Local existence of MHD contact discontinuities - Paola Trebeschi (Università di Brescia)

Abstract:

We prove the local-in-time existence of solutions with a contact discontinuity of the equations of ideal
compressible magnetohydrodynamics (MHD) for 2D planar  flows provided that the Rayleigh-Taylor sign
condition  on the jump of the normal derivative of the pressure is satised at each point
of the initial discontinuity.  MHD contact discontinuities 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. We first prove the well-posedness in Sobolev spaces of the
linearized problem under the Rayleigh-Taylor sign condition satisfied at each point of the unperturbed
discontinuity. Then, the proof of the resolution of the nonlinear problem follows from a suitable tame a
priori estimate in Sobolev spaces for the linearized equations and a Nash-Moser iteration.
This is a joint work with A. Morando and Y. Trakhinin.