Uniform estimate of viscous free-boundary Magnetohydrodynamics with zero vacuum magnetic field

TL;DR

This paper establishes uniform estimates for viscous free-boundary magnetohydrodynamics with zero vacuum magnetic field, enabling the passage to the inviscid limit and solution existence in the zero magnetic field vacuum case.

Contribution

It introduces a Sobolev conormal space approach to obtain uniform regularity estimates, addressing boundary layer issues in the vanishing viscosity limit for free-boundary MHD.

Findings

Achieved uniform estimates independent of viscosity parameter

Proved existence of solutions for inviscid free-boundary MHD with zero vacuum magnetic field

Addressed boundary layer challenges near the free boundary

Abstract

We consider viscous free-boundary magnetohydrodynamics(MHD) under vacuum in $\mathbb{R}^3$, especially when vacuum magnetic field is identically zero. It is a central problem in mathematics to perform vanishing viscosity limit to get a solution of hyperbolic inviscid system. However, boundary layer behavior happens near the free-boundary, so existence time $T^{\varepsilon} \rightarrow 0$ as kinematic viscosity $\varepsilon\rightarrow 0$ in standard sobolev space. Inspired by \cite{NMFR1}, we use sobolev conormal space to derive uniform regularity in viscosity $\varepsilon$. Finally, we get a solution of inviscid free-boundary magnetohydrodynamics when initial magnetic field is zero on the free-boundary and in vacuum.