Existence and stability of viscous shock profiles for 2-D isentropic MHD with infinite electrical resistivity

TL;DR

This paper provides a comprehensive analysis of viscous shock profiles in 2D isentropic MHD with infinite resistivity, classifying all possible profiles and demonstrating their stability through numerical Evans function analysis.

Contribution

It offers the first complete classification of viscous shock profiles in 2D isentropic MHD with infinite resistivity, including stability results for various shock types.

Findings

Profiles include Lax, overcompressive, and undercompressive shocks.

Numerical analysis shows all profiles are linearly and nonlinearly stable.

Under certain conditions, undercompressive shocks are also stable.

Abstract

For the two-dimensional Navier--Stokes equations of isentropic magnetohydrodynamics (MHD) with $\gamma$-law gas equation of state, $\gamma\ge 1$, and infinite electrical resistivity, we carry out a global analysis categorizing all possible viscous shock profiles. Precisely, we show that the phase portrait of the traveling-wave ODE generically consists of either two rest points connected by a viscous Lax profile, or else four rest points, two saddles and two nodes. In the latter configuration, which rest points are connected by profiles depends on the ratio of viscosities, and can involve Lax, overcompressive, or undercompressive shock profiles. For the monatomic and diatomic cases $\gamma=5/3$ and $\gamma= 7/5$, with standard viscosity ratio for a nonmagnetic gas, we find numerically that the the nodes are connected by a family of overcompressive profiles bounded by Lax profiles connecting saddles to nodes, with no undercompressive shocks occurring. We carry out a systematic numerical Evans function analysis indicating that all of these two-dimensional shock profiles are linearly and nonlinearly stable, both with respect to two- and three-dimensional perturbations. For the same gas constants, but different viscosity ratios, we investigate also cases for which undercompressive shocks appear; these are seen numerically to be stable as well