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

• Autores: Milton C. Lopes Filho, Helena J. Nussenzveig Lopes, Steven Schochet
• Localización: Transactions of the American Mathematical Society, ISSN 0002-9947, Vol. 359, Nº 9, 2007, págs. 4125-4142
• Idioma: inglés
• DOI: 10.1090/s0002-9947-07-04309-7
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• Resumen

  • 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.