INITIAL VALUE PROBLEM FOR THE FREE-BOUNDARY MAGNETOHYDRODYNAMICS WITH ZERO MAGNETIC BOUNDARY CONDITION

Abstract

We show local well-posedness of fluid-vacuum free-boundary magnetohydrodynamic (MHD) with both kinematic viscosity and magnetic diffusivity under the gravity force. We consider three-dimensional problem with finite depth and impose zero magnetic field condition on the free boundary and in vacuum. Sobolev-Slobodetskii space (fractional Sobolev space) is used to perform energy estimates. Main difficulty is to control strong nonlinear couplings between velocity and magnetic fields. In [D. Lee, SIAM J. Math. Anal., 49(4):2710-2789, 2017], we send both kinematic viscosity and magnetic diffusivity to zero with same speed to get ideal (inviscid) free-boundary magnetohydrodynamics using the result of this paper.