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Moscow Mathematical Journal

Volume 17, Issue 4, October–December 2017  pp. 699–716.

Classification of Casimirs in 2D Hydrodynamics

Authors:  Anton Izosimov (1) and Boris Khesin (1)
Author institution:(1) Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada


We describe a complete list of Casimirs for 2D Euler hydrodynamics on a surface without boundary: we define generalized enstrophies which, along with circulations, form a complete set of invariants for coadjoint orbits of area-preserving diffeomorphisms on a surface. We also outline a possible extension of main notions to the boundary case and formulate several open questions in that setting.

2010 Math. Subj. Class. Primary: 76M60; Secondary: 76A02, 58B25.

Keywords: Area-preserving diffeomorphisms, enstrophy, Casimir function, coadjoint orbit, vorticity, circulation, hydrodynamical Euler equation, Reeb graph, Morse function.

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