Essential self-adjointness of Liouville operator for 2D Euler point vortices

We analyse the 2-dimensional Euler point vortices dynamics in the Koopman-Von Neumann approach. Classical results provide well-posedness of this dynamics involving singular interactions for a finite number of vortices, on a full-measure set with respect to the volume measure dxN on the phase space, which is preserved by the measurable flow thanks to the Hamiltonian nature of the system. We identify a core for the generator of the one-parameter group of Koopman-Von Neumann unitaries on L2(dxN) associated to said flow, the core being made of observables smooth outside a suitable set on which singularities can occur.