Characterization of Steady Solutions for the 2D Euler Equation
The motion of an ideal fluid on a 2D surface is described by the incompressible Euler equation, which can be regarded as a Hamiltonian system on coadjoint orbits of the symplectic diffeomorphisms group. Using a combinatorial description of these orbits in terms of graphs with some additional structures, we give a characterization of coadjoint orbits which may admit steady solutions of then Euler equation (steady fluid flows). It turns out that when the genus of the surface is at least one, most coadjoint orbits do not admit steady fluid flows, while the set of orbits admitting such flows is a convex polytope. This is a joint work with B.Khesin.