Local Existence of MHD Contact Discontinuities

We prove the local-in-time existence of solutions with a contact discontinuity of
the equations of ideal compressible magnetohydrodynamics (MHD) for two dimensional
planar flows provided that the Rayleigh–Taylor sign condition [∂p/∂N] < 0
on the jump of the normal derivative of the pressure is satisfied 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. This paper is a natural completion of our previous analysis (Morando et al.
in J Differ Equ 258:2531–2571, 2015) where the well-posedness in Sobolev spaces
of the linearized problem was proved under the Rayleigh–Taylor sign condition
satisfied at each point of the unperturbed discontinuity. The proof of the resolution
of the nonlinear problem given in the present paper follows from a suitable tame
a priori estimate in Sobolev spaces for the linearized equations and a Nash–Moser
iteration.