On local existence of MHD contact discontinuities

Alessandro  Morando; Yuri  Trakhinin; Paola Trebeschi

doi:10.3934/dcdss.2016.9.289

Article Contents

2016, Volume 9, Issue 1: 289-313. Doi: 10.3934/dcdss.2016.9.289

This issue Previous Article Nonconforming mixed finite element approximation of a fluid-structure interaction spectral problem Next Article Local well-posedness of free surface problems for the Navier-Stokes equations in a general domain

On local existence of MHD contact discontinuities

1.
DICATAM, Sezione di Matematica, Università di Brescia, Via Valotti, 9, 25133 Brescia
2.
Sobolev Institute of Mathematics, Koptyug av. 4, 630090 Novosibirsk
3.
Dipartimento di Matematica, Università di Brescia, Facoltà di Ingegneria, Via Valotti 9, 25133 Brescia

Received: August 31, 2014

Revised: January 31, 2015

Published: January 2016

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Abstract

We present a recent result [23] for the free boundary problem for contact discontinuities in ideal compressible magnetohydrodynamics (MHD). They are characteristic discontinuities with no flow across the discontinuity for which the pressure, the magnetic field and the velocity are continuous whereas the density and the entropy may have a jump. Under the Rayleigh-Taylor sign condition $[\partial p/\partial N]<0$ on the jump of the normal derivative of the pressure satisfied at each point of the unperturbed contact discontinuity, we prove the well-posedness in Sobolev spaces of the linearized problem for 2D planar MHD flows. This is a necessary step to prove a local-in-time existence theorem [24] for the original nonlinear free boundary problem provided that the Rayleigh-Taylor sign condition is satisfied at each point of the initial discontinuity. The uniqueness of a solution to this problem follows already from the basic a priori estimate deduced for the linearized problem.

Keywords:

Ideal compressible magnetohydrodynamics,
contact discontinuity,
Rayleigh-Taylor sign condition,
well-posedness.

Mathematics Subject Classification: Primary: 76W05; Secondary: 35L50, 35L65.

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