On the Incompressible Limit of Current-Vortex Sheets with or without Surface Tension

Abstract

This is the second part of the two-paper sequence, which aims to present a comprehensive study for current-vortex sheets in ideal compressible magnetohydrodynamics (MHD). The local well-posedness of current-vortex sheets with surface tension has been proved in the first part of the paper sequence [63]. In this paper, we prove the incompressible and zero-surface-tension limits under certain stability conditions. The proof of uniform estimates relies on the analysis of the evolution equation of the free interface via a paradifferential approach, the wave equation of the pressure and a weighted anisotropic structure in vorticity analysis.