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

TL;DR

This paper proves the local well-posedness of axially symmetric incompressible ideal magnetohydrodynamic equations with vacuum, considering a non-collinearity condition and complex magnetic field configurations.

Contribution

It establishes the well-posedness of a free boundary MHD problem with non-trivial vacuum magnetic fields under non-collinearity, a novel analysis in this setting.

Findings

Proved local well-posedness in Sobolev spaces.

Handled non-simply connected vacuum magnetic fields.

Addressed the non-collinearity condition on the free surface.

Abstract

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 non-collinearity condition on the free surface, we prove the local well-posedness of the problem in Sobolev spaces.