Long time behavior of the stochastic 2D Navier-Stokes equations

TL;DR

This paper reviews key results on the long-term behavior of the 2D stochastic Navier-Stokes equations, focusing on invariant measures, and presents recent findings on models with multiplicative noise.

Contribution

It provides a comprehensive review of existing results and introduces new findings on the invariant measures for 2D stochastic Navier-Stokes equations with multiplicative noise.

Findings

Existence and uniqueness of invariant measures for additive noise cases.

New results on invariant measures for multiplicative noise models.

Enhanced understanding of long-term dynamics of stochastic fluid models.

Abstract

We review some basic results on existence and uniqueness of the invariant measure for the two-dimensional stochastic Navier-Stokes equations. A large part of the literature concerns the additive noise case; after revising these models, we consider our recent result, arXiv:2307.03483, with a multiplicative noise.