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Current-vortex sheet is one of the characteristic discontinuities in idealcompressible magnetohydrodynamics (MHD). The motion of current-vortex sheets isdescribed by a free-interface problem of two-phase MHD flows with magneticfields tangential to the interface. This model has been widely used in bothsolar physics and controlled nuclear fusion. This paper is the first part ofthe two-paper sequence, which aims to present a comprehensive study forcompressible current-vortex sheets with or without surface tension. We provethe local well-posedness and the incompressible limit of current-vortex sheetswith surface tension. The key observation is a hidden structure of Lorentzforce in the vorticity analysis which motivates us to establish the uniformestimates in anisotropic-type Sobolev spaces with weights of Mach numberdetermined by the number of tangential derivatives. Besides, we develop arobust framework for iteration and approximation to prove the local existenceof vortex-sheet problems. Our method does not rely on Nash-Moser iteration nortangential smoothing.

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