Well-posedness of axially symmetric incompressible ideal magnetohydrodynamic equations with vacuum under the non-collinearity condition

Abstract

We consider a free boundary problem for the axially symmetric incompressible ideal magnetohydrodynamic equations that describe 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 non-collinearity condition for the plasma and vacuum magnetic fields, we prove the local well-posedness of the problem in Sobolev spaces.