A priori estimates for 3D incompressible current-vortex sheets
Abstract
We consider the free boundary problem for current-vortex sheets in ideal incompressible magneto-hydrodynamics. It is known that current-vortex sheets may be at most weakly (neutrally) stable due to the existence of surface waves solutions to the linearized equations. The existence of such waves may yield a loss of derivatives in the energy estimate of the solution with respect to the source terms. However, under a suitable stability condition satisfied at each point of the initial discontinuity and a flatness condition on the initial front, we prove an a priori estimate in Sobolev spaces for smooth solutions with no loss of derivatives. The result of this paper gives some hope for proving the local existence of smooth current-vortex sheets without resorting to a Nash-Moser iteration. Such result would be a rigorous confirmation of the stabilizing effect of the magnetic field on Kelvin-Helmholtz instabilities, which is well known in astrophysics.
Cite
@article{arxiv.1102.2763,
title = {A priori estimates for 3D incompressible current-vortex sheets},
author = {Jean-Francois Coulombel and Alessandro Morando and Paolo Secchi and Paola Trebeschi},
journal= {arXiv preprint arXiv:1102.2763},
year = {2015}
}