On boundary layers for the Burgers equations in a bounded domain

•The nonlinear interactions due to the quadratic nonlinear term are explicitly analyzed.•The asymptotic analysis is fully analyzed at any order of epsilon.•The strong convergence analysis is done. Previously, the analysis was done only in L1 space. We explicitly did in L2 and H1 spaces using the so-called correctors. To the best of our knowledge, this is new. 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 uε as the viscosity parameter ε gets smaller. The idea is constructing the asymptotic expansions in the order of the ε and validating the convergence of the expansions to the solutions uε as ε → 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 ε using the so-called boundary layer correctors. We also numerically verify the convergences.