Geometry of generalized fluid flows
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 -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.
Cite
@article{arxiv.2206.01434,
title = {Geometry of generalized fluid flows},
author = {Anton Izosimov and Boris Khesin},
journal= {arXiv preprint arXiv:2206.01434},
year = {2023}
}
Comments
30 pages, 4 figures. To appear in Calc. Var. Partial Differential Equations. arXiv admin note: text overlap with arXiv:1705.01603