimensionless velocity and t for the dimensionless time. The solution f = f (t, x, v which is also dimensionless is, at time t the density of ions in position x and with velocity v. Field E is the Electric field and field (/B(x, t is the Strong Magnetic field. We denote by and the directions parallel and perpendicular to this magnetic field. Those two last equations involve neither oscillating coefficients nor any microstructure. Nevertheless, the strong magnetic field induces in the solution high frequency periodic oscillations. Equations (.9 or (. can be cast into the following framework of a singularly perturbed convection equation: u + a u + b u =, x d, t >, (. by setting, in the case of (.9, ( ( v a(x, v, t = and b(x, v =, (. E(t, x v B(t, x and, in the case of (., a(x, v, t = ( v E(t, x ( v and b(x, v =. (.3 v B(t, x Equation of this type are studied in Frénod & Sonnendrücker [6, 7, 8], Frénod & Watbled [9], Frénod, aviart & Sonnendrücker [4], Frénod [, ], Frénod & Hamdache [] Ailliot, Frénod & Monbet [, ], Frénod, Mouton & Sonnendrücker [3] and Frénod, Salvarani & Sonnendrücker [5]..3 A remark concerning periodicity Two-Scale Convergence is well-adapted to the framework of high frequency periodic oscillations (or to cases that can be brought to this framework by an adequate transformation. But, it essentially does not work in non-periodic cases. Even in the case of oscillations with a period depending on the variable describing the macroscopic variation, it does not work. Many questions linked with non periodic homogenization are essentially open.

12 .4 A remark concerning weak-* convergence Here, I give the proof of two important and representative results. They concern the characterization of the weak limit of functions with high frequency periodic oscillations. I will denote, for p =,...,, by L p ( the space of functions defined almost everywhere on such that their p-th power is Lebesgue integrable, by L p #( the space of functions being in L p ( and periodic of period, by C# ( the space of functions being continuous over and periodic of period, by C ( the space of functions being infinitely derivable over, by D( the space of functions being in C ( and compactly supported and by L p (; C# ( the space of functions mapping to C #( such that the p-th power of their norm is Lebesgue integrable. The first result gives the asymptotic behavior, with respect to weak topology, of a periodic function applied in ξ = x/. Lemma. Let ψ L # (. Defining [ψ] by [ψ] (x = ψ( x, then [ψ] ψ(ξ dξ in L ( weak-*. (.4 emark. Convergence (.4 means that for any function φ L (, [ψ] (x φ(x dx ψ(ξ dξ φ(x dx. (.5 Proof. The fist step of the proof consists in noticing that, since the space D( is dense in L (, it is enough to prove [ψ] (x ϕ(x dx ψ(ξ dξ ϕ(x dx. (.6 for any fixed ϕ D(. In a second step, fixing a D(-function ϕ, a real M such that [ M, M] contains the support of ϕ is chosen. Then, the set of points { M, M +, M +,..., M +E(M/, M + (E(M/ +}, where E stands for the integer part, is considered and the integral in (.6 is shared in the following way: [ψ] (x ϕ(x dx = E(M/+ i= M+i M+(i ψ( x ϕ(x dx. (.7 In the third step, since ϕ is regular, it is possible to write that, for any i =,..., E(M/+, for any x [ M + (i, M + i] there exists a c i (x [ M + (i, x] such that ϕ(x = ϕ( M(i + (x + M (i ϕ (c i (x. Clearly, (x + M (i and ϕ (c(x ϕ. (.8

13 Using this in the sum of (.7 yields [ψ] (x ϕ(x dx = E(M/+ + i= E(M/+ i= M+i M+(i M+i M+(i ψ( x dx ϕ( M(i ψ( x (x + M (i ϕ (c i (x dx. (.9 In the last step, using the periodicity of ψ, the first term of (.9 becomes E(M/+ ψ(ξ dξ i= ϕ( M(i, (.3 which, because of iemann like definitions of an integral, converges towards ψ(ξ dξ ϕ(x dx. (.3 as goes to. Beside this using (.8, the second term of (.9 is bounded in the following way: E(M/+ M+i ψ( x i= M+(i (x + M (i ϕ (c i (x dx ( M + ψ(ξ dξ ϕ (M + ψ(ξ dξ ϕ, (.3 and then converges to as goes to. A careful watch to (.7, (.3 and (.3 gives convergence.35. Since, this can be done for any ϕ D(, the Lemma is proved. In view of Lemma., and from the application point of view, it is cleaver to see the L ( weak-* convergence as a way to generalize the concept of average value to functions which have non-periodic oscillations. Hence finding H involved in (.4 - or other similar question - may be translated into a mathematical framework as: "Find an equation satisfied by the weak-* limit of u." This explains why weak-* limit is a key-notion of Homogenization Theory The second result characterizes the asymptotic behavior of a function depending on x and ξ, with a periodic dependence in ξ, and applied in ξ = x/. Notice that here more regularity with respect to ξ is needed than previously. Lemma. Let ψ = ψ(x, ξ L (; C # ( and define [ψ] by setting [ψ] (x = ψ(x, x. Then [ψ] ψ(x, ξ dξ in L ( weak-*. (.33 3

14 emark.3 Convergence (.33 means that for any function φ L (, ( [ψ] (x φ(x dx ψ(x, ξ dξ φ(x dx. (.34 But, for the same reason as previously, if the following convergence ( [ψ] (x ϕ(x dx ψ(x, ξ dξ ϕ(x dx, (.35 is proven for any ϕ D(, it proves the Lemma. Proof. (I chose to restrict the proof to the case when ψ C (; C# ( to avoid technical argument linked with Integration Theory. The first stage of this proof consists in partitioning interval [, ] into m intervals of length /m, for any integer m. Then, fixing any value of m, the indicatrix functions χ i, for i =..., m, of all intervals are considered. They are extended by periodicity over. The center ξ i of each interval is also considered. Using them, function ψ m defined by ψ m (x, ξ = m ψ(x, ξ i χ i (ξ, (.36 i= is built. For every x, ξ ψ m (x, ξ is constant by intervals and, as m tends to infinity, Since, applying Lemma., ψ m (x, ξ ψ(x, ξ uniformly on every compact of. (.37 it may be gotten: [χ i ] χ(ξ dξ = m in L ( weak-*, (.38 [ ψ m ] m ψ(x, ξ i m in L ( weak-*, (.39 i= which is clearly ψ m (x, ξ dξ. Hence, as goes to, [ ψ m ] ψ m (x, ξ dξ in L ( weak-*. (.4 The second stage of the proof consists in fixing ϕ D(, and in showing that for any δ >, it is possible find an, such that for any, ( [ψ] (x ϕ(x dx ψ(x, ξ dξ ϕ(x dx δ. (.4 4

15 The way to get this inequality consists in writing : [ψ] (x ϕ(x dx ( ( [ψ] (x [ ψ m ] (x [ψ] (x [ ψ m ] (x ψ(x, ξ dξ ϕ(x dx ϕ(x dx + ( + ψ m (x, ξ dξ ϕ(x dx + + = ( [ ψ m ] (x ψ(x, ξ dξ ( [ ψ m ] (x ψ m (x, ξ dξ ϕ(x dx ψ m (x, ξ dξ ϕ(x dx dξ ϕ(x dx (.4 ϕ(x dx ( ψ m (x, ξ ψ(x, ξ Because of uniform convergence (.37, it is possible to fix an m such that [ψ] (x [ ψ m ] (x ϕ(x dx δ for any, (.43 3 ( ψ m (x, ξ ψ(x, ξ dξ ϕ(x dx δ 3, (.44 and once this m is fixed, because of (.4, it is possible to fix an such that ( [ ψ m ] (x ψ m (x, ξ dξ ϕ(x dx δ 3 for any. (.45 Using (.43, (.43 and (.45 in (.4 gives the sought formula (.4. Since, this can be done for any ϕ D(, the Lemma is proved. Two-Scale Convergence - Definition and esults. Definitions There are several variants of the Two-Scale Convergence result, fitting more or less for targeted applications and involving various functional spaces (see Nguetseng [, 3], Allaire [3], Amar [4], Casado-Díaz & Gayte [6], Frénod, aviart & Sonnendrücker [4], Nguetseng & Woukeng [5] and Nguetseng & Svanstedt [4]. They are in fact very close to each other in what concerns what they claim and their proofs all follow the same routine based on this two phases : A continuous injection Lemma, A compactness Theorem, emark. In 5, Pak [6] made a important improvement in the Two-Scale Convergence Theory adapting it to manifolds and differential forms. 5

16 I have chosen to present the Two-Scale Convergence in the framework set out in Frénod, aviart & Sonnendrücker [4] since this framework permits to select among the variables the ones that carry oscillations and the others. I begin by giving some notations. Definition. Let be a regular domain in n, L a separable Banach space and L its topological dual space. Let q [, + and p (, + ] being such that /q + /p =. Let C# (n ; L be the space of continuous functions n L and periodic of period with respect to every variable and L p (, L be the space of (equivalence classes for the equivalence relation "= a.e." of measurable functions f : L such that the p-th power of the norm of f: f p L is Lebesgue integrable. Let L p # (n ; L be the space of functions g : n L such that the p-th power of the norm of f: f p L is locally Lebesgue integrable and periodic of period. (L p # (n ; L = (L p # (n ; L. Spaces L q (; L q # (n, L, L q (; C# (n ; L and L p (; L p # (n, L are also considered. Now I give the definition of Two-Scale Convergence. Definition. A sequence (u = (u (x L p (; L is said to Two-Scale converges to a profile U = U(x, ξ L p (; L p # (n, L if, for any function φ = φ(x, ξ L q (; C# (n ; L, the following convergence holds: lim L u (x, φ(x, x L dx = where L.,. L is the duality bracket between L and L. L U(x, ξ, φ(x, ξ L dxdξ, (. [,] n To end this definition subsection, I give two definitions of the Strong Two-Scale Convergence. Definition.3 If p = q =, L is a Hilbert space and U L (; C# (n ; L, the sequence (u = (u (x L (; L is said to Strongly Two-Scale convergences to U = U(x, ξ if lim u (x U(x, x dx =. (. L In (.. L is the norm in L - which can be identified with L - associated with inner product L L of L - which is also the inner product L.,. L of L and the duality bracket L L between L and L. In Definition.3, the assumption that U is continuous with respect to ξ is to insure the ability to compute U(x, x/, which is not insured for a function which is defined only almost everywhere, since the measure of {(x, x/, x n } is in n n. Nguetseng & Woukeng [5] gave a definition of Strong Two-Scale Convergence which involves less regularity: 6

17 Definition.4 If p = q =, L is a Hilbert space and U L (; L # (n ; L, the sequence (u = (u (x L (; L is said to Strongly Two-Scale convergences to U = U(x, ξ if δ >, and Ũ L (; C# (n ; L such that U(x, ξ Ũ(x, ξ dxdξ δ, and (.3 [,] n L u (x Ũ(x, x dx δ L, for every.. Link with Weak Convergence Two-Scale Convergence and weak-* convergence are strongly related. In fact Two-Scale Convergence may be seen are a generalization of weak-* convergence. This link expresses by the following Proposition. Proposition. If a sequence (u L p (; L Two-Scale converges to U L p (; L p # (n ; L, then u U(., ξ dξ weak-* in L p (; L. (.4 [,] n Proof. Choosing test functions φ(x, ξ = φ(x - independent of the oscillating variable ξ - for any ψ L q (; L in the definition of Two-Scale Convergence is possible since L q (; L L q (; C# (n ; L. Doing this yields: lim L u (x, φ(x L dx = [,] n which is nothing but (.4, proving the Proposition. L U(x, ξ, φ(x L dxdξ = ( U(x, ξdξ, φ(x L dx. (.5 [,] n L emark. The last equality in (.5 can be considered as trivial. Nevertheless, I give in this emark the details bringing it. For any fixed integer m, a partition of a subdomain ω m of with mk(m hypercubes of measure /m and a partition of [, ] n with m hypercubes of measure /m are considered. If is compact, ω m = for every m and K(m is constant and equal to the measure of ; if is not compact, (ω m is a sequence of subdomains such that ω m ω m+ for every m, such that the measure of ω m is K(m < + and m N ω m =. φ k, for k =,..., mk(m, stands for the value on the k th hypercube of a piecewise constant function approximating φ and U k,l, for k =,..., mk(m and l =,..., m, is the value on the tensor product of the k th hypercube of ω m and the l th hypercube of [, ] n of a piecewise 7

18 constant function approximating U. L U(x, ξ, φ(x L dxdξ = lim [,] n m + m U k,l mk(m lim m + k= m L (.3 Injection Lemma m = mk(m k=, φ k L = L m = m L U k,l, φ k L = ( [,] n U(x, ξdξ, φ(x L dx. (.6 Now, I turn to the first important ingredient of Two-Scale Convergence which is the fact that taking functions of L q (; C # (n, L in ξ = x/ is a way to inject continuously this space in L q (; L. Lemma. If φ L q (; C # (n ; L, then for all >, function [φ] : L defined by is mesurable and satisfies [φ] (x = φ(x, x (.7 [φ] L q (;L φ L q (;C # (n ;L. (.8 Proof. The first point is to see that φ L q (; C # (n ; L if and only if there exists a set E of measure zero in such that x \ E, ξ φ(x, ξ is continuous and periodic, (.9 ξ [, ] n, x φ(x, ξ is mesurable over, (. x sup φ(x, ξ L is L q (; +. (. ξ [,] n emark.3 This equivalence property is a consequence of integration theory but is not completely obvious. Nevertheless, I take it for granted. In this emark, I just recall milestones that are needed to get it. Denoting L q (; C # (n ; L means that measured space [, completed of Borelian σ algebra, Lebesgue measure] in considered. Since L is a separable Banach space, C # (n ; L is also a separable Banach space. Hence, the following measurable space [C # (n ; L, Borelian σ algebra] may be considered. Then the use of the definition of strictly measurable function C # (n ; L is needed. (A function f is strictly measurable if there exists a set E of measure zero in such that f( \ E is included in a separable subset of C # (n ; L, and, if the inverse image by f of any set of the σ algebra of C # (n, L belongs to σ algebra of. With this, the integration theory can be implemented, involving step functions C # (n, L (which is quite long and characterizations of integrable functions are gotten. 8

19 Among them, there is the following one: A function f : C# (n ; L is integrable if it is strictly measurable and if f C # ( n ;L dx < +. (. Finally, spaces L p (; C # (n ; L and L q (; C # (n ; L can be built. Characterization of L q (; C # (n ; L by (.9, (. and (. can also be gotten. The second point consists in fixing. From (.9, for all fixed, [φ] (x = φ(x, x is well defined on \ E. Now the goal is to prove that [φ] is mesurable. For this purpose, for any fixed integer m, a partition of a subdomain ω m of with mk(m hypercubes of measure /m is considered. (If is compact, ω m = for every m and K(m is constant and equal to the measure of ; if is not compact, (ω m is a sequence of subdomains such that ω m ω m+ for every m, such that the measure of ω m is K(m and m N ω m =. For every i =,..., mk(m, the center x i of the i th hypercube and the indicatrix function ι i of the i th hypercube are considered. With this, function η m defined by is a step function approximating x. η m(x = Considering now function [φ] m defined by clearely, and, since mk(m i= x i ι i(x, (.3 [φ] m(x = φ(x, η m(x, (.4 x \ E, [φ] m(x [φ] (x as m +, (.5 [φ] m(x = φ(x, η m(x = mk(m i= φ(x, x i ι i(x, (.6 and because of (., [φ] m is a finite sum of measurable functions, hence is measurable. Hence [φ] is almost everywhere the limit of a measurable function. Hence it is measurable. The last point is the proof (.8. From (. it is deduced that ( q φ q L q (;C# (n ;L = sup φ(x, ξ L dx < +. (.7 ξ [,] n On the other hand [φ] L q (;L = φ(x, x q L dx ( q sup φ(x, ξ L dx = φ q ξ [,] n L q (;C# (n ;L, (.8 9

20 which is (.8, ending the proof of the Lemma. This Injection Lemma is supplemented by a property giving information on the asymptotic behavior of [φ] as goes to zero. Proposition. Under the same assumption as in Lemma., function [φ] defined by (.7 satisfies: lim [φ] L q (;L = lim φ(x, x q dx = φ(x, ξ q L dxdξ = φ L q (;L q ( n ;L. (.9 L Proof. For any fixed integer m, a partition of [, ] n with m hypercubes of measure /m is considered. the center of the i th hypercube is called ξ i and its indicatrix function, extended by periodicity over n is called χ i. With φ is associated the sequence of function φ m defined by φ m (x, ξ = m φ(x, ξ i χ i (ξ, (. i= The second step consists in considering [χ i ] defined by [χ i ] (x = χ i ( x. (. using Lemma., generalized to the multidimensional setting, the following is deduced: [χ i ] χ i (ξ dξ = [,] n m in L (; weak-*. (. ([χ i ] q χ q i (ξ dξ = [,] n m in L (; weak-*. (.3 Hence lim φ m (x, x q L dx = lim m i= m φ(x, ξ i q L ([χ i] q dx = i= m The goal of the third step is to get, as m, φ(x, ξ i q L dx = [,] n φ m (x, ξ q L dxdξ. (.4 φ m φ in L q (; C # (n ; L. (.5 For this purpose, the function Γ m : defined by Γ m (x = sup φ m (x, ξ φ(x, ξ q, (.6 L ξ [,] n is used. Since ξ φ m (x, ξ φ(x, ξ is piecewise continuous, sup φ m (x, ξ φ(x, ξ q = sup φ m (x, ξ φ(x, ξ L ξ [,] n ξ [,] n Q n q L (.7

21 Hence, Γ m is a supremum on an countable set of measurable functions and, as such, measurable. Moreover, Γ m (x a.e. on, (.8 Γ m (x sup ξ [,] n φ(x, ξ q L, (.9 and sup ξ [,] n φ(x, ξ q L is an integrable function. Hence, invoking the Lebesgue Dominated Convergence Theorem, it may be deduced that, as m : Γ m (x in L (;, (.3 and consequently (.5. The last step consists in using (.4 and (.5 and writing [φ] q L q (;L = φ(x, x ( q dx = φ(x, x L q dx φ m (x, x L q dx L ( + φ m (x, x q dx φ ( m (x, ξ q dxdξ + φ m (x, ξ q dxdξ. (.3 L L L The last term of the right hand side of this formula satisfies ( φ ( m (x, ξ q dxdξ φ(x, ξ q L dxdξ L the second term satisfies ( φ m (x, x q L dx as m +, (.3 φ m (x, ξ q dxdξ as, (.33 L and concerning the first term, φ(x, x q dx φ m (x, x L q dx φ(x, x L q φ m (x, x L q dx L sup φ(x, ξ φ m (x, ξ dx as m +. (.34 ξ [,] n Using these three convergence results in (.3 give (.9 and prove the proposition. q L.4 Two-Scale Convergence criterion Once the Injection Lemma gotten, the following Theorem, which is important for Homogenization issues, may be proven relatively easily. Theorem. If a sequence (u is bounded in L p (; L, i.e. if u L p (;L = ( u (x p p L dx c, (.35

22 for a constant c independent of, then, there exists a profile U L p (; L p # (n ; L such that, up to a subsequence (u Two-Scale converges to U. (.36 Proof. In a first stage of the proof, the Injection Lemma and assumption (.35 is used to get that, for any function φ = φ(x, ξ L q (; C# (n ; L ((/p + (/q =, L u (x, φ(x, x L dx c [φ] L q (,L (.37 Hence the sequence (µ of applications, where c φ L q (;C # (n ;L. (.38 µ : L q (; C# (n ; L φ L u (x, φ(x, x L dx, (.39 is bounded in the dual (L q (; C # (n ; L of L q (; C # (n ; L which is a separable space. emark.4 The norm on (L q (; C # (n ; L is µ = sup φ L q (;C # (n ;L µ, φ φ L q (;C # (n ;L. (.4 Since (µ is the dual of a separable space, extracting a subsequence, there exists an application µ (L q (; C # (n ; L In particular, it implies for any φ L q (; C # (n ; L. µ µ in (L q (; C # (n ; L weak-*. (.4 µ, φ µ, φ, (.4 The beginning of the second stage of the proof consists in making in (.37. The left hand side converges towards µ, φ and, according to Proposition., the right hand side converges towards c φ L q (;L q ( n ;L. Hence, for every function in φ Lq (; C # (n ; L, µ, φ c φ L q (;L q ( n ;L. (.43 Knowing that L q (; C # (n ; L is dense in L q (; L q # (n ; L - whose dual is L p (; L p # (n ; L, applying the iez epresentation Theorem, it may be deduced that there exists U L p (; L p # (n ; L such that (.44 µ, φ = L U(x, ξ, φ(x, ξ L dxdξ, (.45 [,] n

23 and consequently, such that, up to a subsequence, as goes to zero, L u (x, φ(x, x L dx L U(x, ξ, φ(x, ξ L dxdξ, (.46 [,] n which is exactly (.of definition., proving (.36 and so ending the proof..5 Strong Two-Scale Convergence criterion In this section p = q = and L and L are the same separable Hilbert space. In order to go on gradually, I begin by the following very simple result. Lemma. If ψ = ψ(x, ξ L (; C # (n ; L then the sequence of functions ([ψ] L (; C # (n ; L defined by Strongly Two-Scale converges towards ψ. [ψ] (x = ψ(x, x, (.47 Proof. Applying directly (.9 of., it can be gotten: L ψ(x, x, φ(x, x L dx L ψ(x, ξ, φ(x, ξ L dxdξ, (.48 [,] n for all function φ L (; C# (n ; L, meaning that ([ψ] Two-Scale converges towards ψ. Now, in view of (. in definition.3, since [ψ] (x ψ(x, x is completely obvious, the Strong Two-Scale Convergence is insured. As easyly, the following result may also be proven Proposition.3 If ψ is like in Lemma., [ψ] L (;L = ( L ψ(x, x, ψ(x, x L dx ( L ψ(x, ξ, ψ(x, ξ L dxdξ [,] n dx, (.49 L = ψ L (;L # (n ;L. (.5 Now I will give a result, that was already evoked by Lemma., establishing the link between Strong Two-Scale Convergence and Two-Scale Convergence. Theorem. If a sequence (u L (; L Strongly Two-Scale Converges towards U and if U L (; C # (n ; L, then it Two-Scale Converges towards U. 3

24 Proof. Considering the following quantity I = L u (x U(x, x, φ(x, x L dx, (.5 for any function φ L (; C # (n ; L, on the one hand this quantity satisfies ( I u (x U(x, x L ( dx φ(x, x L dx, (.5 as, because of the Strong Two-Scale Convergence. On the other hand, I = L u (x, φ(x, x L dx L U(x, x, φ(x, x L dx, (.53 and according to Lemma., L U(x, x, φ(x, x L dx L U(x, ξ, φ(x, ξ L dxdξ. (.54 [,] n Using (.5 and (.54 it is gotten that, as goes to, L u (x, φ(x, x L dx L U(x, ξ, φ(x, ξ L dxdξ, (.55 [,] n i.e. (u Two-Scale Converges to U, ending the proof. Now, I give the important Theorem concerning Strong Two-Scale Convergence. Theorem.3 If a sequence (u L (; L Two-Scale converges toward a profile U and if U L (; C # (n ; L and if then lim u L (;L = U L (;L ([,] n ;L, (.56 (u Strongly Two-Scale converges to U, (.57 and, for any sequence (v L (; L Two-Scale converging towards a profile V, L u, v L L U(., ξ, V (., ξ L dξ, in D (. (.58 [,] n Proof. The proof of the first part of the Theorem consists just in computing: u (x U(x, x L dx = u (x L dx L u (x, U(x, x L dx + U(x, x L dx, (.59 4

25 and in passing to the limit, as goes to, using the assumptions of the Theorem, lim u (x U(x, x dx = lim u L (x L dx L U(x, ξ, U(x, ξ L dxdξ + U(x, ξ L dxdξ =. (.6 [,] n [,] n In order to prove the second part of the Theorem, for any test function ϕ D( the following quantity is computed: D L u, v L, ϕ D = L u (x, v (x L ϕ(x dx = L U(x, x, v (x L ϕ(x dx L u (x U(x, x, v (x L ϕ(x dx. (.6 Since u (x U(x, x, the second term of the right hand side is such that L u (x U(x, x, v (x L ϕ(x dx, (.6 as goes to. A direct calculation gives the behavior of the first term as goes to : L U(x, x, v (x L ϕ(x dx = L v (x, U(x, x L ϕ(x dx = L v (x, ϕ(xu(x, x L dx L V (x, ξ, ϕ(xu(x, ξ L dxdξ [,] n = L V (x, ξ, U(x, ξ L ϕ(x dxdξ = L U(x, ξ, V (x, ξ L ϕ(x dxdξ, [,] n [,] n (.63 coupling (.6, (.6 and (.63 gives D L u, v L, ϕ D L U(x, ξ, V (x, ξ L ϕ(x dxdξ (.64 [,] n for any test function ϕ D(, as goes to, i.e (.58, ending the proof. 3 Application : Homogenization of linear Singularly Perturbed Hyperbolic Equations Here, I show how to homogenize a linear Singularly Perturbed Hyperbolic Equation with a method based on Two-Scale Convergence. As said in the Introduction, this equation is related to Tokamak Plasma Physic. The setting is a very simplified one. A more general setting may be found in Frénod, aviart & Sonnendrücker [4] despite the presentation is different: In [4], we used Two-Scale Convergence to justify Asymptotic Expansion while here the Two-Scale Convergence based method is used as a constructive Homogenization Method. 5

26 3. Equation of interest and setting The considered equation is the following: u + a u + b u =, (3. u t= = u. (3. This equation is set for u = u (t, x with x d and t [, T, for a given T >. Concerning a, it is assumed that a = a(x does not depend on time t, is very regular and that its divergence a is zero. (Those assumptions can be relaxed but it complicates calculations. Concerning b the following assumptions (which can essentially not be relaxed are done: b = b(x = Mx, where M is a matrix such that trm =, and such that τ e τm is periodic of period. emark 3. According to those assumptions, the divergence b of b is zero, and since X(τ = e τm x is solution to X = MX(= b(x, X( = x, (3.3 τ the characteristics associated with operator (b are periodic of period and preserve the Lebesgue measure. 3. A priori estimate Multiplying equation 3. by u and integrating over d gives ( d u dx d dt =, (3.4 since a u u dx = d a u u dx d a u u dx = d a u u dx =. d (3.5 Integrating 3.4 from to t yields u (t,. dx = d u dx, d (3.6 and consequently T u L ([,T ;L ( d = u dxdt = T u dx. d d (3.7 As a consequence the following result can be claimed. Lemma 3. If u L ( d, then the sequence (u is bounded in L ([, T ; L ( d. Hence, up to a subsequence (u Two-Scale Converges to U = U(t, τ, x L ([, T ; L # ((; L ( d, (3.8 u u = U(., τ,. dτ in L ([, T ; L ( d weak-*. (3.9 6

27 3.3 Weak Formulation with Oscillating Test Functions From any function φ = φ(t, τ, x C ([, T ; C # ((; C ( d it is possible to define [φ] by Since [φ] (t, x = φ(t, t, x. (3. [φ] = [ ] φ + [ ] φ, (3. τ multiplying (3. by [φ] and integrating the result by parts, the following Weak Formulation with Oscillating Test Functions is gotten: T d u ([ ] φ + [ ] φ + a [ φ] + τ b [ φ] dxdt + u φ(,,. dx =. d ( Order Homogenization - Constraint Multiplying Weak Formulation with Oscillating Test Functions (3. by and passing to the limit using the Two-Scale Convergence, we obtain: T d U that is nothing but a weak formulation of ( φ τ + b φ dxdτdt =, (3.3 U τ + b U =. (3.4 This last equation says that U is constant along the characteristics of operator (b. Hence the following Lemma is true. Lemma 3. There exists a function V = V (t, y L ([, T ; L ( d such that U(t, τ, x = V (t, e τm x. emark 3. The result of this Lemma may also be gotten by a direct computations. For instance, (V (t, e τm x + b (V (t, e τm x = τ V (t, e τm x (( e τm Mx + ((e τm Mx V (t, e τm x =. (3.5 7

28 3.5 Order Homogenization - Equation for V From any regular function γ = γ(t, y C ([, T ; C ( d, φ defined by φ(t, τ, x = γ(t, e τm x is regular and satisfies φ τ + b φ =. (3.6 Using such functions in Weak Formulation with Oscillating Test Functions (3. cancels the terms in factor of /: T d u Passing to the limit yields T d U(t, τ, x ([ ] φ + a [ φ] dxdt + u φ(,,. dx =. (3.7 d ( φ (t, τ, x + a(x φ(t, τ, x dxdτdt + u φ(,,. dx =, d (3.8 and using expression of U in terms of V and of φ in terms of γ, since gives T φ γ (t, τ, x = (t, e τm x and φ(t, τ, x = (e τm T γ(t, e τm x, (3.9 ( γ V (t, e τm x d (t, e τm x + e τm a(x γ(t, e τm x dxdτ dt + u (x γ(, x dx =. (3. d In the first integral of the left hand side we make the change of variables (t, τ, x (t, τ, y = e τm x which preserves the Lebesgue measure and which reverse transform is (t, τ, y (t, τ, x = e τm y. It gives or T T Which says: ( γ V (t, y d (t, y + e τm a(e τm y γ(t, y dydτ dt + u (y γ(, y dy =, (3. d d V (t, y ( ( γ (t, y + e τm a(e τm y dτ γ(t, y + dydt d u (y γ(, y dy =, (3. 8

29 Theorem 3. Under assumption of Lemma 3., function V (t, y linked by Lemma 3. with the Two-Scale limit U(t, τ, x of (u is solution to ( V + e σm a(e σm y dσ V =, (3.3 V t= = u. (3.4 emark 3.3 Clearly, the solution of (3.3 and (3.4 is unique. As a consequence, the whole sequence (u converges (Two-Scale towards U, and weak-* towards u 3.6 Order Homogenization - Preparations: equation for U and u Because of the linearity of the problem, it is possible to deduce from (3.3 an equation for U also. Indeed, since U(t, τ, x = (e τm T V (t, e τm x or V (t, e τm x = (e τm T U(t, τ, x, writing (3.3 in y = e τm x, we obtain that ( + e σm a(e σm e τm xdσ V (t, e τm x = ( V (t, e τm x = U (e + τm e σm a(e (σ τm xdσ U = U = U + ( + ( e (τ σm a(e (σ τm xdσ U e σm a(e σm xdσ U, (3.5 the last equality being gotten from periodicity of σ e σm. Now, since ( e σm a(e (σm xdσ does not depend on τ and because of (3.9, integrating (3.5 gives ( u + e σm a(e σm xdσ u =. (3.6 Finally, since u(, x = the following Lemma is true U(, τ, x dτ and U(, τ, x = V (, e τm x = u (e τm x, Lemma 3.3 Under assumption of Lemma 3., the Two-Scale limit U(t, τ, x of (u and its weak-* limit u are solutions to ( U + e σm a(e σm xdσ U =, (3.7 U t= = u (e τm x, (3.8 and ( u + u t= = e σm a(e σm xdσ u =, (3.9 u (e τm x dσ. (3.3 9

30 3.7 Order Homogenization - Strong Two-Scale convergence of U Because (u = u u multiplying (3. by u, we obtain that (u is solution to: and (u = u u, (3.3 (u + a (u + b (u =, (3.3 (u t= = u. (3.33 Hence if u is in L ( d, i.e. if u L 4 ( d, it is possible to do the same for equation (3.3 as for (3. and find that (u Two-Scale converges to a profile, called Z, and that Z is solution to ( Z + e σm a(e σm xdσ Z =, (3.34 Z t= = u (e τm x, (3.35 leading to the conclusion that Z = U or From (3.36, it is easy to get that ((u Two-Scale Converges to U. (3.36 u L ([,T ;L ( d U L ([,T ;L # ((;L ( d, (3.37 as. Indeed, we only need to consider for any δ > the regular function β δ = β δ (x which is such that β δ (x = when x < /δ, β δ (x = when x > /δ + and β δ. Clearly from (3.36, for any δ, T T (u β δ dxdt U β δ dxdτdt, (3.38 d d and as δ, T T d (u β δ dxdt u L ([,T ;L ( d, (3.39 d U β δ dxdτdt U L ([,T ;L # ((;L ( d. (3.4 Moreover, if u is in C ( d then, u C ([, T ; C ( d, U C ([, T ; C # ((; C ( d and V C ([, T ; C ( d. This can be directly deduced from the equations satisfied by those functions. Hence Theorem.3 can be invoked to deduce the next Lemma. 3

31 Lemma 3.4 If u (L L 4 C ( d, then in addition to every already stated results, (u Two-Scale Converges Strongly to U. (3.4 Having this result, we know that (u [U], we now can show more: (u [U] / Two-Scale Converges. 3.8 Order Homogenization - Function W In a first stage, from equation (3., (3.4 and (3.7 we deduce (u [U] + a (u [U] + b (u [U] ( = a e σm a(e σm xdσ [U], (3.4 (u [U] t= =. (3.43 Multiplying this equation by / we obtain ( u [U] ( u [U] + a + ( u b [U] = ( a e σm a(e σm xdσ [U], (3.44 ( u [U] =. (3.45 t= The left hand side of this equation is the same as in (3. but the right hand side is in factor of /. Hence, in a second stage, we introduce a function W = W (t, τ, y such that W = W (t, τ, x = W (t, τ, e τm x, (3.46 satisfies W τ ( + b W = a e σm a(e σm xdσ U. (3.47 Because of (3.47, considering [ W ] = [ W ] (t, x = W (t, t/, x, [ W ] + a [ W ] + b [ W ] [ = W ] [ + [ = W W ] + a [ W ] ] + a [ W ] + b [ W ] ( a e σm a(e σm xdσ [U]. (3.48 3

32 Subtracting (3.48 from (3.4 gives ( u [U] [ W ] ( u [U] + a [ W ] + ( u b [U] [ W ] [ = W ] a [ W ], (3.49 ( u [U] [ W ] = [ W ] t=. (3.5 t= The goal of the third stage is to give an expression of the function W : Function W is solution of (3.47 if and only if W is solution to ( W = a(e τm y e σm a(e (σ+τm ydσ U(t, τ, e τm y. (3.5 τ W τ Beside this, U(t, τ, e τm y = (e τm T ( U(t, τ, e τm y = (e τm T V (t, y, hence W is solution to ( = e τm a(e τm y e (σ+τm a(e (σ+τm y dσ V (t, y = ( e τm a(e τm y e σm a(e σm y dσ V (t, y, (3.5 (using once again periodicity of τ e τm which is: ( τ W (t, τ, y = e σm a(e σm y dσ τ e σm a(e σm y dσ V (t, y. (3.53 This allows us to compute [ W ]. In particular in (3.5, [ W ] t= = and if u is regular (for instance in C ( in addition to assumptions of Lemma 3.4, because of equation (3.7 V satisfies, it is easily gotten that [ W ] a [ W ] C, (3.54 for a constant C not depending on. L ([,T ;L ( d 3.9 Order Homogenization - A priori estimate and convergence Multiplying (3.49 by ((u [U] / [ W ], we get ( d u [U] [ d W ] dx ( u [U] C dt d [ W ] dx, (3.55 from which an estimate can be gotten and expressed in the following Lemma. 3

33 Lemma 3.5 If u (L L 4 C ( d, then in addition to every already stated results, the sequences u [U] [ W ], and consequently u [U], (3.56 are bounded in L ([, T ; L ( d. Then, up to subsequences, ( u [U] Two-Scale Converges to U = U (t, τ, x L ([, T ; L # ((; L ( d, (3.57 ( u [U] [ ] Two-Scale Converges to U W, (3.58 where W is defined in (3.53 and W by ( Order Homogenization - Constraint For any Oscillating Test Function φ = φ(t, τ, x C ([, T ; C# ((; C ( d, it is possible to write the following Weak Formulation: T ( u [U] ([ ] [ d W φ ] + [ ] φ + a [ φ] + τ b [ φ] dxdt T ( [ = W ] a [ W ] [φ] dxdt. (3.59 d Multiplying this equation by and passing to the limit yields the next constrain equation: (U W τ Hence the following Lemma is true. + b (U W =. (3.6 Lemma 3.6 There exists a function V = V (t, y L ([, T ; L ( d such that U (t, τ, x W (t, τ, x = V (t, e τm x or, in other words, such that where W is defined in (3.53. U (t, τ, x = V (t, e τm x + W (t, τ, e τm x, ( Order Homogenization - Equation for V Using now in (3.59 Oscillating Test Function φ(t, τ, x = γ(t, e τm x for any regular function γ = γ(t, y, the terms in factor of cancel and passing to the limit, it gives T ( γ V (t, e τm x d (t, e τm x + e τm a(x γ(t, e τm x dxdτ dt T ( = W a(x W γ(t, e τm xdxdτdt. (3.6 d 33

34 Making in (3.6 the change of variables (t, τ, x (t, τ, y = e τm x gives T ( γ V (t, y d (t, y + e τm a(e τm y γ(t, y dydτ dt T = which is the weak formulation of ( V + e σm a(e σm ydσ V = V t= =. d ( W e τm a(e τm y W γ(t, ydydτdt, (3.63 ( W e τm a(e τm y W dτ, (3.64 (3.65 Now, it remains to express the right hand side of (3.64 using expression (3.53 of W. For this we need to compute the time derivative of V and the Jacobian matrices of ( τ e σm a(e σm y dσ τ e σm a(e σm y dσ and of V (i.e. the Hessian matrix of V. First, using the equation (3.3 satisfied by V, ( V = V = Hence, W + [ [ T ( e σm a(e σm y dσ]] ( V [ V ] e σm a(e σm y dσ. (3.66 dτ = = + ( τ e σm a(e σm y dσ τ e σm a(e σm y dσ ( [ [ T e σm a(e σm y dσ]] ( V dτ ( τ e σm a(e σm y dσ τ [ [ ]] ( e σm a(e σm y dσ ( τ e σm a(e σm y dσ τ ( ( τ [ V ] e σm a(e σm y dσ ( [ V ] e σm a(e σm y dσ τ ( ( e σm a(e σm y dσ dτ e σm a(e σm y dσ ( V dτ e σm a(e σm y dσ e σm a(e σm y dσ dτ. (

35 On another hand, + ( e τm a(e τm y W dτ = ( e τm a(e τm y ( [ [ τ e σm a(e σm y dσ τ + ( e τm a(e τm y ( ( τ [ V ] ([ [ τ e σm a(e σm y dσ τ ( [ V ] ( τ e σm a(e σm y dσ τ e σm a(e σm y dσ τ e σm a(e σm y dσ]] T ( V dτ e σm a(e σm y dσ = e σm a(e σm y dσ]] (e τm a(e τm y ( V dτ As a consequence, the right hand side of (3.64 expresses as ( W ([ [ τ [ [ + ( e τm a(e τm y W dτ = e σm a(e σm y dσ τ ]] ( τ e σm a(e σm y dσ e σm a(e σm y dσ τ ( ( τ [ V ] e σm a(e σm y dσ τ e σm a(e σm y dσ (e τm a(e τm y dτ. ]] (e e σm a(e σm y dσ τm a(e τm y ( (e τm a(e τm y e σm a(e σm y dσ e σm a(e σm y dσ (3.68 ( V dτ e σm a(e σm y dσ dτ. (3.69 The integrand in the last term is the dot product of a symmetric matrix (not depending on τ applied to a vector with the τ-derivative of this same vector; so it is an exact τ-derivative. Consequently, the last term is zero. Beside this, integrating by parts the first piece of the 35