Initial value problem for the free boundary magnetohydrodynamics with zero magnetic boundary condition

TL;DR

This paper proves local existence and uniqueness of solutions for a free boundary magnetohydrodynamics problem with zero magnetic boundary condition, and shows solutions can be extended for small initial data.

Contribution

It establishes the local well-posedness of the free boundary MHD problem with zero magnetic boundary condition using Sobolev-Slobodetskii spaces, and connects to inviscid limits.

Findings

Local existence and uniqueness of solutions

Extension of solutions for small initial data

Foundation for inviscid limit analysis

Abstract

We show local existence and uniqueness of plasma(fluid)-vaccum free boundary problem of magnetohydrodynamic flow in three-dimensional space with infinite depth setting when magnetic field is zero on the free boundary. We use Sobolev-Slobodetskii space which was used in usual free boundary problem in [3,5,6,7,8,9]. We also show that this solution can be extended as long as we want for sufficiently small initial data. Using the result of this paper we will get a unique solution of (kinematic inviscid) - (magnetic non diffusive) free boundary magnetohydrodynamics problem via (kinematic viscosity) - (magnetic diffusivity) limit in [10].