Desingularization of vortex sheets for the 2D Euler equations

TL;DR

The paper develops a desingularization framework for vortex sheets in the 2D Euler equations by regularizing the sheet into a smooth, compactly supported vorticity that concentrates near a fixed analytic curve $oldsymbol{ extGamma}$ with density $oldsymbol{oldsymbol ω}^0$. It constructs a family $oldsymbol{ extomega}_ε^0$ converging to the vortex-sheet measure and proves that the corresponding Euler flows converge to the Birkhoff–Rott evolution of the sheet, on a uniform time interval independent of $ε$. The core method relies on a layer construction that exploits anisotropic regularity: analyticity is required tangentially, while normal regularity is more flexible, and it uses Nishida’s Cauchy–Kovalevskaya framework to obtain a locally well-posed analytic evolution for the regularized system. The work provides a rigorous link between smooth desingularizations and the formal BR dynamics, with careful kernel, geometric, and convergence estimates culminating in a Gronwall-type argument to establish convergence to BR as $ε o 0^+$. These results extend the understanding of vortex-sheet dynamics beyond graph geometries and contribute a robust desingularization mechanism applicable to a broad class of analytic sheet configurations.

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 $ω^0_{\mathrm{sing}}$, which is a signed Radon measure supported on a closed curve. We construct a family of initial vorticities $ω^0_\varepsilon \in C^\infty_c(\mathbb{R}^2)$ converging to $ω^0_{\mathrm{sing}}$ distributionally as $\varepsilon \to 0^+$, and show that the corresponding solutions $ω_\varepsilon(x,t)$ to the 2D incompressible Euler equations converge to the measure defined by the Birkhoff-Rott system with initial datum $ω^0_{\mathrm{sing}}$. 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.

Theorems & Definitions (80)

• Theorem 1.1
• Theorem 2.1
• Remark 2.2
• Proposition 3.1
• Theorem 3.2: Nishida Nishida
• Proposition 3.3
• Proposition 3.4
• Corollary 3.5
• Proposition 3.6
• Remark 3.7