Orbital stability of undercompressive viscous shock waves under $L^1\cap H^4$ perturbation

TL;DR

This paper proves the stability of undercompressive viscous shock waves under certain perturbations, extending previous results for Lax waves by introducing a new vertical estimate that simplifies the analysis.

Contribution

The authors introduce a new vertical estimate that simplifies and extends the stability analysis of viscous shock waves to the undercompressive case.

Findings

Extended stability results to undercompressive shocks.

Simplified the proof using the new vertical estimate.

Generalized previous Lax wave stability results.

Abstract

By the use of a new vertical estimate introduced by the authors in the context of relaxation shocks for shallow water flow, we both simplify and extend the basic $L^1\cap H^3$ stability results of Mascia and Zumbrun for viscous shock waves, in particular extending their results for Lax waves to the undercompressive case.