Numerical proof of stability of viscous shock profiles

TL;DR

This paper presents the first rigorous numerical proof of the stability of viscous shock profiles in isentropic gas dynamics using Evans function computations, covering a wide range of shock strengths.

Contribution

It introduces a novel numerical approach for rigorously verifying the stability of viscous shock profiles across various shock strengths.

Findings

Validated stability for shock Mach numbers from 1.86 to 1669.

Established a framework for automatic rigorous stability verification.

Opened pathways for proving stability of all shocks in specific systems.

Abstract

We carry out the first rigorous numerical proof based on Evans function computations of stability of viscous shock profiles, for the system of isentropic gas dynamics with monatomic equation of state. We treat a selection of shock strengths ranging from the lower stability boundary of Mach number $\approx 1.86 $, below which profiles are known by energy estimates to be stable, to the upper stability boundary of $\approx 1669$, above which profiles are expected to be provable by rigorous asymptotic analysis to be stable. These results open the possibilities of: (i) automatic rigorous verification of stability or instability of individual shocks of general systems, and (ii) rigorous proof of stability of all shocks of particular systems.