Existence of Compressible Current-Vortex Sheets: Variable Coefficients Linear Analysis

Abstract

We study the initial-boundary value problem resulting from the linearization of the equations of ideal compressible magnetohydrodynamics and the Rankine-Hugoniot relations about an unsteady piecewise smooth solution. This solution is supposed to be a classical solution of the system of magnetohydrodynamics on either side of a surface of tangential discontinuity (current-vortex sheet). Under some assumptions on the unperturbed flow, we prove an energy a priori estimate for the linearized problem. Since the tangential discontinuity is characteristic, the functional setting is provided by the anisotropic weighted Sobolev space W21,σ. Despite the fact that the constant coefficients linearized problem does not meet the uniform Kreiss-Lopatinskii condition, the estimate we obtain is without loss of smoothness even for the variable coefficients problem and nonplanar current-vortex sheets. The result of this paper is a necessary step in proving the local-in-time existence of current-vortex sheet solutions of the nonlinear equations of magnetohydrodynamics.