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We consider a free boundary problem for the axially symmetric incompressibleideal magnetohydrodynamic equations that describes the motion of the plasma invacuum. Both the plasma magnetic field and vacuum magnetic field are tangentalong the plasma-vacuum interface. Moreover, the vacuum magnetic field iscomposed in a non-simply connected domain and hence is non-trivial. Under thenon-collinearity condition on the free surface, we prove the localwell-posedness of the problem in Sobolev spaces.

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