Large deviations for the two-dimensional stochastic Navier-Stokes equation with vanishing noise correlation

TL;DR

This paper establishes large deviation principles for the 2D stochastic Navier-Stokes equations under vanishing noise strength and correlation, revealing how the relationship between these parameters affects the mathematical behavior of solutions.

Contribution

It provides new results on large deviations for the 2D Navier-Stokes equations with vanishing noise correlation, depending on the asymptotic relationship between noise strength and correlation length.

Findings

Large deviation principles hold under specific regimes of noise and correlation decay.

Different functional spaces are relevant depending on the relationship between  and ().

The results extend understanding of stochastic fluid dynamics with diminishing noise effects.

Abstract

We are dealing with the validity of a large deviation principle for the two-dimensional Navier-Stokes equation, with periodic boundary conditions, perturbed by a Gaussian random forcing. We are here interested in the regime where both the strength of the noise and its correlation are vanishing, on a length scale $\e$ and $\d(\e)$, respectively, with $0<\e,\ \d(\e)<<1$. Depending on the relationship between $\e$ and $\d(\e)$ we will prove the validity of the large deviation principle in different functional spaces.