## 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 language | English |
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Pages (from-to) | 4125-4142 |

Number of pages | 18 |

Journal | Transactions of the American Mathematical Society |

Volume | 359 |

Issue number | 9 |

DOIs | |

State | Published - 2007 |

## Keywords

- Ideal flow
- Incompressible flow
- Vortex sheets
- Weak solutions