A priori estimates for 3D incompressible current-vortex sheets

TL;DR

This paper establishes a priori estimates for smooth solutions of 3D incompressible current-vortex sheets under stability and flatness conditions, suggesting potential for proving local existence without loss of derivatives.

Contribution

It provides the first a priori Sobolev estimates for current-vortex sheets in 3D, under stability conditions, without derivative loss, advancing the mathematical understanding of these free boundary problems.

Findings

A priori estimates in Sobolev spaces without derivative loss

Stability condition ensures well-posedness

Supports potential proof of local existence of smooth solutions

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.