Well-posedness and Incompressible Limit of Current-Vortex Sheets with Surface Tension in Compressible Ideal MHD

Current-vortex sheet is one of the characteristic discontinuities in ideal compressible magnetohydrodynamics (MHD). The motion of current-vortex sheets is described by a free-interface problem of two-phase MHD flows with magnetic fields tangential to the interface. This paper is the first part of the two-paper sequence, which aims to present a comprehensive study for compressible current-vortex sheets with or without surface tension. In this paper, we prove the local well-posedness and the incompressible limit of current-vortex sheets with surface tension. The key observation is a hidden structure of Lorentz force in the vorticity analysis which motivates us to establish the uniform estimates in anisotropic-type Sobolev spaces with weights of Mach number determined by the number of tangential derivatives. Besides, our framework of iteration and approximation to prove the local existence of vortex-sheet problems does not rely on Nash-Moser iteration. Furthermore, the local existence of current-vortex sheets without surface tension can be proved by taking zero-surface-tension limit under certain stability conditions, which is established in [73] (the second part of the two-paper sequence).