The Stability of Current Vortex Sheets with Transverse Magnetic Field

Abstract

We study a symmetric class of three-dimensional compressible current-vortex sheets for which the basic velocity is tangential to the interface and the basic magnetic field is tangential to the interface but transverse to the velocity. Restricting to perturbations that are independent of the magnetic-field direction, we derive an exact scalar pseudo-differential equation for the interface. If the magneto-acoustic Mach number $M_B:=\frac{|\dot{v}_1^+|}{\bar{C}_B},\quad \bar{C}_B^2=c^2+\frac{|\dot{B}_2|^2}{\dot{\varrho}}$, satisfies $M_B>\sqrt{2}$, we prove unique solvability of the reduced interface equation and the weighted estimate $\gamma^3\|f\|_{H_\gamma^{s+1}}^2\leq C\sum_{\pm}\|\mathcal{K}^{\pm}\|_{L^2(\mathbb{R}_{\pm};H_\gamma^s)}^2,\quad\gamma\geq1$. This result complements the low-$M_B$ linear and nonlinear ill-posedness established by Xie and Zhao in the Journal of Geometric Analysis [32].