Existence and Stability of Compressible and Incompressible Current-Vortex Sheets

Abstract

Recent author’s results in the investigation of current-vortex sheets (MHD tangential discontinuities) are surveyed. A sufficient condition for the neutral stability of planar compressible current-vortex sheets is first found for a general case of the unperturbed flow. In astrophysical applications, this condition can be considered as the sufficient condition for the stability of the heliopause, which is modelled by an ideal compressible current-vortex sheet and caused by the interaction of the supersonic solar wind plasma with the local interstellar medium (in some sense, the heliopause is the boundary of the solar system). The linear variable coefficients problem for nonplanar compressible current-vortex sheets is studied as well. Since the tangential discontinuity is characteristic, the functional setting is provided by the anisotropic weighted Sobolev spaces. The a priori estimate deduced for this problem is a necessary step to prove the local-in-time existence of current-vortex sheet solutions of the nonlinear equations of ideal compressible MHD. Analogous results are obtained for incompressible current-vortex sheets. In the incompressibility limit the sufficient stability condition found for compressible current-vortex sheets describes exactly the half of the whole parameter domain of linear stability of planar discontinuities in ideal incompressible MHD.