The inviscid limit of viscous Burgers at nondegenerate shock formation

TL;DR

This paper analyzes the vanishing viscosity limit of the 1D Burgers equation near shock formation, developing a novel asymptotic expansion to accurately describe solutions up to shock onset.

Contribution

It introduces a matched asymptotic expansion with a fractional spacetime Taylor series, providing sharp viscosity rate estimates near shock formation.

Findings

Derived an asymptotic expansion valid up to shock formation

Established sharp vanishing viscosity rates in various norms

Partially addressed limitations of prior results near shocks

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.