  We consider a free boundary problem for the axially symmetric incompressible
ideal magnetohydrodynamic equations that describes the motion of the plasma in
vacuum. Both the plasma magnetic field and vacuum magnetic field are tangent
along the plasma-vacuum interface. Moreover, the vacuum magnetic field is
composed in a non-simply connected domain and hence is non-trivial. Under the
Rayleigh-Taylor sign condition on the free surface, we prove the local
well-posedness of the problem in Sobolev spaces. Furthermore, we also prove the
local well-posdeness under a more general "stability" assumption for the
initial data, which provided that the Rayleigh-Taylor sign condition is
satisfied at all those points of the initial interface where the
non-collinearity condition fails.

Analysis of PDEs