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

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.

arXiv comment51 pages. Typos corrected. Final version, accepted by CVPDE

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• Local Well-posedness and Incompressible Limit of the Free-Boundary Problem in Compressible Elastodynamics
• Local Well-posedness of the Free-Boundary Incompressible Magnetohydrodynamics with Surface Tension
• Anisotropic Regularity of the Free-Boundary Problem in Compressible Ideal Magnetohydrodynamics
• Incompressible Limit of Compressible Ideal MHD Flows inside a Perfectly Conducting Wall