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 H¹ _p((0, T), H¹ _q) ∩ L_p((0, T), H³ _q) for the velocity field and in an anisotropic space H¹ _p((0, T), L_q) ∩ L_p((0, T), H² _q) for the magnetic fields with 2 < p < ∞, N < q < ∞ and 2/p + N/q < 1. To prove our main result, we used the Lp-Lq 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.