Local free boundary problem for viscous non-homogeneous incompressible magnetohydrodynamics
We consider the motion of viscous non-homogeneous incompressible magnetohydrodynamic (mhd) fluid in a domain bounded by a free surface. In the external domain there exists an electromagnetic field generated by some currents which keeps the mhd flow in the bounded domain. Then on the free surface transmission conditions for electromagnetic fields are imposed. In this paper we prove existence of local regular solutions by the method of successive approximations. The $L_2$ approach is used. This enables us to treat the transmission conditions.