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.