Well-Posedness of the Plasma–Vacuum Interface Problem for Ideal Incompressible MHD

In this paper, we prove the local well-posedness of the plasma–vacuum interface problem for ideal incompressible magnetohydrodynamics under the stability condition: the magnetic field h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf h}$$\end{document} and the vacuum magnetic field h^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{{\mathbf h}}}$$\end{document} are non-collinear on the interface (i.e., |h×h^|>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{\mathbf h}\times {\hat{{\mathbf h}}}|>0$$\end{document}) which was introduced by Trakhinin as a stability condition for the compressible plasma–vacuum interface problem.