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

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.

arXiv comment40 pages

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• Local Well-posedness for the Motion of a Compressible Gravity Water Wave with Vorticity
• Local Well-posedness of the Free-Boundary Incompressible Magnetohydrodynamics with Surface Tension
• Zero Surface Tension Limit of the Free-Boundary Problem in Incompressible Magnetohydrodynamics
• Well-posedness for Moving Interfaces with Surface Tension in Ideal Compressible MHD
• Nonlinear Stability of MHD Contact Discontinuities with Surface Tension
• Local Well-posedness of the Free-Boundary Problem in Compressible Resistive Magnetohydrodynamics
• Anisotropic Regularity of the Free-Boundary Problem in Compressible Ideal Magnetohydrodynamics
• A priori estimates and a blow-up criterion for the incompressible ideal MHD equations with surface tension and a closed free surface
• Incompressible Limit of Compressible Ideal MHD Flows inside a Perfectly Conducting Wall
• Well-posedness for moving interfaces in anisotropic plasmas