On well-posedness of the free boundary problem for ideal compressible MHD equations and Maxwell equations in vacuum

We survey results on the well-posedness of the free interface problem when an interface separates a perfectly conducting inviscid fluid (e.g., plasma) from a vacuum. The fluid flow is governed by the equations of ideal compressible magnetohydrodynamics (MHD). Unlike the classical statement, when the vacuum magnetic field obeys the div-curl system of pre-Maxwell dynamics, we do not neglect the displacement current in the vacuum region and consider the Maxwell equations for electric and magnetic fields. With boundary conditions on the interface this forms a nonlinear hyperbolic problem with a characteristic free boundary. The statement of this free boundary problem comes from the relativistic setting where the displacement current in vacuum cannot be neglected. We also briefly discuss the recent result showing the stabilizing effect of surface tension.