2D vortex motion of an incompressible ideal fluid: the Koopman-von Neumann approach

An incompressible ideal fluid in the two-dimensional torus (i.e. the Euler equation in a rectangle with periodic boundary conditions) is considered. The flow for a vorticity field concentrated in any finite number of points is analyzed. A compound Poisson measure Pi, invariant for this flow, is introduced. The Hilbert space L-2(Pi) and the properties of the corresponding L-2-flow are investigated. In particular it is proven that the corresponding generator is Markov unique.