On the well-posedness of a linearized plasma-vacuum interface problem in ideal compressible MHD

TL;DR

This paper investigates the mathematical well-posedness of a linearized plasma-vacuum interface problem in ideal compressible MHD, identifying conditions under which the problem is well-posed and establishing key estimates.

Contribution

It analyzes the linearized problem's well-posedness in two cases, providing new a priori estimates for the variable coefficient problem in anisotropic Sobolev spaces.

Findings

Well-posedness in the gas dynamical case with negative pressure derivative jump

Existence of well-posedness in the purely MHD case with non-zero, non-parallel magnetic fields

Established a priori estimates in anisotropic weighted Sobolev spaces

Abstract

We study the initial-boundary value problem resulting from the linearization of the plasma-vacuum interface problem in ideal compressible magnetohydrodynamics (MHD). We suppose that the plasma and the vacuum regions are unbounded domains and the plasma density does not go to zero continuously, but jumps. For the basic state upon which we perform linearization we find two cases of well-posedness of the "frozen" coefficient problem: the "gas dynamical" case and the "purely MHD" case. In the "gas dynamical" case we assume that the jump of the normal derivative of the total pressure is always negative. In the "purely MHD" case this condition can be violated but the plasma and the vacuum magnetic fields are assumed to be non-zero and non-parallel to each other everywhere on the interface. For this case we prove a basic a priori estimate in the anisotropic weighted Sobolev space $H^1_*$ for the variable coefficient problem.