TY - JOUR
T1 - Geometry of generalized fluid flows
AU - Izosimov, Anton
AU - Khesin, Boris
N1 - Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2024/1
Y1 - 2024/1
N2 - 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 L2 -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.
AB - 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 L2 -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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U2 - 10.1007/s00526-023-02612-5
DO - 10.1007/s00526-023-02612-5
M3 - Article
AN - SCOPUS:85177232028
SN - 0944-2669
VL - 63
JO - Calculus of Variations and Partial Differential Equations
JF - Calculus of Variations and Partial Differential Equations
IS - 1
M1 - 3
ER -