On the free boundary problem of magnetohydrodynamics

Let a capillary liquid drop electrically conducting be set in a rigid perfectly conducting container C. In C there is vacuum, the domain occupied by the liquid drop is strictly contained in a bounded domain. In the present paper we are concerned with the simplest free boundary problem of magnetohydrodynamics. It consists of finding a bounded variable domain filled with a viscous incompressible capillary conducting fluid, together with the vector velocity, a scalar pressure fields, and the magnetic field satisfying the system of equations of magnetohydrodynamics in the domain. The boundary is the free surface of the fluid that is subject to surface tension beside the electrodynamical tensions. It is assumed that the fluid is surrounded by a vacuum region and that the domain full spaceis independent of time and bounded by a perfectly conducting surface S. The magnetic field should be found not only in interior but also in the external domain. It is assumed that both domains are simply connected.