Stabilization Effect of Magnetic Fields on Two-Dimensional Compressible Current-Vortex Sheets

Abstract

We analyze the linear stability of rectilinear compressible current-vortex sheets in two-dimensional isentropic magnetohydrodynamics, which is a free boundary problem with the boundary being characteristic. In the case when the magnitude of the magnetic field has no jump on the current-vortex sheets, we find a necessary and sufficient condition of linear stability for the rectilinear current-vortex sheets, showing that magnetic fields exert a stabilization effect on compressible vortex sheets. In addition, a loss of regularity with respect to the source terms, both in the interior domain and on the boundary, occurs in a priori estimates of solutions to the linearized problem for a rectilinear current-vortex sheet, as the Kreiss–Lopatinskii determinant associated with this linearized boundary value problem has roots on the boundary of frequency spaces. In this study, the construction of symmetrizers for a reduced differential system, which has poles at which the Kreiss–Lopatinskii condition may fail simultaneously, plays a crucial role in the a priori estimates.