The Euler equation describes the flow of an incompressible fluid on a Riemannian manifold, which we assume to be a 2-dimensional closed orientable surface. The 2D Euler equation can be viewed as a geodesic equation on the symplectomorphisms group, or, alternatively, as a Hamiltonian system on its coadjoint orbits. Using a combinatorial description of the latter coadjoint orbits in terms of graphs with some additional structures, we give a characterization of those orbits which may admit steady solutions of the Euler equation (steady fluid flows) for an appropriate choice of a Riemannian metric. It turns out that when the genus of the surface is at least one, most coadjoint orbits do not admit steady fluid flows, and the set of orbits admitting such flows is, in a certain sense, a convex polytope.
The talk is based on a joint work with Boris Khesin.