Local Well-posedness of the Free-Boundary Problem in Compressible Resistive Magnetohydrodynamics

TL;DR

This paper establishes local well-posedness for the free-boundary problem in compressible resistive magnetohydrodynamics, demonstrating existence, uniqueness, and stability of solutions under specific physical and mathematical conditions.

Contribution

It introduces a novel approach combining Lagrangian coordinates and tangential smoothing to prove well-posedness for a complex MHD free-boundary problem with magnetic diffusion.

Findings

Proves local well-posedness in Sobolev spaces for the MHD free-boundary problem.

Shows magnetic diffusion and elliptic estimates control magnetic field, pressure, and Lorentz force.

Utilizes Christodoulou-Lindblad elliptic estimates with magnetic diffusion for energy control.

Abstract

We prove the local well-posedness in Sobolev spaces of the free-boundary problem for compressible inviscid resistive isentropic MHD system under the Rayleigh-Taylor physical sign condition, which describes the motion of a free-boundary compressible plasma in an electro-magnetic field with magnetic diffusion. We use Lagrangian coordinates and apply the tangential smoothing method introduced by Coutand-Shkoller to construct the approximation solutions. One of the key observations is that the Christodoulou-Lindblad type elliptic estimate together with magnetic diffusion not only gives the common control of magnetic field and fluid pressure directly, but also controls the Lorentz force which is a higher order term in the energy functional.