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

Burst of Point Vortices and Non-Uniqueness of 2D Euler Equations

“…To the best of our knowledge, this is a novelty in this context, so we briefly discuss it below (subsection 4.2) and leave it for future developments. The results of this paper might also contribute to answer to the question of whether the Liouville operator (the generator of Koopman's group of unitaries) associated to the N point vortices evolution is essentially self-adjoint on certain classes of observables, a problem left open in [1,14]. We also mention [2, Section III], in which existence of arbitrarily large collapsing configurations of vortices was left open, and where some interesting consequences are conjectured.…”

Uniform approximation of 2D Navier-Stokes equations with vorticity creation by stochastic interacting particle systems

“…including self-interaction terms induced from the boundary effects; this is necessary for the system to satisfy in a weak form (as in [44]) the 2-dimensional Euler equations. We refer to [38, chapter 4] for a general introduction to the topic, to [15,22,25,26,30] on the issue of wellposedness of the dynamics and vortex collisions, and to [23,24,[27][28][29]36] on the statistical mechanics point of view. Self-interactions diverge logarithmically at ∂D, and this prevents us to include them in our model because the Brownian part of the dynamics (which we must include to model viscosity) might drive particles onto the boundary causing blow-up of the dynamics at finite time.…”

Infinitesimal Invariance of Completely Random Measures for 2D Euler Equations

“…However, it is possible to exhibit configurations of vortices leading to collapse: we refer to [18] for details. How these phenomena relate to uniqueness in the extension of the generator A on cylinder function is an interesting problem, for which we refer to [6,16] (see also Section 5 for further remarks on general M ∼ [0, q, ν]).…”