Nonlinear stability and existence of two-dimensional compressible current-vortex sheets

TL;DR

This paper proves the local-in-time existence and nonlinear stability of two-dimensional current-vortex sheets in ideal compressible MHD under a specific weak stability condition, highlighting the stabilizing role of magnetic fields.

Contribution

It introduces a new stability condition for current-vortex sheets and establishes their existence and stability using Nash-Moser iteration, advancing understanding of MHD interface stability.

Findings

Weak linear stability condition identified

Energy estimates with derivative loss established

Nonlinear stability proved via Nash-Moser iteration

Abstract

We are concerned with nonlinear stability and existence of two-dimensional current-vortex sheets in ideal compressible magnetohydrodynamics. This is a nonlinear hyperbolic initial-boundary value problem with characteristic free boundary. It is well-known that current-vortex sheets may be at most weakly (neutrally) stable due to the existence of surface waves solutions that yield a loss of derivatives in the energy estimate of the solution with respect to the source terms. We first identify a sufficient condition ensuring the weak stability of the linearized current-vortex sheets problem. Under this stability condition for the background state, we show that the linearized problem obeys an energy estimate in anisotropic weighted Sobolev spaces with a loss of derivatives. Based on the weakly linear stability results, we then establish the local-in-time existence and nonlinear stability of current-vortex sheets by a suitable Nash-Moser iteration, provided the stability condition is satisfied at each point of the initial discontinuity. This result gives a new confirmation of the stabilizing effect of sufficiently strong magnetic fields on Kelvin-Helmholtz instabilities.