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

2007; American Mathematical Society; Volume: 359; Issue: 09 Linguagem: Inglês

10.1090/s0002-9947-07-04309-7

1088-6850

Milton C. Lopes Filho, Helena J. Nussenzveig Lopes, Steven Schochet,

Computational Fluid Dynamics and Aerodynamics

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.