Large deviations principle for 2D Navier-Stokes equations with space time localised noise

TL;DR

This paper establishes a large deviations principle for the 2D Navier-Stokes equations driven by localized space-time noise, extending ergodicity results to probabilistic deviations in a bounded domain.

Contribution

It proves a large deviations principle for stochastic 2D Navier-Stokes equations with localized noise, strengthening previous ergodicity conditions.

Findings

Proves large deviations principle for 2D Navier-Stokes with localized noise

Extends ergodicity results to probabilistic deviation estimates

Uses uniform irreducibility and Feller properties in proof

Abstract

We consider a stochastic 2D Navier-Stokes equation in a bounded domain. The random force is assumed to be non-degenerate and periodic in time, its law has a support localised with respect to both time and space. Slightly strengthening the conditions in the pioneering work about exponential ergodicity by Shirikyan [Shi15], we prove that the stochastic system satisfies Donsker-Varadhan typle large deviations principle. Our proof is based on a criterion of [JNPS15] in which we need to verify uniform irreducibility and uniform Feller property for the related Feynman-Kac semigroup.