Local Well-posedness of the Incompressible Current-Vortex Sheet Problems

TL;DR

This paper establishes the local well-posedness of incompressible current-vortex sheet problems, demonstrating stabilization effects of surface tension and magnetic fields, and analyzing the vanishing surface tension limit.

Contribution

It proves well-posedness in Sobolev spaces under specific conditions, including the Syrovatskij condition, without requiring the interface to be a graph.

Findings

Surface tension and magnetic fields stabilize current-vortex sheet motion.

Well-posedness holds without the interface being a graph.

Vanishing surface tension limit is established under certain conditions.

Abstract

We prove the local well-posedness of the incompressible current-vortex sheet problems in standard Sobolev spaces under the surface tension or the Syrovatskij condition, which shows that both capillary forces and large tangential magnetic fields can stabilize the motion of current-vortex sheets. Furthermore, under the Syrovatskij condition, the vanishing surface tension limit is established for the motion of current-vortex sheets. These results hold without assuming the interface separating the two plasmas being a graph.