Analysis of Mixed Elliptic and Parabolic Boundary Layers with Corners

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