J. London Math. Soc. (2) 72 (2005) 391–409 C
2005 London Mathematical Society
doi:10.1112/S0024610705006824
HOMOGENIZATION OF A NONLINEAR
CONVECTION-DIFFUSION EQUATION WITH RAPIDLY
OSCILLATING COEFFICIENTS AND STRONG CONVECTION
EDUARD MARUSˇIC´-PALOKA and ANDREY L. PIATNITSKI
Abstract
A Cauchy problem for a nonlinear convection-diffusion equation with periodic rapidly oscillating
coefficients is studied. Under the assumption that the convection term is large, it is proved that
the limit (homogenized) equation is a nonlinear diffusion equation which shows dispersion effects.
The convergence of the homogenization procedure is justified by using a new version of a two-scale
convergence technique adapted to rapidly moving coordinates.
1. Introduction
This paper is devoted to homogenization of a model semilinear parabolic equation of
convection-diffusion type with periodic, rapidly oscillating coefficients. The material
is stratified, that is oscillations are allowed in all but one direction. We assume that
the convection term is large, which is related to the self-similar diffusive scaling in
the equation. For this strong convection term we do not suppose that the convection
velocity is divergence free, nor that the effective drift is zero. As described in [8,
Chapter 2] in such a situation the convection might dominate the diffusion, and
we cannot expect nontrivial convergence of the family of solutions uε(t, x) for a
fixed spatial frame x but only in moving coordinates x + Bε(t). Due to the choice
of scaling, in appropriate moving coordinates the homogenized problem shows the
diffusive dynamics. As a consequence of the presence of strong nonlinear convection,
the dispersion effects appear, that is the diffusion coefficients of the limit quasilinear
problem depend on the convection velocity (see, for example [2] or [11] for the
formal asymptotic explanation of the dispersion). To prove the result we adapt the
two-scale convergence method introduced by Nguetseng [12] and Allaire [1], to the
case of rapidly moving coordinates, and we combine it with the appropriate choice
of test functions depending on the solution of the adjoint auxiliary problem (5).
Also, since classical theorems on compactness of embedding of Sobolev spaces in
bounded domains do not apply in the whole space, in our case the compactness
result for the family of solutions is not a straightforward consequence of a priori
estimates. We show that the uniform localization of solutions holds in moving
coordinates, and, in this way, we gain the compactness in the moving coordinates.
The problem under consideration appears, for instance, when studying the long-
term behaviour of the nonlinear convection-diffusion model in stratified periodic
media. In this case, letting ε = 1/
√
T and making self-similar rescaling, we arrive
at our homogenization problems. The desired long-term behaviour can now be
described in terms of the effective characteristics of this problem.
Received 11 October 2004.
2000 Mathematics Subject Classification 35B27, 35B40, 35C20, 35K55.

392 eduard marusˇic´-paloka and andrey l. piatnitski
Previously, homogenization problems for linear convection-diffusion models with
a zero mean drift were considered in [4] and then in many other works. The
case of divergence-free convection term has been widely studied in the existing
literature; for instance, [3, 7] dealt with equations with small diffusion coefficients.
The homogenization result for general linear periodic convection-diffusion operators
with nontrivial effective drift was obtained in [13].
2. Setting of the problem
We study the asymptotic behaviour of solutions of the Cauchy problem
∂uε
∂t
− div(Aε(t, x)∇uε) + ε−1bε(t, x, uε) · ∇uε = 0 in ]0, T [×Rn , (1)
uε(0, x) = ϕ(x), x ∈ Rn , (2)
with Aε(t, x) = A(t, x′/ε) and
bε(t, x, v) =
(
a
(
t,
x′
ε
)
, h
(
t,
x′
ε
)
f(v)
)
, (3)
where x = (x′, xn ) ∈ Rn , x′ ∈ Rn−1, xn ∈ R. The matrix function A(t, y) =
[Aij (t, y)], vector function a(t, y) = (a1(t, y), . . . , an−1(t, y)) and scalar function
h(t, y) are assumed to be 1-periodic in y = (y1, . . . , yn−1), and thus can be identified
with the corresponding functions on the (n − 1)-dimensional torus denoted by Y .
We suppose throughout this paper that the following hold.
(1) Coefficients A, a and h are of class C2per(Y ).
(2) The nonlinearity f ∈ C2(R).
(3) The initial condition ϕ ∈ C∞0 (Rn ).
(4) The diffusion tensor A is uniformly positive definite, that is there exists a
constant c0 > 0 such that for any (t, y) ∈ ]0, T [ × Y and ξ ∈ Rn
ξ ·A ξ =
n
∑
i,j=1
Aij (t, y) ξi ξj
c0 |ξ|2. (4)
Since the convection term here is not divergence free, the following auxiliary
problem plays an important role in further analysis:
divy (At ∇y z + a z) = 0 in Y, (5)
where we use the following notation for partial differential operators with respect
to y.
divy v =
n−1
∑
α=1
∂vα
∂yα
, ∇y χ =
(
∂χ
∂y1
, . . . ,
∂χ
∂yn−1
)
, (At ∇y z)α (t, y)
=
n−1
∑
β=1
Aαβ (t, y)
∂z
∂yβ
,
and we always assume that any function depending on y is Y periodic, that is
defined on the torus Y = Tn−1. Equation (5) is linear and has a nontrivial solution
in the space of Y -periodic functions. Furthermore, under the normalization
∫
Y
z(t, y) dy = 1 (6)