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

TL;DR

This paper establishes the local well-posedness of the free boundary problem in viscous incompressible magnetohydrodynamics, demonstrating existence, uniqueness, and regularity of solutions in a mathematically rigorous framework.

Contribution

It provides the first rigorous proof of local well-posedness for free boundary MHD flows with electromagnetic transmission conditions in a general domain setting.

Findings

Solutions exist and are unique in maximal regularity classes.

Velocity regularity exceeds magnetic field regularity by one order.

The approach uses Lp-Lq maximal regularity for Stokes and magnetic equations.

Abstract

In this paper, we consider the motion of incompressible magnetohydrodynamics (MHD) with resistivity in a domain bounded by a free surface. The free boundary problem for MHD is an important problem not only for mathematical fluid dynamics but also some application to the field of engineering In fact, when a thermonuclear reaction is caused artificially, a high-temperature plasma is sometimes subjected to a magnetic field and held in the air, and the boundary of the fluid at this time is a free one. In this paper, 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 the maximal regularity class, and in particular, the regularity class of velocity fields is one more higher than the regularity class of magnetic fields. To prove our main result, we use Lp-Lq maximal regularity theorem for the Stokes equations with free boundary conditions and for the magnetic field equations with transmission conditions.