  We consider the free boundary problem for the plasma-vacuum interface in
ideal compressible magnetohydrodynamics (MHD). In the plasma region the flow is
governed by the usual compressible MHD equations, while in the vacuum region we
consider the pre-Maxwell dynamics for the magnetic field. At the
free-interface, driven by the plasma velocity, the total pressure is continuous
and the magnetic field on both sides is tangent to the boundary. The
plasma-vacuum system is not isolated from the outside world, because of a given
surface current on the fixed boundary that forces oscillations.
  Under a suitable stability condition satisfied at each point of the initial
interface, stating that the magnetic fields on either side of the interface are
not collinear, we show the existence and uniqueness of the solution to the
nonlinear plasma-vacuum interface problem in suitable anisotropic Sobolev
spaces. The proof is based on the results proved in the companion paper
arXiv:1112.3101, about the well-posedness of the homogeneous linearized problem
and the proof of a basic a priori energy estimate. The proof of the resolution
of the nonlinear problem given in the present paper follows from the analysis
of the elliptic system for the vacuum magnetic field, a suitable tame estimate
in Sobolev spaces for the full linearized equations, and a Nash-Moser
iteration.
