Local Well-Posedness for Free Boundary Problem of Viscous Incompressible Magnetohydrodynamics

In this paper, we consider the motion of incompressible magnetohydrodynamics (MHD) with resistivity in a domain bounded by a free surface. An electromagnetic field generated by some currents in an external domain keeps an MHD flow in a bounded domain. On the free surface, free boundary conditions for MHD flow and transmission conditions for electromagnetic fields are imposed. We proved the local well-posedness in the general setting of domains from a mathematical point of view. The solutions are obtained in an anisotropic space H-p(1) ((0,T), H-q(1)) boolean AND L-p((0,T), H-q(3) ) for the velocity field and in an anisotropic space H-p(1) ((0,T), L-q) boolean AND L-p((0,T), H-q(2)) for the magnetic fields with 2 < p < infinity, N < q < infinity and 2/p + N/q < 1. To prove our main result, we used the L-p-L-q maximal regularity theorem for the Stokes equations with free boundary conditions and for the magnetic field equations with transmission conditions, which have been obtained by Frolova and the second author.