On boundary layers for the Burgers equations in a bounded domain

Abstract

As a simplified model derived from the Navier-Stokes equations, we consider the viscous Burgers equations in a bounded domain with two-point boundary conditions. We investigate the singular behaviors of their solutions u(epsilon) as the viscosity parameter epsilon gets smaller. The idea is constructing the asymptotic expansions in the order of the epsilon and validating the convergence of the expansions to the solutions as epsilon -> 0. In this article, we consider the case where sharp transitions occur at the boundaries, i.e. boundary layers, and we fully analyze the convergence at any order of epsilon using the so-called boundary layer correctors. We also numerically verify the convergences.