Geometry of generalized fluid flows

Anton Izosimov; Boris Khesin

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arXiv · 2206.01434

Geometry of generalized fluid flows

Anton Izosimov·Boris Khesin

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Abstract

The Euler equation of an ideal (i.e. inviscid incompressible) fluid can be regarded, following V.Arnold, as the geodesic flow of the right-invariant $L^2$-metric on the group of volume-preserving diffeomorphisms of the flow domain. In this paper we describe the common origin and symmetry of generalized flows, multiphase fluids (homogenized vortex sheets), and conventional vortex sheets: they all correspond to geodesics on certain groupoids of multiphase diffeomorphisms. Furthermore, we prove that all these problems are Hamiltonian with respect to a Poisson structure on a dual Lie algebroid, generalizing the Hamiltonian property of the Euler equation on a Lie algebra dual.

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Anton Izosimov, Boris Khesin. 2022-06-03. Geometry of generalized fluid flows. https://arxiv.org/abs/2206.01434

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