abstract: This talk is concerned with the free boundary problem for 2D current-vortex sheets in ideal incompress- ible 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 1 have introduced an asymptotic quadratically nonlinear integro-differential equation for the amplitude of small perturbations of the planar discontinuity. We study such amplitude equation and prove its nonlinear well-posedness under a stability condition given in terms of a longitudinal strain of the fluid along the discontinuity. We first present the problem and discuss some known results about the stability of current-vortex sheets; then we give some new results on the well-posedness of the Cauchy problem associated to the amplitude equation. This is a joint work with P. Secchi and P. Trebeschi.
1: Hunter, J. K. and Thoo, J. B., On the weakly nonlinear Kelvin-Helmholtz instability of tangential discontinuities in MHD, J. Hyperbolic Differ. Equ., 8 (4), 2011, 691–726.