The inviscid limit of viscous Burgers at nondegenerate shock formation

Abstract

We study the vanishing viscosity limit of the one-dimensional Burgers equation near nondegenerate shock formation. We develop a matched asymptotic expansion that describes small-viscosity solutions to arbitrary order up to the moment the first shock forms. The inner part of this expansion has a novel structure based on a fractional spacetime Taylor series for the inviscid solution. We obtain sharp vanishing viscosity rates in a variety of norms, including $L^\infty$. Comparable prior results break down in the vicinity of shock formation. We partially fill this gap.