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

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

2005-02-10

arxiv.org/abs/math/0502215v1

Transactions of the A. M. S., v. 359 (2007), 4125--4142

Mathematics

Analysis of PDEs

22 pages

Scientific paper

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.

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