Local Well-Posedness of the Plasma-Vacuum Interface Problem for the Ideal Incompressible Mhd

In this article, we consider the plasma-vacuum interface problem for the incompressible ideal MHD. The plasma magnetic field is tangential to the interface while the vacuum magnetic field vanishes. We shall prove the stability of the interface under the Taylor sign condition. By deriving the evolution equation of the interface in the Eulerian coordinates, we are able to identify different stability mechanisms which correspond to the hyperboliticity of this evolution equation. Once the optimal regularity of the interface is obtained, all the other quantities can be estimated in the Eulerian coordinates. Therefore, we do not need the change of coordinates or the use of the Alinhac’s good unknowns.