Well-posedness of the plasma-vacuum interface problem

TL;DR

This paper proves the well-posedness of the plasma-vacuum interface problem in ideal compressible MHD under a stability condition, establishing existence and uniqueness of solutions using advanced mathematical techniques.

Contribution

It introduces a rigorous proof of well-posedness for the nonlinear plasma-vacuum interface problem with a new stability condition and detailed Sobolev space analysis.

Findings

Existence and uniqueness of solutions under stability conditions

Development of energy estimates for the nonlinear problem

Application of Nash-Moser iteration for solution construction

Abstract

We consider the free boundary problem for the plasma-vacuum interface in ideal compressible magnetohydrodynamics (MHD). In the plasma region the flow is governed by the usual compressible MHD equations, while in the vacuum region we consider the pre-Maxwell dynamics for the magnetic field. At the free-interface, driven by the plasma velocity, the total pressure is continuous and the magnetic field on both sides is tangent to the boundary. The plasma-vacuum system is not isolated from the outside world, because of a given surface current on the fixed boundary that forces oscillations. Under a suitable stability condition satisfied at each point of the initial interface, stating that the magnetic fields on either side of the interface are not collinear, we show the existence and uniqueness of the solution to the nonlinear plasma-vacuum interface problem in suitable anisotropic Sobolev spaces. The proof is based on the results proved in the companion paper arXiv:1112.3101, about the well-posedness of the homogeneous linearized problem and the proof of a basic a priori energy estimate. The proof of the resolution of the nonlinear problem given in the present paper follows from the analysis of the elliptic system for the vacuum magnetic field, a suitable tame estimate in Sobolev spaces for the full linearized equations, and a Nash-Moser iteration.