A criterion for the equivalence of the Birkhoff-Rott and Euler descriptions of vortex sheet evolution

Lopes Filho, Milton; Nussenzveig Lopes, Helena; Schochet, Steven

doi:10.1090/S0002-9947-07-04309-7

A criterion for the equivalence of the Birkhoff-Rott and Euler descriptions of vortex sheet evolution
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by Milton C. Lopes Filho, Helena J. Nussenzveig Lopes and Steven Schochet PDF

Trans. Amer. Math. Soc. 359 (2007), 4125-4142 Request permission

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.

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