Global current-vortex sheets in the two-dimensional ideal incompressible MHD

TL;DR

This paper proves the global nonlinear stability of current-vortex sheets in 2D ideal incompressible MHD with a strong horizontal magnetic field, a first for such free boundary problems in inviscid fluids.

Contribution

It establishes the first global existence result for free boundary problems in ideal incompressible rotational MHD, highlighting the stabilizing role of a strong magnetic field.

Findings

Global stability of current-vortex sheets demonstrated

Magnetic field stabilizes free boundary in ideal MHD

Novel energy estimate techniques developed

Abstract

The magnetohydrodynamic current-vortex sheet is a free boundary problem involving a moving free surface separating two plasma regions. We prove the global nonlinear stability of current-vortex sheet in the two dimensional ideal incompressible magnetohydrodynamics under the strong horizontal background magnetic field. This appears to be the first result on the global solutions of the free boundary problems for the ideal (inviscid and non-resistive) incompressible rotational fluids. The strong magnetic field plays a crucial role in the global in time stabilization effect. The proof relies on the understanding of the interplay between the dynamics of the fluids inside the domain and on the free interface, a design of multiple-level energy estimates with different weights, and the inherent structures of the problem.