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

Abstract

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