Vortex sheets and diffeomorphism groupoids

Anton Izosimov, Boris Khesin

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

In 1966 V. Arnold suggested a group-theoretic approach to ideal hydrodynamics in which the motion of an inviscid incompressible fluid is described as the geodesic flow of the right-invariant L2-metric on the group of volume-preserving diffeomorphisms of the flow domain. Here we propose geodesic, group-theoretic, and Hamiltonian frameworks to include fluid flows with vortex sheets. It turns out that the corresponding dynamics is related to a certain groupoid of pairs of volume-preserving diffeomorphisms with common interface. We also develop a general framework for Euler–Arnold equations for geodesics on groupoids equipped with one-sided invariant metrics.

Original languageEnglish (US)
Pages (from-to)447-501
Number of pages55
JournalAdvances in Mathematics
Volume338
DOIs
StatePublished - Nov 7 2018

Keywords

  • Diffeomorphism groups
  • Euler equations
  • Hamiltonian systems
  • Ideal hydrodynamics
  • Lie algebroid
  • Vortex sheets

ASJC Scopus subject areas

  • Mathematics(all)

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