Stability of the linearized MHD-Maxwell free interface problem

Davide Catania; Marcello D'Abbicco; Paolo Secchi

doi:10.3934/cpaa.2014.13.2407

Article Contents

2014, Volume 13, Issue 6: 2407-2443. Doi: 10.3934/cpaa.2014.13.2407

This issue Previous Article Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation Next Article Weak solutions to the equations of stationary magnetohydrodynamic flows in porous media

Stability of the linearized MHD-Maxwell free interface problem

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

Received: November 2013

Revised: April 2014

Published: November 2014

Abstract / Introduction Related Papers Cited by

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 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.

Keywords:

Ideal compressible magneto-hydrodynamics,
Maxwell equations,
plasma-vacuum interface,
characteristic free boundary,
nonuniformly characteristic boundary.

Mathematics Subject Classification: Primary: 76W05; Secondary: 35Q35, 35L50, 76E17, 76E25, 35R35, 76B03.

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