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

TL;DR

This paper investigates the behavior of current-vortex sheets in ideal compressible MHD, focusing on the incompressible and zero-surface-tension limits, using advanced analytical techniques to establish stability and uniform estimates.

Contribution

It provides the first rigorous proof of the incompressible and zero-surface-tension limits for current-vortex sheets in ideal compressible MHD under certain stability conditions.

Findings

Established uniform estimates for the limits

Analyzed the evolution of the free interface using paradifferential calculus

Proved stability conditions for the limits

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.