  We are concerned with nonlinear stability and existence of two-dimensional
current-vortex sheets in ideal compressible magnetohydrodynamics. This is a
nonlinear hyperbolic initial-boundary value problem with characteristic free
boundary. It is well-known that current-vortex sheets may be at most weakly
(neutrally) stable due to the existence of surface waves solutions that yield a
loss of derivatives in the energy estimate of the solution with respect to the
source terms. We first identify a sufficient condition ensuring the weak
stability of the linearized current-vortex sheets problem. Under this stability
condition for the background state, we show that the linearized problem obeys
an energy estimate in anisotropic weighted Sobolev spaces with a loss of
derivatives. Based on the weakly linear stability results, we then establish
the local-in-time existence and nonlinear stability of current-vortex sheets by
a suitable Nash-Moser iteration, provided the stability condition is satisfied
at each point of the initial discontinuity. This result gives a new
confirmation of the stabilizing effect of sufficiently strong magnetic fields
on Kelvin-Helmholtz instabilities.
