Global regularity for a Birkhoff-Rott-alpha approximation of the dynamics of vortex sheets of the 2D Euler equations

TL;DR

This paper introduces an alpha-regularization of the Birkhoff-Rott equation derived from Euler-alpha equations, demonstrating that smooth, self-avoiding vortex sheets remain smooth over time under this regularization.

Contribution

It provides a novel alpha-regularization approach for vortex sheet dynamics, ensuring global regularity for initially smooth sheets.

Findings

Smooth vortex sheets stay smooth for all time under the regularized dynamics.

The regularization preserves the smoothness of vortex sheets with integrable vorticity density.

The approach guarantees global regularity for the vortex sheet evolution.

Abstract

We present an alpha-regularization of the Birkhoff-Rott equation, induced by the two-dimensional Euler-alpha equations, for the vortex sheet dynamics. We show that initially smooth self-avoiding vortex sheet remains smooth for all times under the alpha-regularized dynamics, provided the initial density of vorticity is an integrable function over the curve with respect to the arc-length measure.