Problem of magnetohydrodynamics for two liquids and its generalization for multidimensional space.

We consider a problem of magnetohydrodynamics in a bounded domain that describes the motion of two viscous incompressible liquids separated by a free surface. We proof the local in time solvability in Sobolev-Slobodeсkij spaces. While linearizing the equations of magnetohydrodynamics it is sufficient to consider independently the linearized problem of hydrodynamics and the equations for the magnetic field, since the binded terms are non-linear. A part of the talk will be devoted to the theorem of maximal $L_p-L_q$ regularity for the linearized equations for the electromagnetic field with the conjugation conditions on the separation surface (a joint work with Y. Shibata).