# American Institute of Mathematical Sciences

November  2014, 13(6): 2407-2443. doi: 10.3934/cpaa.2014.13.2407

## Stability of the linearized MHD-Maxwell free interface problem

 1 DICATAM, Mathematical Division, University of Brescia, Via Valotti, 9, 25133 Brescia, Italy, Italy 2 Dipartimento di Matematica, Facoltà di Ingegneria, Università di Brescia, Via Valotti, 9, 25133 Brescia

Received  November 2013 Revised  April 2014 Published  July 2014

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 Maxwell system for the electric and the magnetic fields, in order to investigate the well-posedness of the problem, in particular in relation with the electric field in vacuum. 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.

Under suitable stability conditions satisfied at each point of the plasma-vacuum interface, we derive a basic a priori estimate for solutions to the linearized problem in the Sobolev space $H^1_{\tan}$ with conormal regularity. The proof follows by a suitable secondary symmetrization of the Maxwell equations in vacuum and the energy method.

An interesting novelty is represented by the fact that the interface is characteristic with variable multiplicity, so that the problem requires a different number of boundary conditions, depending on the direction of the front velocity (plasma expansion into vacuum or viceversa). To overcome this difficulty, we recast the vacuum equations in terms of a new variable which makes the interface characteristic of constant multiplicity. In particular, we don't assume that plasma expands into vacuum.
Citation: Davide Catania, Marcello D'Abbicco, Paolo Secchi. Stability of the linearized MHD-Maxwell free interface problem. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2407-2443. doi: 10.3934/cpaa.2014.13.2407
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