Analysis of Mixed Elliptic and Parabolic Boundary Layers with Corners

Abstract

We study the asymptotic behavior at small diffusivity of the solutions, uε, to a convection-diffusion equation in a rectangular domain Ω. The diffusive equation is supplemented with a Dirichlet boundary condition, which is smooth along the edges and continuous at the corners. To resolve the discrepancy, on ∂Ω, between uε and the corresponding limit solution, u0, we propose asymptotic expansions of uε at any arbitrary, but fixed, order. In order to manage some singular effects near the four corners of Ω, the so-called elliptic and ordinary corner correctors are added in the asymptotic expansions as well as the parabolic and classical boundary layer functions. Then, performing the energy estimates on the difference of uε and the proposed expansions, the validity of our asymptotic expansions is established in suitable Sobolev spaces.