# Large deviations and entropy production in viscous fluid flows

## TL;DR

This paper investigates large deviations and entropy production in viscous fluid flows by analyzing particle trajectories in a stochastic 2D Navier-Stokes system, establishing a large deviation principle and properties of entropy production.

## Contribution

It introduces a new criterion for the level-3 large deviation principle in Markov processes and applies it to particle motion in stochastic fluid flows, linking entropy production to trajectory analysis.

## Key findings

- Empirical measures satisfy the large deviation principle with a good rate function.
- The law of the particle component has a positive smooth density in finite time.
- Time-averaged entropy production is bounded along trajectories.

## Abstract

We study the motion of a particle in a random time-dependent vector field defined by the 2D Navier-Stokes system with a noise. Under suitable non-degeneracy hypotheses we prove that the empirical measures of the trajectories of the pair (velocity field, particle) satisfy the LDP with a good rate function. Moreover, we show that the law of a unique stationary solution restricted to the particle component possesses a positive smooth density with respect to the Lebesgue measure in any finite time. This allows one to define a natural concept of the entropy production, and to show that its time average is a bounded function of the trajectory. The proofs are based on a new criterion for the validity of the level-3 LDP for Markov processes and an application of a general result on the image of probability measures under smooth maps to the laws associated with the motion of the particle.