Uniform Estimate of Viscous Free-Boundary Magnetohydrodynamics with Zero Vacuum Magnetic Field

Abstract

We consider viscous free-boundary magnetohydrodynamics (MHD) under vacuum in R-3, especially when the vacuum magnetic field is identically zero. It is a central problem in mathematics to perform the vanishing viscosity limit to get a solution of a hyperbolic inviscid system. However, boundary layer behavior happens near the free boundary, so the existence time T-epsilon -> as the kinematic viscosity epsilon -> 0 in standard Sobolev space. Inspired by [N. Masmoudi and F. Rousset, Arch. Ration. Mech. Anal., 223 (2017), pp. 301-417], we use Sobolev conormal space to derive uniform regularity in viscosity E. Finally, we get a solution of inviscid free-boundary MHD when the initial magnetic field is zero on the free boundary and in vacuum.