Analysis of Mixed Elliptic and Parabolic Boundary Layers with Corners

Abstract

We study the asymptotic behavior at small diffusivity of the solutions,$u^{\epsilon }$, to a convection-diffusion equation in a rectangular domain $\Omega$. The diffusiveequation is supplemented with a Dirichlet boundary condition, which is smoothalong the edges and continuous at the corners. To resolve the discrepancy, on $\partial \Omega$, between $u^{\epsilon }$ and the corresponding limit solution, $u^0$, we propose asymptotic expansionsof $u^{\epsilon }$ at any arbitrary, but fixed, order. In order to manage some singulareffects near the four corners of $\Omega$, 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 $u^{\epsilon }$ and the proposed expansions, the validity of our asymptotic expansions isestablished in suitable Sobolev spaces.