22/11/2017 Approximate current-vortex sheets near the onset of instability

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-di erential 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.