Large Deviations and Entropy Production in Viscous Fluid Flows

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.

Lagrangian chaos and scalar advection in stochastic fluid mechanics

“…Although the strong Feller property played an important role in early investigations on uniqueness of stationary measures for infinite-dimensional Markov processes [52], it is now well understood that uniqueness holds in settings where the strong Feller property is unknown (and perhaps likely to be false -see, e.g., the discussion in [64]). A relevant example is the very recent work [71], published after this work was completed, which showed that .u t ; x t / has a unique stationary measure when .u t / is governed by Navier-Stokes with a spatially smooth, almost-surely bounded noise, and so for this model assertion (i) is known to hold. However, the degeneracy of the noise model considered is too restricting to establish the strong Feller property, and so it remains open whether C > 0 for the model in [71].…”

The complex Ginzburg-Landau equation perturbed by a force localised both in physical and Fourier spaces

“…The criterion of [KNS20a] is applied in [BGN20] to the system of 3D primitive equations of meteorology and oceanology with a noise only in the temperature equation, and in [Ner19], to the NS system in unbounded domains. In [JNPS19], the controllability approach is further developed to establish a Donsker-Varadhan type large deviations principle for the Lagrangian trajectories of the NS system.…”

The complex Ginzburg-Landau equation perturbed by a force localised both in physical and Fourier spaces

“…The criterion of [KNS20a] is applied in [BGN20] to the system of 3D primitive equations of meteorology and oceanology with a noise only in the temperature equation, and in [Ner19], to the NS system in unbounded domains. In [JNPS19], the controllability approach is further developed to establish a Donsker-Varadhan type large deviations principle for the Lagrangian trajectories of the NS system.…”