Local existence of MHD contact discontinuities

TL;DR

This paper establishes the local-in-time existence of solutions with contact discontinuities in 2D ideal compressible MHD flows, under the Rayleigh-Taylor sign condition, using a Nash-Moser iteration scheme.

Contribution

It proves the nonlinear local existence of MHD contact discontinuities satisfying the Rayleigh-Taylor condition, extending previous linearized analysis to the nonlinear regime.

Findings

Existence of solutions with contact discontinuities under Rayleigh-Taylor condition

Application of Nash-Moser iteration for nonlinear problem

Extension of previous linearized well-posedness results

Abstract

We prove the local-in-time existence of solutions with a contact discontinuity of the equations of ideal compressible magnetohydrodynamics (MHD) for 2D planar flows provided that the Rayleigh-Taylor sign condition $[\partial p/\partial N]<0$ on the jump of the normal derivative of the pressure is satisfied at each point of the initial discontinuity. MHD contact discontinuities are characteristic discontinuities with no flow across the discontinuity for which the pressure, the magnetic field and the velocity are continuous whereas the density and the entropy may have a jump. This paper is a natural completion of our previous analysis [Morando A., Trakhinin Y., Trebeschi P., J. Differential Equations 258:2531--2571, 2015] where the well-posedness in Sobolev spaces of the linearized problem was proved under the Rayleigh-Taylor sign condition satisfied at each point of the unperturbed discontinuity. The proof of the resolution of the nonlinear problem given in the present paper follows from a suitable tame a priori estimate in Sobolev spaces for the linearized equations and a Nash-Moser iteration.