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

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

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Pages (from-to)4125-4142
Number of pages18
JournalTransactions of the American Mathematical Society
Volume359
Issue number9
DOIs
StatePublished - 2007

Keywords

  • Ideal flow
  • Incompressible flow
  • Vortex sheets
  • Weak solutions

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