Well-posedness of the linearized problem for contact MHD discontinuities

TL;DR

This paper proves the well-posedness of the linearized free boundary problem for contact discontinuities in ideal 2D compressible MHD flows under the Rayleigh-Taylor sign condition, ensuring stability and mathematical rigor.

Contribution

It establishes the well-posedness of the linearized contact discontinuity problem in 2D MHD under specific physical conditions, advancing theoretical understanding.

Findings

Well-posedness in Sobolev spaces for the linearized problem

Validation of the Rayleigh-Taylor sign condition for stability

Mathematical framework for contact discontinuities in MHD

Abstract

We study the free boundary problem for contact discontinuities in ideal compressible magnetohydrodynamics (MHD). They 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. Under the Rayleigh-Taylor sign condition $[\partial p/\partial N]<0$ on the jump of the normal derivative of the pressure satisfied at each point of the unperturbed contact discontinuity, we prove the well-posedness in Sobolev spaces of the linearized problem for 2D planar MHD flows.