On the shock wave spectrum for isentropic gas dynamics with capillarity

TL;DR

This paper investigates the spectral stability of shock waves in isentropic gas dynamics with capillarity, demonstrating stability for small and moderate amplitudes and analyzing the transition from monotone to oscillatory profiles.

Contribution

It introduces novel spectral energy estimates and conducts a comprehensive Evans function analysis to assess stability across different shock and capillarity regimes.

Findings

Small-amplitude shocks are spectrally stable.

Monotone shocks have no unstable real spectrum.

Transition to oscillatory profiles does not induce instabilities.

Abstract

We consider the stability problem for shock layers in Slemrod's model of an isentropic gas with capillarity. We show that these traveling waves are monotone in the weak capillarity case, and become highly oscillatory as the capillarity strength increases. Using a spectral energy estimate we prove that small-amplitude monotone shocks are spectrally stable. We also show, through the use of a novel spectral energy estimate, that monotone shocks have no unstable real spectrum regardless of amplitude; this implies that any instabilities of these monotone traveling waves, if they exist, must occur through a Hopf-like bifurcation, where one or more conjugate pairs of eigenvalues cross the imaginary axis. We then conduct a systematic numerical Evans function study, which shows that monotone and mildly oscillatory profiles in an adiabatic gas are spectrally stable for moderate values of shock and capillarity strengths. In particular, we show that the transition from monotone to non-monotone profiles does not appear to trigger any instabilities.