Structural stability of shock waves and current-vortex sheets in shallow water magnetohydrodynamics

TL;DR

This paper investigates the stability of shock waves and current-vortex sheets in shallow water magnetohydrodynamics, establishing conditions for their local-in-time existence, uniqueness, and stability based on fluid height changes and symmetrization techniques.

Contribution

It provides the first stability criteria for shock waves and current-vortex sheets in SMHD, linking stability to fluid height variations and applying a novel symmetrization approach.

Findings

Shock waves are stable if fluid height increases across the shock.

Current-vortex sheets are stable under a specific symmetrization condition.

The stability conditions are derived using analogies with elastodynamics.

Abstract

We study the structural stability of shock waves and current-vortex sheets in shallow water magnetohydrodynamics (SMHD) in the sense of the local-in-time existence and uniqueness of discontinuous solutions satisfying corresponding jump conditions. The equations of SMHD form a symmetric hyperbolic system which is formally analogous to the system of 2D compressible elastodynamics for particular nonphysical deformations. Using this analogy and the recent results in [Morando A., Trakhinin Y., Trebeschi P. Math. Ann. (2019), https://doi.org/10.1007/s00208-019-01920-6] for shock waves in 2D compressible elastodynamics, we prove that shock waves in SMHD are structurally stable if and only if the fluid height increases across the shock front. For current-vortex sheets the fluid height is continuous whereas the tangential components of the velocity and the magnetic field may have a jump. Applying a so-called secondary symmetrization of the symmetric system of SMHD equations, we find a condition sufficient for the structural stability of current-vortex sheets.