Characterization of steady solutions to the 2D Euler equation

TL;DR

This paper characterizes the conditions under which steady solutions exist for the 2D Euler equations on surfaces, introducing a novel topological framework involving Reeb graphs and circulation functions.

Contribution

It introduces a new topological criterion for steady flows based on the vorticity function's Reeb graph and describes the structure of coadjoint orbits and Casimir invariants in 2D Euler hydrodynamics.

Findings

Necessary and sufficient conditions for steady flows are established.

The set of coadjoint orbits with steady flows forms a convex polytope.

A complete list of Casimirs, including generalized enstrophies, is provided.

Abstract

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. For this we introduce the notion of an antiderivative (or circulation function) on a measured graph, the Reeb graph associated to 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 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.