Abstract
We consider viscous free-boundary magnetohydrodynamics (MHD) under vacuum in $\mathbb{R}^3$, 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 $T^{\varepsilon} \rightarrow 0$ as the kinematic viscosity $\varepsilon\rightarrow 0$ 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 $\varepsilon$. Finally, we get a solution of inviscid free-boundary MHD when the initial magnetic field is zero on the free boundary and in vacuum.
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