Level-3 large deviations for the white-forced 2D Navier-Stokes system in a bounded domain

TL;DR

This paper establishes large deviations principles for the two-dimensional Navier-Stokes system with white noise in a bounded domain, providing a rigorous probabilistic framework for understanding rare events in this fluid dynamics model.

Contribution

It introduces a novel approach to proving level-2 and level-3 large deviations principles for the stochastic Navier-Stokes equations with non-degenerate noise, extending existing theories.

Findings

Proved level-2 and level-3 LDPs for the system.

Derived explicit rate functions via Donsker-Varadhan formulas.

Developed an improved criterion and approximation scheme for large-time asymptotics.

Abstract

We study the large deviations principle (LDP) of Donsker-Varadhan type for the white-forced Navier-Stokes system in a bounded domain. Under the assumption that the noise is non-degenerate, we establish level-2 and level-3 LDPs with rate functions given by the Donsker-Varadhan formulas. The proof relies on an improved version of Kifer's criterion, a lift argument inspired from [DV83], an improved abstract result on the large-time asymptotics of generalized Markov semigroups, and a delicate approximation scheme utilizing the resolvent operators of the Markov semigroup.