Spectral stability of shock profiles for hyperbolically regularized systems of conservation laws

TL;DR

This paper proves spectral stability of shock profiles in hyperbolically regularized conservation laws for small shock amplitudes, extending previous results from parabolic and relaxation systems.

Contribution

It establishes spectral stability of shock profiles in hyperbolically regularized systems, generalizing prior stability results to a broader class of conservation laws.

Findings

Spectral stability holds for small shock amplitudes.

The Evans function has only one zero at zero, indicating stability.

Extends stability results from parabolic and relaxation systems.

Abstract

We report a proof that under natural assumptions shock profiles viewed as heteroclinic travelling wave solutions to a hyperbolically regularized system of conservation laws are spectrally stable, if the shock amplitude is sufficiently small. This means that an associated Evans function $\mathcal{E}:\Lambda\rightarrow\mathbb{C}$ with $\Lambda\subset\mathbb{C}$ an open superset of the closed right half plane $\mathbb{H}^+\equiv\{\kappa\in\mathbb{C}:\text{Re}\,\kappa \geq 0\}$, has only one zero, namely a simple zero at $0$. The result is analogous to the one obtained in [FS02] and [PZ04] for parabolically regularized systems of conservation laws, and also distinctly extends findings on hyperbolic relaxation systems in [PZ04], [MZ09], [Ued09] .