On Compressible Current–Vortex Sheets

Abstract

Recent results for ideal compressible current-vortex sheets (tangential MHD discontinuities) are surveyed. A sufficient condition for the weak stability of planar current-vortex sheets is first found for a general case of the unperturbed flow. In astrophysics, this condition can be interpreted as the sufficient condition for the macroscopic stability of the heliopause. The crucial role in finding this stability condition is played by a new symmetric form of the MHD equations. The linear variable coefficients problem for nonplanar current-vortex sheets is studied as well. The fact that the Kreiss-Lopatinski condition is satisfied only in a weak sense yields losses of derivatives in a priori estimates. We prove an a priori tame estimate that is a necessary step to show the local-in-time existence of “stable” current-vortex sheet solutions to the nonlinear equations of ideal compressible MHD by a suitable NashMoser-type iteration scheme. Since the tangential discontinuity is characteristic, the functional setting is provided by the anisotropic weighted Sobolev spaces H ∗ .