We study 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\ud Rayleigh–Taylor sign condition [∂p/∂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

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 [∂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, Trakhinin, Trebeschi in J Differential Equations 258: 2015) where the wellposedness 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.

Abstract. We consider the free boundary problem for current-vortex sheets in ideal incompressible magneto-hydrodynamics. It is known that current-vortex sheets may be at most weakly (neutrally) stable due to the existence of surface waves solutions to the linearized equations. The existence of such waves may yield a loss of derivatives in the energy estimate of the solution with respect to the source terms. However, under a suitable stability condition satisfied at each point of the initial discontinuity and a flatness condition on the initial front, we prove an a priori estimate in Sobolev spaces for smooth solutions with no loss of derivatives. The result of this paper gives some hope for proving the local existence of smooth current-vortex sheets without resorting to a Nash-Moser iteration. Such result would be a rigorous confirmation of the stabilizing effect of the magnetic field on Kelvin-Helmholtz instabilities, which is well known in astrophysics.

Abstract. We study the free boundary problem for the plasma-vacuum interface in ideal incompressible magnetohydrodynamics (MHD). In the vacuum region the magnetic field is described by the div-curl system of pre-Maxwell dynamics, while at the interface the total pressure is continuous and the magnetic field is tangent to the boundary. Under a suitable stability condition satisfied at each point of the plasma-vacuum interface, we prove the well-posedness of the linearized problem in Sobolev spaces.

The purpose of this paper is to give a compactness-continuity result for the solution of a nonlinear Dirichlet problem in terms of its domain variation. The topology in the family of domains is given by the Hausdorff metric and continuity is obtained under capacity conditions. A generalisation of Sverak's result in iV-dimensions is deduced as a particular case.