Global regularity and convergence of 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 for vortex sheet dynamics, demonstrating convergence to weak Euler solutions and preserving regularity of initial vortex configurations over time.

Contribution

It provides a new regularization approach for vortex sheet evolution, proving convergence of Euler-alpha solutions and preservation of regularity for various initial conditions.

Findings

Solutions of Euler-alpha converge to weak Euler solutions.

Regularity of vortex sheets is preserved over time.

Weak Euler-alpha and BR-alpha descriptions are equivalent for certain initial conditions.

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 the convergence of the solutions of Euler-alpha equations to a weak solution of the Euler equations for initial vorticity being a finite Radon measure of fixed sign, which includes the vortex sheets case. We also show that, provided the initial density of vorticity is an integrable function over the curve with respect to the arc-length measure, (i) an initially Lipschitz chord arc vortex sheet (curve), evolving under the BR-alpha equation, remains Lipschitz for all times, (ii) an initially Holder C^{1,beta}, 0 <= beta < 1, chord arc curve remains in C^{1,beta} for all times, and finally, (iii) an initially Holder C^{n,beta}, n <= 1, 0 < beta < 1, closed chord arc curve remains so for all times. In all these cases the weak Euler-alpha and the BR-alpha descriptions of the vortex sheet motion are equivalent.