Existence of approximate current-vortex sheets near the onset of instability

TL;DR

This paper proves the local-in-time existence of smooth solutions with optimal regularity for an amplitude equation modeling the near-onset dynamics of 2D current-vortex sheets in ideal MHD, near the stability threshold.

Contribution

It establishes the existence of solutions with preserved regularity for the amplitude equation, improving previous results that had regularity loss.

Findings

Existence of smooth solutions with optimal regularity.

Solutions preserve initial regularity over time.

Addresses dynamics near the stability-instability transition.

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. The local-in-time existence of smooth solutions to the Cauchy problem for such amplitude equation was already proven, under a suitable stability condition. However, the solution found there has a loss of regularity (of order two) from the initial data. In the present paper, we are able to obtain an existence result of solutions with optimal regularity, in the sense that the regularity of the initial data is preserved in the motion for positive times.