Leapfrogging vortex rings for the three-dimensional incompressible Euler equations

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Abstract

A classical problem in fluid dynamics concerns the interaction of multiple vortex rings sharing a common axis of symmetry in an incompressible, inviscid three-dimensional fluid. In 1858, Helmholtz observed that a pair of similar thin, coaxial vortex rings may pass through each other repeatedly due to the induced flow of the rings acting on each other. This celebrated configuration, known as leapfrogging, has not yet been rigorously established. We provide a mathematical justification for this phenomenon by constructing a smooth solution of the 3D Euler equations exhibiting this motion pattern.

Original languageEnglish
Pages (from-to)1-115
JournalCommunications on Pure and Applied Mathematics
Early online date6 May 2024
DOIs
Publication statusE-pub ahead of print - 6 May 2024

Bibliographical note

Publisher Copyright:
© 2024 The Authors. Communications on Pure and Applied Mathematics published by Courant Institute of Mathematics and Wiley Periodicals LLC.

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics

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