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

Abstract

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 Hp1((0,T),Hq1)∩Lp((0,T),Hq3) for the velocity field and in an anisotropic space Hp1((0,T),Lq)∩Lp((0,T),Hq2) for the magnetic fields with 2p∞, Nq∞ and 2/p+N/q1. 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.

Keywords

• free boundary problem
• transmission condition
• magnetohydorodynamics
• local well-posedness
• <i>L</i>_<i>p</i>-<i>L</i>_<i>q</i> maximal regularity