Essential Self-Adjointness of Liouville Operator for 2D Euler Point Vortices

TL;DR

This paper investigates the mathematical properties of the 2D Euler point vortices system using the Koopman-Von Neumann framework, establishing essential self-adjointness of the associated Liouville operator and identifying a core for its generator.

Contribution

It provides a rigorous analysis of the Liouville operator's self-adjointness for vortex dynamics, extending the understanding of the Koopman approach in singular Hamiltonian systems.

Findings

Identifies a core for the generator of the Koopman-Von Neumann unitaries.

Proves essential self-adjointness of the Liouville operator.

Shows the flow preserves the volume measure on phase space.

Abstract

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 $dx^N$ 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 $L^2(dx^N)$ associated to said flow, the core being made of observables smooth outside a suitable set on which singularities can occur.