Stability of viscous shocks in isentropic gas dynamics

TL;DR

This paper investigates the spectral stability of viscous shock solutions in isentropic gas dynamics, extending known stability regimes, ruling out real unstable eigenvalues, and providing numerical evidence for stability at high Mach numbers.

Contribution

The authors extend stability analysis for viscous shocks, introduce a spectral energy estimate, and numerically demonstrate stability for large shocks up to Mach number 3000.

Findings

Small-amplitude shocks are spectrally stable in an extended parameter regime.

No purely real unstable eigenvalues exist for any shock strength.

Large-amplitude shocks are numerically stable up to Mach number approximately 3000.

Abstract

In this paper, we examine the stability problem for viscous shock solutions of the isentropic compressible Navier--Stokes equations, or $p$-system with real viscosity. We first revisit the work of Matsumura and Nishihara, extending the known parameter regime for which small-amplitude viscous shocks are provably spectrally stable by an optimized version of their original argument. Next, using a novel spectral energy estimate, we show that there are no purely real unstable eigenvalues for any shock strength. By related estimates, we show that unstable eigenvalues are confined to a bounded region independent of shock strength. Then through an extensive numerical Evans function study, we show that there is no unstable spectrum in the entire right-half plane, thus demonstrating numerically that large-amplitude shocks are spectrally stable up to Mach number $M\approx 3000$ for $1 \le \gamma \leq 3$. This strongly suggests that shocks are stable independent of amplitude and the adiabatic constant $\gamma$. We complete our study by showing that finite-difference simulations of perturbed large-amplitude shocks converge to a translate of the original shock wave, as expected.