Existence of Local Solutions to a Free Boundary Problem for Compressible Viscous Magnetohydrodynamics

The motion of viscous compressible magnetohydrodynamics (MHD) is considered in a domain bounded by a free boundary. The motion interacts through the free surface with an electromagnetic field located in a domain that is exterior to the free surface and bounded by a given fixed surface. Some data for the electromagnetic fields are prescribed on this fixed boundary. On the free surface, jumps in magnetic and electric fields are assumed. We prove the local existence of solutions by the method of successive approximations using Sobolev–Slobodetskii spaces.