One-dimensional stability of parallel shock layers in isentropic magnetohydrodynamic

TL;DR

This paper investigates the one-dimensional stability of parallel isentropic magnetohydrodynamic shock layers across various physical parameters, using a combination of asymptotic estimates and numerical Evans function computations, and finds that these shocks are stable.

Contribution

It extends previous stability analyses to the full range of physical parameters for isentropic MHD shocks, including different shock types and large-amplitude limits, with novel numerical techniques.

Findings

Shocks are stable across all examined parameters.

The shock layer is purely gas-dynamical, independent of magnetic field.

Stability persists despite changes in shock type and amplitude.

Abstract

Extending investigations of Barker, Humpherys, Lafitte, Rudd, and Zumbrun for compressible gas dynamics and Freist\"uhler and Trakhinin for compressible magnetohydrodynamics, we study by a combination of asymptotic ODE estimates and numerical Evans function computations the one-dimensional stability of parallel isentropic magnetohydrodynamic shock layers over the full range of physical parameters (shock amplitude, strength of imposed magnetic field, viscosity, magnetic permeability, and electrical resistivity) for a $\gamma$-law gas with $\gamma\in [1,3]$. Other $\gamma$-values may be treated similarly, but were not checked numerically. Depending on magnetic field strength, these shocks may be of fast Lax, intermediate (overcompressive), or slow Lax type; however, the shock layer is independent of magnetic field, consisting of a purely gas-dynamical profile. In each case, our results indicate stability. Interesting features of the analysis are the need to renormalize the Evans function in order to pass continuously across parameter values where the shock changes type or toward the large-amplitude limit at frequency $\lambda=0$ and the systematic use of winding number computations on Riemann surfaces.