Well-Posedness of Free Boundary Problem in Non-relativistic and Relativistic Ideal Compressible Magnetohydrodynamics

TL;DR

This paper proves the local well-posedness of a free boundary problem in ideal compressible magnetohydrodynamics, including both non-relativistic and relativistic cases, under specific physical conditions.

Contribution

It establishes the first well-posedness result for the free boundary problem with zero total pressure in ideal compressible MHD, uniform in light speed.

Findings

Existence and uniqueness of solutions under Rayleigh--Taylor condition

Use of anisotropic Sobolev spaces and Nash--Moser scheme

Results are uniform in the light speed parameter

Abstract

We consider the free boundary problem for non-relativistic and relativistic ideal compressible magnetohydrodynamics in two and three spatial dimensions with the total pressure vanishing on the plasma--vacuum interface. We establish the local-in-time existence and uniqueness of solutions to this nonlinear characteristic hyperbolic problem under the Rayleigh--Taylor sign condition on the total pressure. The proof is based on certain tame estimates in anisotropic Sobolev spaces for the linearized problem and a modification of the Nash--Moser iteration scheme. Our result is uniform in the light speed and appears to be the first well-posedness result for the free boundary problem in ideal compressible magnetohydrodynamics with zero total pressure on the moving boundary.