Local Well-Posedness for the Magnetohydrodynamics in the Different Two Liquids Case

We consider the free boundary problem of MHD in multi-dimensional case. This describes motion two incompressible fluids separated by a closed interface under action magnetic field. is overdetermined, and we find an equivalent system equations which uniquely solvable locally time Lp-Lq maximal regularity class, where 1<p,q<∞ 2/p+N/q<1. As result, original two-phase for time.