HOMOGENIZATION OF CONVECTION-DIFFUSION EQUATION IN INFINITE CYLINDER

The paper deals with a periodic homogenization problem for a non-stationary convection-diffusion equation stated in a thin infinite cylindrical domain with homogeneous Neumann boundary condition on the lateral boundary. It is shown that homogenization result holds in moving coordinates, and that the solution admits an asymptotic expansion which consists of the interior expansion being regular in time, and an initial layer. Introduction. The goal of the paper is to study the asymptotic behaviour of a solution to an initial boundary problem for a convection-diffusion equation defined in a thin infinite cylinder with homogeneous Neumann condition on its lateral boundary. We assume that the coefficients of the equation are periodic in the axial direction of the cylinder and that the period is of the same order as the cylinder diameter. The corresponding parabolic operator takes the form ∂tu− div ( a (x ε ) ∇u ) + 1 ε ( b (x ε ) ,∇u ) ; (1) here ε is a small positive parameter, and we assume the standard uniform ellipticity conditions on a(y) and the boundedness of the entries of a(y) and b(y). Notice that the scaling factor 1/ε is natural for the convection term. Indeed, if one wants to consider the long-term behaviour of a convection-diffusion process described by the equation ∂su− div ( a(y)∇u ) + ( b(y),∇u ) = 0 in a fixed infinite cylinder, then making the diffusive change of variables x = εy, t = εs leads to a convection-diffusion problem for operator (1) in a thin cylinder. 2000 Mathematics Subject Classification. Primary: 35B27, 35B40, 35K20; Secondary: 35B25.

Introduction. The goal of the paper is to study the asymptotic behaviour of a solution to an initial boundary problem for a convection-diffusion equation defined in a thin infinite cylinder with homogeneous Neumann condition on its lateral boundary. We assume that the coefficients of the equation are periodic in the axial direction of the cylinder and that the period is of the same order as the cylinder diameter. The corresponding parabolic operator takes the form here ε is a small positive parameter, and we assume the standard uniform ellipticity conditions on a(y) and the boundedness of the entries of a(y) and b(y). Notice that the scaling factor 1/ε is natural for the convection term. Indeed, if one wants to consider the long-term behaviour of a convection-diffusion process described by the equation 112 IRYNA PANKRATOVA AND ANDREY PIATNITSKI Closely related problems for a convection-diffusion equation defined in the whole space have been considered in [6] and [1]. It was proved in particular that the homogenization takes place in moving coordinates (x, t) −→ x − (b/ε)t, t with a constant vectorb.
Homogenization problems for divergence form operators and systems in thin bounded domains have been investigated by many authors, we mention here the works [7], [10] and [11]. General homogenization theory results for parabolic equations can be found in [5] and [12].
In the paper we first prove uniform in ε a priori estimates for the solution. This requires integration in weighted spaces where the solution of the periodic adjoint cell problem is used as a weight. Then we construct the leading terms of the asymptotic expansion in moving coordinates, determine the effective speed, and obtain the estimates for the rate of convergence. Additional difficulty appearing in the problem under consideration is the dimension reduction issue. Indeed, the solutions of the original problem belong to variable Sobolev spaces, which makes the convergence analysis rather involved.
The paper is organized as follows. Section 1 contains the problem setup. In Section 2 we deal with a priori estimates and study auxiliary parabolic cell problems. In Section 3 we construct formal asymptotic expansion which includes the corresponding initial layers, the presence of the initial layer allows us to satisfy the initial condition in higher order approximations. Section 4 is devoted to the convergence analysis.
1. Problem statement. Let Q be a bounded domain in R d−1 with the Lipschitz boundary ∂Q. For any ε > 0, we denote by G ε a thin infinite cylinder R × ε Q with the axis directed along x 1 . The lateral boundary of the cylinder G ε is denoted by Σ ε = R × ∂(εQ). We study the following non-stationary convection-diffusion equation: where and n stands for the exterior unit normal on Σ ε . We suppose that the following conditions are fulfilled: .., d, are 1-periodic functions with respect to y 1 ; (H3) The matrix a(y) satisfies the uniform ellipticity condition, that is there exists a positive constant Λ such that, for almost all x ∈ R d , We are interested in the asymptotic behaviour of u ε (t, x), as ε → 0. Notice that, for any ε > 0, the existence and the uniqueness of a generalized solution to problem (2) is given by classical theory (see, e.g., [8]).

A priori estimates.
In what follows we denote Y = [0, 1) × Q, By the Krein-Rutman theorem and the Harnack inequality, the adjoint periodic problem has a positive solution p * (y) ∈ C(Y ) ∩ H 1 (Y ) such that We fix the choice of p * by the normalization condition Y p * (y) dy = 1.
The goal of this section is to obtain a priori estimates for a non-stationary convectiondiffusion equation stated in a thin infinite cylinder. Namely, we consider the following non-homogeneous problem: Multiplying the equation in (7) by p * (x/ε) u ε (x) and integrating the resulting relation by parts over G ε , we obtain Here we use the notations Taking into account the definition of p * (y) we get Using the positive definiteness of the matrix a(y), bounds (6), and the Cauchy-Bunyakovsky inequality, one can obtain for any γ > 0. By the trace theorem with constants C 1 , C 2 independent of ε. Consequently, for a sufficiently small γ, Integrating with respect to t and applying the Grönwall lemma and the positiveness of the function p * , one can see that where the constant C does not depend on ε, and depends only on Λ, d and Q.
For the justification procedure we also need a priori estimates in the case of righthand side being the divergence of a bounded vector-function. Namely, consider the following problem: with F (t, x) such that Multiplying equation in (9) by p * ( x ε )u ε (x) and integrating by parts over G ε , we obtain Exploiting the Couchy-Bunyakovsky inequality and taking into account (6) one gets The integral I ε 2 is estimated as follows where Y n = (n, n + 1] × Q. Finally, combining the obtained estimates for I ε 1 and I ε 2 with (10), and using the Grönwall's lemma, for t ∈ (0, T ] one has 2.2. Auxiliary results. In the sequel we will need the information about the asymptotic behaviour of solutions to parabolic equations, as t → ∞. Consider the 116 IRYNA PANKRATOVA AND ANDREY PIATNITSKI initial boundary value problem where Y = [0, 1) × Q.
Lemma 2.1. Suppose conditions (H1)−(H3) are fulfilled and ψ(y) ∈ L 2 (Y ). Then there exists a unique weak solution v to problem (12), and it stabilizes to a constant v ∞ at the exponential rate, as τ → ∞, that is with positive constants C 0 and γ depending only on Λ, d and Q. Moreover, ∇ y v stabilizes exponentially to 0, as τ → ∞: The where p * (y) solves problem (5).
Proof. Let us consider two functions By the maximum principle, M (τ ) decreases and m(τ ) increases. In view of the linearity of the problem, without loss of generality we assume that m(τ 0 ) = 0.
Since v ≥ 0, then we can use the Harnack inequality to obtain the estimate Consequently, where C and γ depend only on Λ, d and Q.
Let us calculate the constant v ∞ . To this end we multiply the equation in (12) by p * and integrate by parts over the set (0, τ ) × Y . As a result we obtain the following equality: Y v(τ, y) p * (y) dy = Y ψ(y) p * (y) dy.
Since v converges uniformly to the constant v ∞ , as τ → ∞, then it follows from the last equality that Now we prove estimate (14). Note that the function w = v − v ∞ solves the same equation as v, and satisfies the initial condition w(0, y) = ψ − v ∞ . Multiplying the equation by w, integrating by parts and applying the Cauchy-Bunyakovski inequality gives and, consequently, choosing η < 2Λ 2 and using (13), we obtain The next lemma generalizes the result of Lemma 2.1 to the non-homogeneous case. Consider the boundary value problem Under the assumptions being made, a solution of problem (15) satisfies the estimates 118

IRYNA PANKRATOVA AND ANDREY PIATNITSKI
Here C depends on Λ, d and Q. The constant v ∞ is determined by , y), ∇p * (y)) dy dτ p * being a solution of (5).
Proof. First of all we represent the functions on the right-hand side of (15) as the sums of functions with finite supports, that is where v m solves the problem Notice that, in view of the uniqueness of the solution, v m (τ, y) = 0 for τ ∈ [0, m). Then, multiplying the equation in (18) by v m and integrating over (m−1, m+2)×Y , we obtain By Lemma 2.1, v m satisfies estimate (13) for some constant v ∞ m , whereγ = min{γ, γ 1 }. In view of the maximum principle, v ∞ m ≤ Ce −γ1m .

Let us show that
The exponential decay of ∇v can be proved in much the same way as in the homogeneous case.

Formal inner expansion.
Following the ideas in [6] and [1], we are looking for an approximate solution in the form where v k , k ≥ 1, are unknown functions which are 1-periodic in y 1 , the constantb 1 is to be determined. Substituting (19) into (2) and collecting power-like terms in front of ε −1 in the equation and of ε 0 in the boundary condition, we obtain the following periodic problem for the unknown function v 1 : (20) we guarantee that a solution to problem (20) exists. The specific form of the righthand side of (20) suggests the following representation of v 1 : where a Y -periodic function N 1 solves the problem

IRYNA PANKRATOVA AND ANDREY PIATNITSKI
Similarly, we get the problem for v 2 The compatibility condition for (23) gives rise to the Cauchy problem for u 0 where the constant a hom 11 is defined by The positiveness of a hom 11 has been proved in [9]. The form of the right-hand side of the equation in (23) suggests the following representation for the solution v 2 : with y 1 -periodic function N 2 being a solution of the problem A N 2 (y) = a 11 (y) + ∂ yi (a i1 (y)N 1 (y)) + a 1j (y)∂ yj N 1 (y) Similarly, we obtain a boundary value problem for v 3 From the compatibility condition for (26) we derive the equation for u 1 : Naturally, v 3 can be represented as the sum with N 3 being a y 1 -periodic solution of the cell problem Arguing as above, one can derive the equation for u 2 Notice that determining initial conditions for u 1 and u 2 requires constructing initial layer correctors, which is done in Section 3.2. Finally, as an inner approximate solution we take first three terms of (19) 3.2. Initial layers. The leading term u 0 (t, x 1 ) satisfies the initial condition u 0 (0, x 1 ) = ϕ(x 1 ). We introduce the initial layer functions, which will allow us to satisfy the initial condition up to the second power of ε. Consider the function φ 1 (τ, y) which is a solution to the problem By Lemma 2.1, φ 1 stabilizes to a constant φ 1 , as τ → ∞, at the exponential rate. The constant φ 1 can be calculated as follows We use this constant to set the initial value for u 1 : u 1 (0, x 1 ) = φ 1 ϕ (x 1 ). In this way Similarly, we introduce φ 2 (τ, y) such that: The constant to which φ 2 stabilizes, as τ → ∞, we denote by φ 2 and set u 2 (0, In this way the boundary value problems for u 1 and u 2 take the form Proof. In order to estimate the norm (in the appropriate space) of the difference u ε − U ε between the exact and the approximate solutions, we calculate first A ε (u ε − U ε ) and B ε (u ε − U ε ), and then make use of a priori estimates (8), (11). Straightforward computations yield By a priori estimates (8) and (11), In order to estimate R ε 1 , R ε 2 and R ε 3 , we analyze properties of the solutions u 0 , u 1 and u 2 of problems (24), (34) and (35). For u 0 the well-known integral Poisson formula takes place: Here θ is the unit step function, that is θ(t) = 1 for t ≥ 0, and θ(t) = 0 when t < 0. Moreover, similar formula is valid for any derivative of u 0 with respect to x 1 : Bearing in mind that ϕ has finite support, one can see that where α depends on T . Similarly, the following integral representation of ∂ (k) x1 u 1 is valid: 4a hom 11 (t−τ ) dξ = I 1 + I 2 .
Let us estimate I 2 .
(41) Bearing in mind the boundedness of the coefficients a ij , b j , properties of N 1 and N 2 as the solutions of (22), (25), and bounds (39)-(41), one can check the validity of (38). Note that the exponential decay of the initial layer functions is used while estimating the corresponding terms.