Infinitesimal invariance of completely Random Measures for 2D Euler Equations

Grotto, Francesco; Peccati, Giovanni

doi:10.1090/tpms/1178

Authors: Francesco Grotto and Giovanni Peccati
Journal: Theor. Probability and Math. Statist. 107 (2022), 15-35
MSC (2020): Primary 47B33, 54C40, 14E20; Secondary 46E25, 20C20
DOI: https://doi.org/10.1090/tpms/1178
Published electronically: November 8, 2022
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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 vortices dynamics. We also discuss difficulties arising when attempting to produce a solution flow for Euler’s equations preserving independently scattered random measures.

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