Existence of Multi-dimensional Contact Discontinuities for the Ideal Compressible Magnetohydrodynamics

TL;DR

This paper proves the local existence and uniqueness of multi-dimensional contact discontinuities in ideal compressible MHD without the Rayleigh--Taylor sign condition, using Lagrangian formulation and special unknowns, for both 2D and 3D cases.

Contribution

It establishes the well-posedness of contact discontinuities in ideal compressible MHD without additional sign conditions, solving two open questions in the field.

Findings

Proves local existence and uniqueness in Sobolev spaces.

Removes the Rayleigh--Taylor sign condition requirement.

Constructs solutions as inviscid limits of viscous approximations.

Abstract

We establish the local existence and uniqueness of multi-dimensional contact discontinuities for the ideal compressible magnetohydrodynamics (MHD) in Sobolev spaces, which are most typical interfacial waves for astrophysical plasmas and prototypical fundamental waves for hyperbolic systems of conservation laws. Such waves are characteristic discontinuities for which there is no flow across the discontinuity surface while the magnetic field crosses transversely, which lead to a two-phase free boundary problem where the pressure, velocity and magnetic field are continuous across the interface whereas the entropy and density may have jumps. To overcome the difficulties of possible nonlinear Rayleigh--Taylor instability and loss of derivatives, here we use crucially the Lagrangian formulation and Cauchy's celebrated integral (1815) for the magnetic field. These motivate us to define two special good unknowns; one enables us to capture the boundary regularizing effect of the transversal magnetic field on the flow map, and the other one allows us to get around the troublesome boundary integrals due to the transversality of the magnetic field. In particular, our result removes the additional assumption of the Rayleigh--Taylor sign condition required by Morando, Trakhinin and Trebeschi (\emph{J. Differential Equations} \textbf{258} (2015), no. 7, 2531--2571; \emph{Arch. Ration. Mech. Anal.} \textbf{228} (2018), no. 2, 697--742) and holds for both 2D and 3D and hence gives a complete answer to the two open questions raised therein. Moreover, there is {\it no loss of derivatives} in our well-posedness theory. The solution is constructed as {\it the inviscid limit} of solutions to some suitably-chosen nonlinear approximate problems for the two-phase compressible viscous non-resistive MHD.