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

Abstract

The paper is concerned with 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 [On the weakly nonlinear Kelvin–Helmholtz instability of tangential discontinuities in MHD, J. Hyperbolic Differ. Equations 8(4) (2011) 691–726] 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 shown in [Approximate current-vortex sheets near the onset of instability, J. Math. Pures Appl. 105(4) (2016) 490–536; Existence of approximate current-vortex sheets near the onset of instability, J. Hyperbolic Differ. Equations]. 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.