On Well-posedness of the Two-dimensional MHD–Maxwell Free Interface Problem

We study the well-posedness of the 2D plasma–vacuum free interface problem in ideal compressible magnetohydrodynamics (MHD) taking into account the influence of the displacement current in vacuum. That is, in the plasma region the planar flow is governed by the 2D ideal compressible MHD equations, while in the vacuum region we consider the Maxwell system for electric and magnetic fields.

By using a suitable secondary symmetrization of the Maxwell system, we prove the well-posedness of the corresponding variable coefficient linearized problem, provided that at least one of the two unperturbed magnetic fields, in the plasma or in the vacuum region, is nonzero at each point of the interface. We also briefly discuss the proof of the local well-posedness theorem for the original nonlinear free boundary problem under the same nonzero requirement for the magnetic fields at the initial interface.