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Vortex sheets and diffeomorphism groupoids

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Izosimov,  Anton
Max Planck Institute for Mathematics, Max Planck Society;

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arXiv:1705.01603.pdf
(Preprint), 500KB

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Citation

Izosimov, A., & Khesin, B. (2018). Vortex sheets and diffeomorphism groupoids. Advances in Mathematics, 338, 447-501. doi:10.1016/j.aim.2018.09.015.


Cite as: http://hdl.handle.net/21.11116/0000-0003-AD76-6
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 $L^2$-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.