Infinitesimal Invariance of Completely Random Measures for 2D Euler Equations

TL;DR

This paper demonstrates that completely random measures are infinitesimally invariant for 2D Euler equations, extending previous results and addressing challenges in defining solution flows with such measures.

Contribution

It establishes the infinitesimal invariance of completely random measures for 2D Euler equations, generalizing prior Gaussian and vortex dynamics results.

Findings

Complete random measures are infinitesimally invariant for 2D Euler.

Space regularity of these measures falls outside well-posedness regimes.

Discusses difficulties in constructing solution flows preserving these measures.

Abstract

We consider suitable weak solutions of 2-dimensional Euler equations on bounded domains, and show that the class of completely random measures is infinitesimally invariant for the dynamics. Space regularity of samples of these random fields falls outside of the well-posedness regime of the PDE under consideration, so it is necessary to resort to stochastic integrals with respect to the candidate invariant measure in order to give a definition of the dynamics. Our findings generalize and unify previous results on Gaussian stationary solutions of Euler equations and point vortex dynamics. We also discuss difficulties arising when attempting to produce a solution flow for Euler's equations preserving independently scattered random measures.