Leapfrogging vortex rings for the 3-dimensional incompressible Euler equations

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A classical problem in fluid dynamics concerns the interaction of multiple vortex rings sharing a common axis of symmetry in an incompressible, inviscid 3-dimensional fluid. Helmholtz (1858) 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
Number of pages67
JournalCommunications on Pure and Applied Mathematics
Publication statusAcceptance date - 8 Mar 2024

Bibliographical note

Submitted to ArXiv on 7/7/2022; updated on 11/1/2023 (v.3)


  • Euler equation
  • leapfrogging

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