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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: Milton C. Lopes Filho, Helena J. Nussenzveig Lopes and Steven Schochet
Journal: Trans. Amer. Math. Soc. 359 (2007), 4125-4142
MSC (2000): Primary 76B03; Secondary 35Q35, 76B47
Published electronically: April 11, 2007
MathSciNet review: 2309179
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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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Additional Information

Milton C. Lopes Filho
Affiliation: Departamento de Matemática, IMECC-UNICAMP, Cx. Postal 6065, Campinas SP 13081-970, Brazil
Email: mlopes@ime.unicamp.br

Helena J. Nussenzveig Lopes
Affiliation: Departamento de Matemática, IMECC-UNICAMP, Cx. Postal 6065, Campinas SP 13081-970, Brazil
Email: hlopes@ime.unicamp.br

Steven Schochet
Affiliation: School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978 Israel
Email: schochet@post.tau.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04309-7
PII: S 0002-9947(07)04309-7
Keywords: Vortex sheets, incompressible flow, ideal flow, weak solutions
Received by editor(s): March 4, 2005
Published electronically: April 11, 2007
Article copyright: © Copyright 2007 American Mathematical Society