Homogenization of periodic non self-adjoint problems with large drift and potential

ESAIM: Control, Optimisation and Calculus of Variations

Vol. 13, N^o 4, 2007, pp. 735–749 www.esaim-cocv.org

DOI: 10.1051/cocv:2007030

HOMOGENIZATION OF PERIODIC NON SELF-ADJOINT PROBLEMS WITH LARGE DRIFT AND POTENTIAL

Gr´ egoire Allaire

and Rafael Orive

Abstract. We consider the homogenization of both the parabolic and eigenvalue problems for a singularly perturbed convection-diffusion equation in a periodic medium. All coefficients of the equation may vary both on the macroscopic scale and on the periodic microscopic scale. Denoting byεthe period, the potential or zero-order term is scaled asε⁻²and the drift or first-order term is scaled asε⁻¹. Under a structural hypothesis on the first cell eigenvalue, which is assumed to admit a unique minimum in the domain with non-degenerate quadratic behavior, we prove an exponential localization at this minimum point. The homogenized problem features a diffusion equation with quadratic potential in the whole space.

Mathematics Subject Classification. 35B27, 35K57, 35P15, 74Q10.

Received July 11, 2005. Revised June 29, 2006.

Published online July 20, 2007.

1. Introduction

We study problems of homogenization associated to the following singularly perturbed second order non self-adjoint elliptic operator with locally periodic coefficients

A^ε≡ −div (A(x, x/ε)∇) +ε⁻¹b(x, x/ε)· ∇+ε⁻²c(x, x/ε), (1.1) where the coefficients A(x, y) = {a_ij(x, y)}1≤i,j≤N, b(x, y) = {b_i(x, y)}1≤i≤N, c(x, y) are real and bounded functions defined for x∈ Ω⊂R^N and y ∈ T^N (the unit torus). In other words, the coefficients are periodic of period (0,1)^N with respect to the variable y. Furthermore, the matrix A(x, y) is possibly non-symmetric, uniformly positive definite, while the vectorb(x, y) and the scalar potential c(x, y) do not satisfy any further assumption (in particularcis not necessarily non-negative andbis not divergence-free nor of zero mean value).

We study the asymptotic behavior, asεgoes to zero, of the following parabolic problem ∂u_ε

∂t +A^εu_ε= 0 in (0, T)×Ω,

u_ε(t, x) = 0 (t, x)∈(0, T)×∂Ω, u_ε(0, x) =u⁰_ε(x) in Ω,

(1.2)

Keywords and phrases. Homogenization, non self-adjoint operators, convection-diffusion, periodic medium.

1 Centre de Math´ematiques Appliqu´ees, ´Ecole Polytechnique, 91128 Palaiseau Cedex, Paris, France;

[email protected]

2 Departamento de Matem´aticas, Universidad Aut´onoma de Madrid, 28049 Madrid, Spain;[email protected]

c EDP Sciences, SMAI 2007 Article published by EDP Sciences and available at http://www.esaim-cocv.org or http://dx.doi.org/10.1051/cocv:2007030

and of the eigenvalue problem

A^εp^ε=λ^εp^ε in Ω, p^ε= 0 on∂Ω. (1.3)

The novelty in the present work is that (1.1) is a non self-adjoint operator. The case of self-adjoint operators was previously analyzed in [2, 7, 8, 15].

In order to homogenize both (1.2) and (1.3), the main idea is to use an auxiliary non self-adjoint eigenvalue problem in the space of periodic functions (see Sect. 2). It is called exponential spectral problems (see [6,12,13]) and it is similar to the well-known Bloch spectral problem.

Our main assumption is that the first exponential eigenvalue has a single minimum at a pointx₀∈Ω with a non-degenerate quadratic behavior nearx₀ (see Sect. 2). Then, we prove that the first eigenfunction of (1.3), is approximately given as the product of an oscillating function (with periodε) and an homogenized eigenfunction concentrating exponentially atx₀ (with characteristic lengthscale√

ε). The homogenized problem is posed on the whole space R^N with a quadratic potential linked to the Hessian of the first exponential eigenvalue (see Th. 3.5). We also prove a similar result for the parabolic equation (1.2) if the initial data are well prepared (see Th. 3.1). In other words, we build a sequence of approximate solutions of (1.2) which are defined as the product of the same oscillating function (with periodε) and of the solution of a parabolic homogenized equation, concentrating exponentially atx₀ with characteristic lengthscale√

ε.

There are several motivations for studying the homogenization of (1.2) and (1.3). The first problem can be seen as a convection-diffusion-reaction equation. In particular, in the limit case of indicator-type potentials (i.e. c = +∞ in a subdomain of T^N and c = 0 elsewhere), we recover perforated domains with periodic holes supporting Dirichlet boundary conditions. In such a case the term of order ε⁻² disappears from the operator (1.1) although there is still a singular perturbation due to the presence of Dirichlet holes. This is a typical model of convection-diffusion in porous media. The self-adjoint case (for the spectral problem) was studied in [21].

The ground state asymptotic (i.e., characterizing the limit of the first eigenvalue and eigenfunction asεgoes to 0) plays an important role when studying the long time behavior of solutions of the corresponding parabolic equation. Namely, the first eigenvalue governs the rate of decay (or growth) of solutions while the limit profile of the solutions is determined by the first eigenfunction. Another motivation for studying the eigenvalue problem is the modelling of the so-called criticality problem for the neutron diffusion equation (which allows to compute the power distribution in a nuclear reactor core, seee.g., [3]).

Apart from the already quoted papers [2, 7, 8], there are many other related works. For example, when the coefficients are not rapidly oscillating (i.e. they depend on x and not on y), the study of (1.1) is a problem of singular perturbation without homogenization [17]. The result of [17] is that, if c(x) has a unique global minimum pointx₀ ∈Ω then the first eigenfunction p^ε₁(x) is exponentially small everywhere except at x₀ and the logarithmic asymptotic behavior ofp^ε₁is given. A similar logarithmic asymptotic of the ground state for an operator with locally periodic coefficients was obtained in [18]. On the other hand, when the coefficients are purely periodically oscillating functions (i.e. they depend ony but not onx), the homogenization of (1.2) and (1.3) is also quite well understood, and more precise results are obtained in [3, 5, 13, 14]. In the case b≡0 and c≡0 we also refer to [20]. Similar problems, with a different scaling, are also studied using probabilistic tools in [10, 11].

The rest of the paper is organized as follows: in Section 2 we present the non-self adjoint exponential eigenvalue problem and our assumptions on the cell ground state. In Section 3 we state our main homogenization results. In Sections 4 and 5 we prove the convergence of the homogenization process for the parabolic problem and the eigenvalue problem, respectively.

2. Notations and assumptions

In order to formulate our assumptions on the operatorA^εwe introduce an auxiliary eigenvalue problem: the so-called exponential cell eigenproblem. Followinge.g. [6, 12, 13], for any parameter vectorθ∈R^N and for any

point x∈Ω, we introduce the followingθ-exponential periodic problem: find (λ(x, θ), ψ_x,θ) the first eigenpair

of

−div_y(A(x, y)∇yψ_x,θ) +b(x, y)· ∇yψ_x,θ+c(x, y)ψ_x,θ =λ(x, θ)ψ_x,θ,

y→ψ_x,θ(y)e^−2πθ·y (0,1)^N-periodic. (2.1) Here and in the sequel we denote by∇y the derivative in the oscillating variable and by∇xthe derivative with respect to the macroscopic variable. Analogously, denoting by A^∗ the adjoint matrix of A, we consider the adjoint problem: find (λ^∗(x, θ), ψ^∗_x,θ) the first eigenpair of

−div_y

A^∗(x, y)∇_yψ^∗_x,θ

−div_y(b(x, y)ψ^∗_x,θ) +c(x, y)ψ^∗_x,θ=λ^∗(x, θ)ψ_x,θ^∗ ,

y→ψ^∗_x,θ(y)e^2πθ·y (0,1)^N-periodic. (2.2)

Of course, in (2.1) and (2.2), ψ_x,θ and ψ_x,θ^∗ are functions of the variabley (x∈Ω is just a parameter, like θ).

Equations (2.1) and (2.2) are posed on the whole spaceR^N, but, due to the “exponential periodic” boundary condition, they can be restricted to the unit cell (0,1)^N. In order to find a convenient formulation of the boundary condition we change the unknown as

ψ_x,θ(y) = e^2πθ·yφ_x,θ(y) and ψ^∗_x,θ(y) = e^−2πθ·yφ^∗_x,θ(y),

whereφ_x,θ andφ^∗_x,θ are now (0,1)^N-periodic functions. This allows us to rewrite an equivalent form of (2.1) on the unit torusT^N (a compact manifold)

−(div_y+ 2πθ) (A(x, y)(∇_y+ 2πθ)φ_x,θ) +b(x, y)(∇_y+ 2πθ)φ_x,θ+c(x, y)φ_x,θ =λ(x, θ)φ_x,θ, (2.3) and similarly for the adjoint problem (2.2). Remark that there is noi or imaginary numbers in (2.3) which is the main difference with the Bloch cell problems. In particular (2.3) is not self-adjoint, even ifAis symmetric andb= 0. However, since (2.3) is a real-valued problem and the corresponding Green operator is compact, one can apply the Krein-Rutman theorem which is at the basis of the next result. By an obvious generalization of Lemma 3 of [13], we get the following result:

Lemma 2.1. For allθ∈R^N andx∈Ω, there exist a first eigenvalue and eigenfunction ψ_x,θ to(2.1)andψ_x,θ^∗ to(2.2). The first eigenvalue of(2.1)and(2.2)is the same, i.e. λ(x, θ) =λ^∗(x, θ). It is real, of smallest modulus among all eigenvalues and of geometric and algebraic multiplicity equal to one. The first eigenfunctions ψ_x,θ andψ^∗_x,θ can be chosen positive in T^N.

Furthermore, for allx∈Ω, the functionθ→λ(x, θ)is concave fromR^N intoRand admits a maximumλ_∞(x) which is obtained for a unique θ =θ_∞(x). Denoting by ψ_∞(x, y) the corresponding eigenfunction ψ_x,θ∞(x)(y) (and similarly for the adjoint), it is characterized by the following relation

T^N

β(x, y) dy= 0, (2.4)

where the divergence-free vector β is defined by

β(x, y) =ψ_∞^∗ ψ_∞b(x, y) +ψ_∞A^∗∇_yψ_∞^∗ (x, y)−ψ^∗_∞A∇_yψ_∞(x, y). (2.5) Remark 2.2. The characterization (2.4) is actually equivalent to the optimality condition∇θλ(x, θ_∞(x)) = 0 [13]. In the sequel a case of special interest will beθ_∞(x) = 0 (see Rem. 3.2). It is clear that if the operator (1.1) is self-adjoint,i.e. if the matrixA is symmetric and the vectorbis equal to 0, then ψ_x,0^∗ =ψ_x,0 which implies β ≡ 0 and θ_∞(x) = 0. Another sufficient condition to have θ_∞(x) = 0 is the following central symmetry with respect to the center of the unit cell (−1/2; 1/2)^N. If we assume that A(−y) = A(y), c(−y) = c(y) and b(−y) =−b(y), then ψ^∗_x,0(−y) = ψ_x,0^∗ (y) and ψ_x,0(−y) =ψ_x,0(y), so thatβ is an odd function of y and condition (2.4) is satisfied forθ_∞(x) = 0.

Remark 2.3. Since the eigenfunctions are positive and continuous on T^N, there exists a positive constantC (depending onxandθ) such thatψ_x,θ(y)≥C >0 andψ_x,θ^∗ (y)≥C >0 uniformly iny∈T^N. As is usual we normalize the eigenfunctions by imposing

T^N

|φx,θ(y)|²dy= 1 and

T^N

φ_x,θ(y)φ^∗_x,θ(y) dy=

T^N

ψ_x,θ(y)ψ^∗_x,θ(y) dy= 1. (2.6)

A consequence of the simplicity ofλ(x, θ) and of the above normalization is that the eigenvalue and eigenfunc- tions have the same differentiability property as the coefficients with respect tox.

Our main assumptions onλ_∞(x) =λ(x, θ_∞(x)) with respect to its dependence onxare:

Hypothesis H1. The functionx→λ_∞(x) has a unique global minimum pointx₀ in the interior of Ω.

Hypothesis H2. The coefficientsa_ij(x, y),b_j(x, y) andc(x, y) are of classC² in Ω×T^N, and the Taylor series forλ_∞(x) aboutx₀has non-degenerate (positive definite) quadratic form

λ_∞(x) =λ_∞(x₀) + Λ_ij(x−x₀)_i(x−x₀)_j+o(|x−x₀|²), Λ_ijξ_iξ_j ≥c|ξ|² (c >0), (2.7) for any vectorξ∈R^N, where

Λ_ij≡ 1 2

∂²λ_∞

∂x_i∂x_j(x₀).

Without loss of generality we shall assume in the sequel that x₀ = 0 and we denote ψ_∞(y) ≡ ψ_∞(0, y), θ_∞=θ_∞(0) and λ_∞=λ_∞(0).

Remark 2.4. Hypothesis H1 ensures the concentration of ψ_∞(x, x/ε) in the neighborhood of x₀ while Hy- pothesisH2allows to characterize, in the vicinity ofx₀, the asymptotic behavior of its profile.

Notations. For any function φ(x, y) defined on R^N ×T^N, we denote by φ^ε the function φ(x, x/ε). For a variablez∈R^N, we denote byL²(R⁺;L²(R^N;|z|²)) the space defined by

L²(R⁺;L²(R^N;|z|²)) ={v(t, z)∈L²(R⁺×R^N) such that |z|v _L²_(R⁺_×R^N₎<∞}.

3. Main results

Let Ω ⊂ R^N be an open set (bounded or not). Let 0 < T < +∞ be a final time. We first consider the following parabolic problem ⎧

⎪⎨

⎪⎩

∂u_ε

∂t +A^εu_ε= 0 in (0, T)×Ω, u_ε(t, x) = 0 (x, t)∈∂Ω×(0, T), u_ε(0, x) =u⁰_ε(x) in Ω,

(3.1) withA^εdefined in (1.1). Hille-Yosida-Phillip’s theorem guarantees that (3.1) generates a semigroup of contrac- tions. Thus, for any initial datau⁰_εwhich belongs toL²(Ω), (3.1) has a unique weak solutionu_ε=u_ε(x, t) such that u_ε∈C([0, T];L²(Ω))∩L²((0, T);H₀¹(Ω)).

Theorem 3.1. AssumeH1andH2and that the initial data u⁰_ε∈L²(Ω) is of the form u⁰_ε(x) =ψ_∞(x, x/ε)w⁰(x/√

ε), (3.2)

with w⁰∈L²(R^N). The solution of(3.1)can be written as u_ε(t, x) = e^−λ∞ε2tw_ε(t/ε, x/√

ε)ψ_∞(x, x/ε), (3.3)

wherew^ε(s, z)is a function which converges weakly inL²(R⁺;H¹(R^N))∩L^∞(R⁺;L²(R^N))∩L²(R⁺;L²(R^N;|z|²)) and strongly in L²([0, S];L²(R^N))(for any finite timeS) to the solution wof the homogenized problem

∂w

∂t −div(A^eff∇w) +

Λ_ijz_iz_j+d^eff

w= 0 inR⁺×R^N, w(0, z) =w⁰(z) inR^N,

(3.4)

with

d^eff =

T^N

ψ^∗_∞b∇xψ_∞−ψ_∞^∗ div_y(A∇xψ_∞)−ψ^∗_∞div_x(A∇yψ_∞)

(0, y)dy, (3.5)

and the matrix A^eff ={a^eff_ij}1≤i,j≤N is given by a^eff_ij =

T^N

ψ_∞ψ^∗_∞A(0, y)∇(χj+y_j)· ∇(χi+y_i)dy, (3.6)

where the functionsχ_i are defined fori= 1, . . . , N by:

−div_y

ψ_∞ψ^∗_∞A(0, y)∇_y(χ_i+y_i)

+β(0, y)· ∇_y(χ_i+y_i) = 0 in T^N,

y→χ_i(y) T^N-periodic, (3.7)

with the vector β defined in (2.5).

Remark 3.2. The initial data (3.2) is well prepared. Indeed, recall that ψ_∞(x, x/ε) = e^{2πθ∞·x/ε}φ_∞(x, x/ε),

where φ_∞(x, y)is (0,1)^N-periodic in the variabley. This implies that ψ_∞ has an exponential behavior, except ifθ_∞= 0. In the special caseθ_∞= 0, the initial data is uniformly bounded. Furthermore, there is no need to have well prepared initial data in such a case since for any sequenceu⁰_ε, uniformly bounded inL²(Ω), one can extract a subsequence which has the same limit behavior (in the sense of two-scale convergence) as (3.2).

Remark 3.3. The periodic cell problems (3.7) are well posed for 1≤i≤N. Indeed, by Fredholm’s alternative, there exists a unique solution (up to an additive constant) of (3.7) if and only if

T^N

β(0, y)· ∇y(χ_i+y_i)dy = 0.

This is satisfied sinceβ defined in (2.5) is divergence free and satisfies condition (2.4).

Remark 3.4. Theorem 2 still holds true if we add a source term of orderε⁻¹. Indeed, we can add to (3.1) the following source term

f_ε(t, x) = 1

εe^−λε∞2tf(t/ε, x, x/√ ε, x/ε),

wheref(t, x, z, y)∈L²(R⁺×Ω×R^N ×T^N). In such a case, the homogenized equation (3.4) has an additional source term which is f^eff(t, z) defined by

f^eff(t, z) =

T^Nf(t,0, z, y)ψ_∞^∗(0, y)dy.

Also, we can add to equation (3.1) a non-linear term ε⁻¹g(x, x/√

ε, x/ε, u_ε) where g(x, z, y, ξ) is a continu- ous function, homogeneous of degree one and uniformly Lipschitz with respect to the variable ξ. Hence, the homogenized equation (3.4) has an additional non-linear termg^eff(z, w) defined by

g^eff(z, w) =

T^Ng(0, z, y, ψ_∞(0, y)w)ψ_∞^∗ (0, y)dy.

Next, for a bounded open set Ω, we consider the following eigenvalue problem A^εp^ε=λ^εp^ε in Ω,

p^ε= 0 on∂Ω, (3.8)

with A^ε defined in (1.1). As is well known, for each fixed ε > 0 this operator has compact inverse in L²(Ω) (possibly non self-adjoint) and satisfies the maximum principle. Thus, by the Krein-Rutman theorem, we have a first eigenvalueλ^ε₁ of multiplicity one and the corresponding eigenfunctionp^ε₁ can be chosen positive in Ω.

Theorem 3.5. Under assumption H1andH2we have

λ^ε₁=λ_∞+εµ₁+o(ε) with lim

ε→0

o(ε) ε = 0, and the corresponding eigenvector p^ε₁ admits the representation

p^ε₁(x) =q^ε₁(x/√

ε)ψ_∞(x, x/ε), (3.9)

whereq₁^ε(z)∈H¹(R^N), up to a renormalization, converges, as εgoes to zero, to a limit q₁(z) which is the first eigenfunction associated to the first eigenvalue µ₁ of the homogenized spectral problem

−div(A^eff∇q) +

Λ_ijz_iz_j+d^eff

q=µq in R^N, (3.10)

with d^eff andA^eff defined by (3.5)and(3.6), respectively.

Remark 3.6. The spectral problem (3.10) has a purely punctual spectrum since Λ_ijz_iz_j is a non-degenerate quadratic potential. This is the so-called harmonic oscillator problem (seee.g. [7]).

4. Parabolic problem

This section is devoted to the proof of Theorem 3.1. We introduce a new unknown in the spirit of [8, 13], v_ε(t, x) = e^{λε∞2 t} u_ε(t, x)

ψ_∞(x, x/ε), (4.1)

which is well defined since ψ_∞ is positive as already mentioned in Remark 2.3. We replaceu_ε by v_ε in (3.1) and we recall thatψ_∞(x, y) andψ_∞^∗ (x, y) are the first eigenfunction of (2.1) and (2.2), respectively. After some algebra we obtain that (3.1) is equivalent to

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩

εσ^ε∂v_ε

∂t −εdiv (α^ε∇v_ε) +β^ε· ∇v_ε +σ^ελ_∞(x)−λ_∞

ε v_ε+εB^ε· ∇v_ε+ Σ^εv_ε= 0 in R⁺×Ω, v_ε(t, x) = 0 (t, x)∈R⁺×∂Ω,

v_ε(0, x) = u⁰_ε(x)

ψ_∞(x, x/ε) in Ω,

(4.2)

where

σ^ε(x) =ψ_∞ψ_∞^∗ (x, x/ε), α^ε(x) =ψ_∞^∗ ψ_∞A(x, x/ε), β^ε(x) =β(x, x/ε), (4.3) and

B^ε(x) =ψ_∞A^∗∇xψ^∗_∞(x, x/ε)−ψ_∞^∗ A∇xψ_∞(x, x/ε),

Σ^ε(x) = Σ^ε₀(x) +εΣ^ε₁(x), (4.4)

with

Σ^ε₀(x) =ψ^∗_∞b∇xψ_∞(x, x/ε)−ψ_∞^∗ div_yA∇xψ_∞(x, x/ε)−ψ^∗_∞div_xA∇yψ_∞(x, x/ε), Σ^ε₁(x) =−ψ^∗_∞div_xA∇xψ_∞(x, x/ε).

Remark that σ^ε converges weakly- in L^∞(R^N) to 1 because of the normalization condition (2.6), while Σ^ε₀ converges weakly-inL^∞(R^N) tod^eff defined by (3.5).

In order to eliminate theεscaling in front of the second-order operator in (4.2) and to obtain uniforma priori estimates, we rescale the space and time variable by introducing

z= x

√ε ∈Ω^ε=ε^−1/2Ω, s= t

ε, (4.5)

and consider the function

w_ε(s, z) =v_ε(t, x), (s, z)∈R⁺×Ω^ε. (4.6) This yields

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ σ^ε∂w_ε

∂s +A^εw_ε+σ^ελ_∞(√

εz)−λ_∞

ε w_ε+√

εB^ε· ∇wε+Σ^εw_ε= 0 inR⁺×Ω^ε, w_ε(s, z) = 0 (s, z)∈R⁺×∂Ω^ε,

w_ε(0, z) =w⁰_ε(z) =u⁰_ε(√ εz)

ψ^ε_∞(z) in Ω^ε,

(4.7)

where the operatorA^ε is defined by

A^ε≡ −div (α^ε(z)∇) +ε^−1/2β^ε(z)· ∇. (4.8) In (4.7) and (4.8), we use the notationφ^ε(z) =φ(√

εz, z/√

ε) for any function φ(x, y) defined onR^N ×T^N. Now, we can obtain the followinga prioriestimate.

Lemma 4.1. Let 0< S <+∞be a final time. Assume that w^ε satisfies(4.7)withw⁰_ε∈L²(Ω^ε). Then, w_{ε L}^∞_((0,S);L²_(Ω^ε₎₎+ ∇w_{ε L}²_((0,S)×Ω^ε₎^N+ |z|w_{ε L}²_((0,S)×Ω^ε₎≤C(S) w⁰_{ε L}²_(Ω^ε₎, (4.9) and, for any function ϕ(s, z)in the spaceL²([0, S];H¹(R^N))∩L²([0, S];L²(R^N;|z|²)),

S 0

Ω^ε

∂w_ε

∂s ϕdzds

≤C(S) w⁰_{ε L}²_(Ω^ε₎

ϕ _L2([0,S];H¹(R^N))+ ϕ _L2([0,S];L²(R^N;|z|²))

, (4.10)

with C(S)>0 independent ofε.

Proof. First, we multiply equation (4.7) byw_εand we integrate by parts to obtain 1

2

Ω^ε

σ^ε(z)|w_ε(S, z)|²dz−1 2

Ω^ε

σ^ε(z)|w⁰_ε(z)|²dz

+ S 0

Ω^ε

A^εw_ε(s, z)·w_ε(s, z)dzds+ S

0

Ω^ε

σ^ελ_∞(√

εz)−λ_∞

ε |w_ε(s, z)|²dzds

+√ ε

S 0

Ω^ε

B^ε(z)· ∇w_ε(s, z)w_ε(s, z)dzds+ S

0

Ω^ε

Σ^ε(z)|w_ε(s, z)|²dzds= 0.

By virtue of our smoothness assumption on the coefficients, the vector B^ε is uniformly bounded and Cauchy- Schwartz inequality yields

S 0

Ω^ε

B^ε(z)· ∇wε(s, z)w_ε(s, z)dzds≤c wε L²((0,S)×Ω^ε) ∇wε L²((0,S)×Ω^ε).

Analogously, the functionΣ^ε₀ andΣ^ε₁ are uniformly bounded. Therefore, S

0

Ω^ε

|Σ^ε(z)| |wε(s, z)|²dzds≤c wε 2

L²((0,S)×Ω^ε).

The non-self adjoint operatorA^εdefined in (4.8) can be partly written in divergence form. Indeed, by (4.3) we haveβ^ε(z) =β(√

εz, z/√

ε) where the vectorβ, defined by (2.5), has zero average in y and is divergence free by virtue of Lemma 2.1. Thus there exists a skew-symmetric matrixD(x, y), defined by its entries

d_ij(x, y) = (−∆y)⁻¹ ∂β_i

∂y_j(x, y)−∂β_j

∂y_i(x, y)

, (4.11)

such that div_yD=−β (see [13] if necessary) and the operatorA^ε can be written as A^εϕ=−div

(α^ε(z) +D^ε(z))∇ϕ +√

εdiv_xD^ε(z)· ∇ϕ. (4.12) The hypotheses on the coefficients imply that div_xD^ε(z) is uniformly bounded. Thus, since D^ε is a skew- symmetric matrix and the matrixα^εdefined in (4.3) is coercive, we get

S 0

Ω^ε

A^εw_ε(s, z)·w_ε(s, z)dzds≥c ∇wε 2

L²((0,S)×Ω^ε)−c√

ε wε L²((0,S)×Ω^ε) ∇wε L²((0,S)×Ω^ε).

On the other hand, hypothesis H2 implies that there exists two positive constants 0< c₁< c₂ such that c₁|z|²≤ λ_∞(√

εz)−λ_∞

ε ≤c₂|z|² ∀z∈Ω^ε. (4.13)

Thus, using also the fact thatC > σ^ε(z)> c >0, we deduce

Ω^ε

|w_ε(S, z)|²dz+ S 0

Ω^ε

|∇w_ε(s, z)|²dzds+

S 0

Ω^ε

|z|²|w_ε(s, z)|²dzds≤C

Ω^ε

|w⁰_ε(z)|²dz+C S 0

Ω^ε

|w_ε(s, z)|²dzds.

Thus, by Gronwall’s Lemma, we immediately obtain that S

0

Ω^ε

|w_ε(s, z)|²dz≤e^cS

Ω^ε

|w_ε⁰(z)|²dz.

which yields estimate (4.9).

Now, we multiply equation (4.7) by ϕ∈L²(R⁺;H¹(R^N))∩L²(R⁺;L²(R^N;|z|²)). Then,

S 0

Ω^ε

σ^ε∂w_ε

∂s ϕ(s, z)dzds ≤

S 0

Ω^ε

A^εw_ε(s, z)ϕ(s, z)dzds +√

ε S 0

Ω^ε

B^ε(z)· ∇w_ε(s, z)ϕ(s, z)dzds +

1

2 _S

0

Ω^ε

Σ^ε(z)w_ε(s, z)ϕ(s, z)dzds +

_S

0

Ω^ε

σ^ε(z)λ_∞(√

εz)−λ_∞

2ε w_ε(s, z)ϕ(s, z)dzds . By (4.12) and the hypotheses of the coefficients, we get

S 0

Ω^ε

A^εw_ε(s, z)ϕ(s, z)dzds

≤ c ∇w_{ε L}²_((0,S)×Ω^ε₎ ∇ϕ _L²_((0,S)×Ω^ε₎+c√

ε ∇w_{ε L}²_((0,S)×Ω^ε₎ ϕ _L²_((0,S)×Ω^ε₎.

Since the vectorB^εandΣ^ε are uniformly bounded, Cauchy-Schwartz inequality yields

S

0

Ω^ε

B^ε(z)· ∇w_ε(s, z)ϕ(s, z)dzds

≤c ∇w_{ε L}2((0,S)×Ω^ε) ϕ _L2((0,S)×Ω^ε),

1 2

_S

0

Ω^ε

Σ^ε(z)w_ε(s, z)ϕ(s, z)dzds

≤c w_{ε L}²_((0,S)×Ω^ε₎ ϕ _L2((0,S)×Ω^ε). On the other hand, the hypothesis H2 implies (4.13) and, therefore,

_S

0

Ω^ε

σ^ε(z)λ_∞(√

εz)−λ_∞

2ε w_ε(s, z)ϕ(s, z)dzds

≤c w_{ε L}2((0,S);L²(Ω^ε;|z|²)) ϕ _L2((0,S);L²(Ω^ε;|z|²)). Thus, using the fact thatC > σ^ε(z)> c >0 and the estimate (4.9), we conclude the proof of (4.10).

As a consequence of the previousa prioriestimates and using the Aubin-Lions compactness lemma, we have the following compactness result.

Lemma 4.2. Let w_ε be a solution of(4.7). Assume that w⁰_ε is uniformly bounded inL²(R^N). Then, up to as subsequence, there exists a limitw(s, z)∈L^∞(R⁺;L²(R^N))∩L²(R⁺;H¹(R^N))∩L²(R⁺;L²(R^N;|z|²))such that w_ε converges weakly- in this space tow, and strongly inL²([0, S];L²(R^N))for any finite timeS <+∞.

Proof. By the a priori estimate (4.9), up to a subsequence, w_ε converges weakly- to a limit w in the space L^∞(R⁺;L²(R^N))∩L²(R⁺;H¹(R^N))∩L²(R⁺;L²(R^N;|z|²)). On the other hand, it is well-known thatH¹(R^N)∩

L²(R^N;|z|²) is included in L²(R^N) with compact embedding. In view of estimate (4.10), the time deriva- tive ^∂w_∂t^ε is uniformly bounded in the dual space of the Hilbert spaceL²([0, S];H¹(R^N))∩L²([0, S];L²(R^N;|z|²)).

Then, using the classical Aubin-Lions compactness lemma (see [19]), the sequence w_ε is relatively compact in

L²([0, S];L²(R^N)) for any finite timeS.

We now briefly recall the notion of two-scale convergence (see [1, 16]). Below, we denote byδ >0 the period which is going to 0, since we shall apply this result not for εbut rather for √

ε.

Proposition 4.3. Letδ >0denote a sequence of positive reals going to zero. Letw_δ be a bounded sequence in L²(Ω). There exist a subsequence, still denoted byδ, and a limitw(x, y)∈L²(Ω;L²(T^N))such thatw_δ two-scale converges to win the sense that

δ→0lim

Ω

w_δ(x)φ(x, x/δ)dx=

Ω

T^N

w(x, y)φ(x, y)dxdy,

for all functionsφ(x, y)∈L²(Ω;C(T^N)).

Furthermore, if w_δ is a bounded sequence that converges weakly to a limit w in H¹(Ω). Then, w_δ two-scale converges tow, and there exists a functionw₁(x, y)∈L²(Ω;H¹(T^N))such that, up to a sequence,∇w_δ two-scale converges to ∇_xw(x) +∇_yw₁(x, y).

Proof of Theorem 3.1. Equation (4.7) is a combined problem of homogenization and singular perturbations:

the coefficients are oscillating with a period√

ε, and they concentrate to 0 with respect to their first macroscopic argument. Therefore, we expect that the limit problem of (4.7) is precisely the homogenized problem (3.4). To prove this statement and study the asymptotic behavior of (4.7), we follow the methodology of [7, 8].

We multiply (4.7) by a test function φ_ε(s, z) = φ(s, z) +√

εφ₁(s, z, z/√

ε) where φ(s, z) andφ₁(s, z, y) are smooth test functions defined onR⁺×R^NandR⁺×R^N×T^N, respectively, and with compact support inR⁺×R^N. For sufficiently smallε, the support ofφ_εis included in Ω^ε for all times. Integrating by parts yields

∞ 0

R^N

σ^ε(z)w_ε∂φ_ε

∂s dzds+ ∞ 0

R^N

A^εw_εφ_εdzds+ ∞ 0

R^N

σ^ελ_∞(√

εz)−λ_∞

ε w_εφ_εdzds

+√ ε

∞ 0

R^N

B^ε(z)· ∇wεφ_εdzds+ ∞ 0

R^N

Σ^ε(z)w_εφ_εdzds= 0. (4.14)

In view of thea prioriestimate (4.9), we get

√ε ∞ 0

R^N

B^ε(z)· ∇w_ε(s, z)φ_ε(s, z)dzds→0,

and, since Σ^ε= Σ^ε₀+εΣ^ε₁,

ε ∞ 0

R^N

Σ^ε₁(z)w_ε(s, z)φ_ε(s, z)dzds→0. (4.15)

On the compact and fixed space support of φ_ε, the values of the smooth coefficients at points (√ εz, z/√

ε) are uniformly close to their values at (0, z/√

ε). Thus, since by virtue of Lemma 4.2 any average in time of w_ε converges strongly to the same average ofwin L²(R^N), we get

∞ 0

R^N

σ^ε(z)w_ε∂φ_ε

∂sdzds→ ∞ 0

R^N

w∂φ

∂sdzds, ∞

0

R^N

Σ^ε₀(z)w_ε(s, z)φ_ε(s, z)dzds→ ∞ 0

R^N

d^effw(s, z)φ(s, z)dzds.

By assumptions H2 we get ∞

0

R^N

σ^ελ_∞(√

εz)−λ_∞

ε w_εφ_εdzds= ∞ 0

R^N

σ^εΛ_ijz_iz_jw_εφdzds + o(1),

and the same argument of strong convergence inL²(R^N) of time averages ofw_εimplies ∞

0

R^N

σ^ελ_∞(√

εz)−λ_∞

ε w_εφ_εdzds→ ∞ 0

R^N

Λ_ijz_iz_jwφdzds.

Finally, passing to the two-scale limit (see Prop. 4.3), there exists a functionw₁(s, z, y) defined inR⁺×R^N×T^N, such that

ε→0lim ∞ 0

R^N

A^εw_εφ_εdzds= ∞ 0

R^N

T

ψ_∞ψ^∗_∞a_ij(0, y) ∂w

∂x_j +∂w₁

∂y_j

∂φ

∂x_i +∂φ₁

∂y_i

dydzds

+ ∞

0

R^N

T

d_ij(0, y) ∂w

∂x_j +∂w₁

∂y_j

∂φ

∂x_i +∂φ₁

∂y_i

dydzds

where the skew-symmetric matrix D = {d_ij} is defined by (4.11). Regrouping all terms we obtain a limit variational formulation for the couple (w, w₁). In order to eliminate w₁, we take φ ≡ 0 and, integrating by parts, we obtain thatw₁ satisfies

− ∂

∂y_i

ψ_∞ψ_∞^∗ a_ij(0, y) ∂w

∂x_j +∂w₁

∂y_j

− ∂

∂y_i

d_ij(0, y) ∂w

∂x_j +∂w₁

∂y_j

= 0 inR⁺×R^N ×T.

Since div_yD=β, we get thatw₁is the weak solution of

−div_y(ψ_∞ψ^∗_∞A(0, y) (∇xw+∇yw₁)) +β(0, y) (∇xw+∇yw₁) = 0 in R⁺×R^N×T. (4.16)