A Class Of Viscous Conservation Laws With Boundary Layer

This paper is concerned with the asymptotic limiting behavior of the solution to the onedimensional quasi-linear scalar viscous equation with a weak boundary layer and an expansive strong boundary layer. It is proved that the boundary layers are nonlinearly stable for viscous conservation laws with genuinely-nonlinear fluxes. The proofs are based on the methods of matched asymptotic analysis and basic energy estimates, which depend crucially on the structure of the underlying boundary layers. This article is organized as follows. The first chapter describes the question, gives the equation model and the main conclusions. In the second chapter, using the matching asymptotic analysis method constructs the approximate solution. The third Chapter is stability analysis, which is the most important part of the article theoretical analysis. We prove proposition 1 and Theorem 1 by using priori estimates.