# American Institute of Mathematical Sciences

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February  2016, 9(1): 289-313. doi: 10.3934/dcdss.2016.9.289

## 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, Russian Federation 3 Dipartimento di Matematica, Università di Brescia, Facoltà di Ingegneria, Via Valotti 9, 25133 Brescia

Received  September 2014 Revised  February 2015 Published  December 2015

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.
Citation: Alessandro Morando, Yuri Trakhinin, Paola Trebeschi. On local existence of MHD contact discontinuities. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 289-313. doi: 10.3934/dcdss.2016.9.289
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