Data dependence of approximate current-vortex sheets near the onset of instability

TL;DR

This paper investigates the well-posedness of an amplitude equation modeling the dynamics of 2D current-vortex sheets near instability onset, demonstrating continuous dependence on initial data.

Contribution

It proves the strong norm continuous dependence of solutions on initial data, completing the well-posedness analysis for the amplitude equation.

Findings

Solutions depend continuously on initial data in strong norm.

Established well-posedness of the amplitude equation.

Confirmed stability properties near the 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 the amplitude equation was already shown. In the present paper we prove the continuous dependence in strong norm of solutions on the initial data. This completes the proof of the well-posedness of the problem in the classical sense of Hadamard.