Viscous Boundary Layers and Their Stability (I)

Abstract

This paper is concerned with the asympootic limiting behavior of solutions to one-dimensional quasilinear scalar viscous equations for small viscosity in the presence of boundaries. We consider only non-characteristic boundary conditions. The main goals are to understand the evolution of viscous boundary layers, to construct the leading asymptotic ansatz which is uniformly valid up to the boundaries, and to obtain rigorously the uniform convergence to smooth solution of the associated inviscid hyperbolic equations away from the boundaries.