TY - JOUR

T1 - Characterization of steady solutions to the 2D Euler equation

AU - Izosimov, Anton

AU - Khesin, Boris

N1 - Funding Information:
This work was partially supported by the NSERC [Grant RGPIN-2014-05036 to B.K. and A.I.] and

PY - 2017/12/1

Y1 - 2017/12/1

N2 - Steady fluid flows have very special topology. In this paper we describe necessary and sufficient conditions on the vorticity function of a 2D ideal flow on a surface with or without boundary, for which there exists a steady flow among isovorticed fields. (Here the necessary condition is for any metric, while the sufficient condition is for an appropriate one.) For this we introduce the notion of an antiderivative (or circulation function) on a measured graph, the Reeb graph associated with the vorticity function on the surface, while the criterion is related to the total negativity of this antiderivative. It turns out that given topology of the vorticity function, the set of coadjoint orbits of the symplectomorphism group admitting steady flows with this topology (and a certain metric) forms a convex polytope. As a byproduct of the proposed construction, we also describe a complete list of Casimirs for the 2D Euler hydrodynamics: we define generalized enstrophies which, along with circulations, form a complete set of invariants for coadjoint orbits of area-preserving diffeomorphisms on a surface.

AB - Steady fluid flows have very special topology. In this paper we describe necessary and sufficient conditions on the vorticity function of a 2D ideal flow on a surface with or without boundary, for which there exists a steady flow among isovorticed fields. (Here the necessary condition is for any metric, while the sufficient condition is for an appropriate one.) For this we introduce the notion of an antiderivative (or circulation function) on a measured graph, the Reeb graph associated with the vorticity function on the surface, while the criterion is related to the total negativity of this antiderivative. It turns out that given topology of the vorticity function, the set of coadjoint orbits of the symplectomorphism group admitting steady flows with this topology (and a certain metric) forms a convex polytope. As a byproduct of the proposed construction, we also describe a complete list of Casimirs for the 2D Euler hydrodynamics: we define generalized enstrophies which, along with circulations, form a complete set of invariants for coadjoint orbits of area-preserving diffeomorphisms on a surface.

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U2 - 10.1093/imrn/rnw230

DO - 10.1093/imrn/rnw230

M3 - Article

AN - SCOPUS:85036536568

VL - 2017

SP - 7459

EP - 7503

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 24

ER -