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A Criterion for the Equivalence of the Birkhoff-Rott and Euler Descriptions of Vortex Sheet Evolution

M. C. Lopes Filho, H. J. Nussenzveig Lopes, S. Schochet

TL;DR

The paper tackles when the explicit Birkhoff-Rott description of vortex-sheet evolution is equivalent to the 2D incompressible Euler weak formulation. It introduces a weak BR framework for arclength-parametrized sheets and proves a sharp equivalence: BR solutions correspond to Euler weak solutions precisely when the sheet is a regular curve and the vorticity density is in $L^2(ds)$. The authors establish the result via a two-step comparison of BR and vorticity identities and prove sharpness with the Prandtl-Munk example, showing the necessity of the $L^2$ condition. They also discuss the implications for the regularity requirements of vortex-sheet dynamics and outline open problems related to nonregular sheets and spiral structures.

Abstract

In this article we consider the evolution of vortex sheets in the plane both as a weak solution of the two dimensional incompressible Euler equations and as a (weak) solution of the Birkhoff-Rott equations. We begin by discussing the classical Birkhoff-Rott equations with respect to arbitrary parametrizations of the sheet. We introduce a notion of weak solution to the Birkhoff-Rott system and we prove consistency of this notion with the classical formulation of the equations. Our main purpose in this paper is to present a sharp criterion for the equivalence of the weak Euler and weak Birkhoff-Rott descriptions of vortex sheet dynamics.

A Criterion for the Equivalence of the Birkhoff-Rott and Euler Descriptions of Vortex Sheet Evolution

TL;DR

The paper tackles when the explicit Birkhoff-Rott description of vortex-sheet evolution is equivalent to the 2D incompressible Euler weak formulation. It introduces a weak BR framework for arclength-parametrized sheets and proves a sharp equivalence: BR solutions correspond to Euler weak solutions precisely when the sheet is a regular curve and the vorticity density is in . The authors establish the result via a two-step comparison of BR and vorticity identities and prove sharpness with the Prandtl-Munk example, showing the necessity of the condition. They also discuss the implications for the regularity requirements of vortex-sheet dynamics and outline open problems related to nonregular sheets and spiral structures.

Abstract

In this article we consider the evolution of vortex sheets in the plane both as a weak solution of the two dimensional incompressible Euler equations and as a (weak) solution of the Birkhoff-Rott equations. We begin by discussing the classical Birkhoff-Rott equations with respect to arbitrary parametrizations of the sheet. We introduce a notion of weak solution to the Birkhoff-Rott system and we prove consistency of this notion with the classical formulation of the equations. Our main purpose in this paper is to present a sharp criterion for the equivalence of the weak Euler and weak Birkhoff-Rott descriptions of vortex sheet dynamics.

Paper Structure

This paper contains 10 sections, 4 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Vortex sheet equations in weak form
  3. Weak forms for the vorticity equation
  4. The Birkhoff-Rott system
  5. Weak form of the Birkhoff-Rott system
  6. Equivalence of weak formulations
  7. Regular Curves
  8. The Equivalence Theorem
  9. Sharpness
  10. Final Remarks

Key Result

Lemma 2.1

Let \mu_0 = \gamma_0\delta_{\mathcal{C}_0} be a Radon measure with support on a smooth curve \mathcal{C}_0. Let \mu = \mu(\cdot,t) be the transport by v of the measure \mu_0. Let y = y(\eta,t) be a parametrization of \mathcal{C}_t and denote where s=s(\eta,t) is arclength with respect to a reference point. Then there exists a=a(\eta,t) such that the following equations are satisfied:

Theorems & Definitions (9)

  • Definition 2.1
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Theorem 2.2
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • proof