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

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 am-
plitude equation was already shown by the authors. In the present paper we prove the continuous de-
pendence 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.