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

Abstract

We study the free boundary problem for two-dimensional 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. For this amplitude equation, a local-in-time existence theory for smooth solutions to the Cauchy problem was established earlier by the authors under a suitable stability condition. However, the solution found therein enjoyed a loss of regularity (of order two) in comparison to the regularity of the initial data. In this work, we are able to prove an existence result with optimal regularity, in the sense that the regularity of the initial data is preserved in the evolution for positive times.