Existence, Nonlinear Stability and Incompressible Limit of Current-Vortex Sheets with or without Surface Tension in Compressible Ideal MHD

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. First, we prove local well-posedness ofcurrent-vortex sheets with surface tension by developing a robust frameworkthat does not rely on Nash-Moser iteration nor tangential smoothing. Second,the energy estimates are uniform in Mach number and are also uniform insurface-tension coefficient under suitable stability conditions. Thus, wepresent a comprehensive study within one attempt, including well-posedness,nonlinear structural stability and incompressible limit of current-vortexsheets with or without surface tension. Our result demonstrates that either suitable magnetic fields or surfacetension could suppress the analogue of Kelvin-Helmholtz instability forcompressible vortex sheets. The key observation is a hidden structure ofLorentz force in the vorticity analysis which motivates us to establish theuniform estimates in some anisotropic Sobolev spaces with suitable weights ofMach number determined by the number of tangential derivatives. Moreover, forisentropic two-phase flows whose density functions are close to the sameconstant, we can drop the boundedness assumption (with respect to Mach number)on high-order time derivatives by paralinearizing the evolution equation of thefree interface. To our knowledge, this is the first result that rigorouslyjustifies the incompressible limit of compressible vortex sheets.