Well-Posedness for the Free-Boundary Ideal Compressible Magnetohydrodynamic Equations with Surface Tension

TL;DR

This paper proves local existence and uniqueness of solutions for the free-boundary ideal compressible MHD equations with surface tension in 3D, using a modified Nash--Moser scheme and anisotropic Sobolev spaces.

Contribution

It introduces a novel approach combining tame estimates, boundary regularity from surface tension, and an $ ext{varepsilon}$-regularization to establish well-posedness.

Findings

Successful application of Nash--Moser iteration to free-boundary MHD equations

Development of tame estimates leveraging boundary regularity

Construction of solutions via $ ext{varepsilon}$-regularization and limit passage

Abstract

We establish the local existence and uniqueness of solutions to the free-boundary ideal compressible magnetohydrodynamic equations with surface tension in three spatial dimensions by a suitable modification of the Nash--Moser iteration scheme. The main ingredients in proving the convergence of the scheme are the tame estimates and unique solvability of the linearized problem in the anisotropic Sobolev spaces $H_*^m$ for $m$ large enough. In order to derive the tame estimates, we make full use of the boundary regularity enhanced from the surface tension. The unique solution of the linearized problem is constructed by designing some suitable $\varepsilon$--regularization and passing to the limit $\varepsilon\to 0$.