The Existence of Current-Vortex Sheets in Ideal Compressible Magnetohydrodynamics

We prove the local-in-time existence of solutions with a surface of current-vortex sheet (tangential discontinuity) of the equations of ideal compressible magnetohydrodynamics in three space dimensions provided that a stability condition is satisfied at each point of the initial discontinuity. This paper is a natural completion of our previous analysis (Trakhinin in Arch Ration Mech Anal 177:331–366, 2005) where a sufficient condition for the weak stability of planar current-vortex sheets was found and a basic a priori estimate was proved for the linearized variable coefficients problem for nonplanar discontinuities. The original nonlinear problem is a free boundary hyperbolic problem. Since the free boundary is characteristic, the functional setting is provided by the anisotropic weighted Sobolev spaces $${H^m_*}$$ . The fact that the Kreiss–Lopatinski condition is satisfied only in a weak sense yields losses of derivatives in a priori estimates. Therefore, we prove our existence theorem by a suitable Nash–Moser-type iteration scheme.

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