Approximate current-vortex sheets near the onset of instability

TL;DR

This paper analyzes the behavior of 2D current-vortex sheets near the critical point of instability, introducing an amplitude equation and proving its nonlinear well-posedness under specific stability conditions.

Contribution

It provides the first rigorous analysis of the amplitude equation governing near-critical current-vortex sheet dynamics in ideal MHD.

Findings

Proves nonlinear well-posedness of the amplitude equation

Establishes stability conditions based on fluid strain

Advances understanding of transition to instability in MHD sheets

Abstract

The paper is concerned with the free boundary problem for 2D current-vortex sheets in ideal incompressible magneto-hydrodynamics near the transition point between the linearized stability and instability. In order to study the dynamics of the discontinuity near the onset of the instability, Hunter and Thoo have introduced an asymptotic quadratically nonlinear integro-differential equation for the amplitude of small perturbations of the planar discontinuity. We study such amplitude equation and prove its nonlinear well-posedness under a stability condition given in terms of a longitudinal strain of the fluid along the discontinuity.