On well-posedness of the plasma-vacuum interface problem: the case of non-elliptic interface symbol

TL;DR

This paper investigates the well-posedness of the plasma-vacuum interface problem in magnetohydrodynamics when the interface symbol is non-elliptic, establishing conditions for stability and ill-posedness.

Contribution

It extends the analysis of plasma-vacuum interface problems to non-elliptic cases, providing a priori estimates and identifying instability conditions.

Findings

Proves an $L^2$ estimate under the Rayleigh-Taylor sign condition.

Constructs an ill-posedness example for non-elliptic interface symbols.

Shows instability when both non-collinearity and Rayleigh-Taylor conditions fail.

Abstract

We consider the plasma-vacuum interface problem in a classical statement when in the plasma region the flow is governed by the equations of ideal compressible magnetohydrodynamics, while in the vacuum region the magnetic field obeys the div-curl system of pre-Maxwell dynamics. The local-in-time existence and uniqueness of the solution to this problem in suitable anisotropic Sobolev spaces was proved in [P. Secchi, Y. Trakhinin, Nonlinearity 27 (2014), 105-169], provided that at each point of the initial interface the plasma density is strictly positive and the magnetic fields on either side of the interface are not collinear. The non-collinearity condition appears as the requirement that the symbol associated to the interface is elliptic. We now consider the case when this symbol is not elliptic and study the linearized problem, provided that the unperturbed plasma and vacuum non-zero magnetic fields are collinear on the interface. We prove a basic a priori $L^2$ estimate for this problem under the (generalized) Rayleigh-Taylor sign condition $[\partial q/\partial N]<0$ on the jump of the normal derivative of the unperturbed total pressure satisfied at each point of the interface. By constructing an Hadamard-type ill-posedness example for the frozen coefficients linearized problem we show that the simultaneous failure of the non-collinearity condition and the Rayleigh-Taylor sign condition leads to Rayleigh-Taylor instability.