Desingularization of vortex sheets for the 2D Euler equations

TL;DR

This paper introduces a method to regularize vortex sheets in the 2D Euler equations by constructing smooth approximations that converge to the singular initial data, enabling analysis of vortex sheet dynamics.

Contribution

The authors develop a novel regularization technique for vortex sheets using smooth, compactly supported vorticities that converge to the singular initial data, facilitating the study of vortex sheet evolution.

Findings

Smooth vorticity approximations converge to the Birkhoff-Rott vortex sheet dynamics.

The regularization exploits anisotropic effects of Kelvin-Helmholtz instability.

The method provides a framework for analyzing vortex sheet evolution in Euler flows.

Abstract

We show how to regularize vortex sheets by means of smooth, compactly supported vorticities that asymptotically evolve according to the Birkhoff-Rott vortex sheet dynamics. More precisely, consider a vortex sheet initial datum $\omega^0_{\mathrm{sing}}$, which is a signed Radon measure supported on a closed curve. We construct a family of initial vorticities $\omega^0_\varepsilon \in C^\infty_c(\mathbb{R}^2)$ converging to $\omega^0_{\mathrm{sing}}$ distributionally as $\varepsilon \to 0^+$, and show that the corresponding solutions $\omega_\varepsilon(x,t)$ to the 2D incompressible Euler equations converge to the measure defined by the Birkhoff-Rott system with initial datum $\omega^0_{\mathrm{sing}}$. The regularization relies on a layer construction designed to exploit the key observation that the Kelvin-Helmholtz instability has a strongly anisotropic effect: while vorticities must be analytic in the "tangential" direction, the way layers can be arranged in the "normal" direction is essentially arbitrary.