# Fokker-Planck equation for dissipative 2D Euler equations with cylindrical noise

## TL;DR

This paper advances the understanding of dissipative 2D Euler equations with cylindrical noise by analyzing the associated infinite-dimensional Fokker-Planck equation, focusing on probabilistic bounds without energy estimates.

## Contribution

It introduces new a priori estimates for the Fokker-Planck equation in infinite dimensions, applicable to solutions with limited regularity and random initial conditions.

## Key findings

- Improved results on 2D Euler equations with cylindrical noise.
- Establishment of probabilistic bounds without energy estimates.
- Application to Gibbsian measures based on renormalized kinetic energy.

## Abstract

After a short review of recent progresses in 2D Euler equations with random initial conditions and noise, some of the recent results are improved by exploiting a priori estimates on the associated infinite dimensional Fokker-Planck equation. The regularity class of solutions investigated here does not allow energy- or enstrophy-type estimates, but only bounds in probability with respect to suitable distributions of the initial conditions. This is a remarkable application of Fokker-Planck equations in infinite dimensions. Among the example of random initial conditions we consider Gibbsian measures based on renormalized kinetic energy.