Ergodicity for stochastic equation of Navier--Stokes type

TL;DR

This paper investigates the long-term behavior and ergodic properties of a 2D stochastic Navier--Stokes system with degenerate noise, identifying conditions for unique invariant measures and analyzing a related finite-dimensional stochastic system.

Contribution

It provides new insights into the ergodicity and invariant measures of degenerate stochastic Navier--Stokes equations and a simplified finite-dimensional model, highlighting the effects of noise strength.

Findings

Existence of invariant measures under certain conditions.

Uniqueness and stability of invariant measures depend on noise and initial data.

Finite-dimensional model exhibits phase transition based on drift strength.

Abstract

In the first part of the note we analyze the long time behaviour of a two dimensional stochastic Navier--Stokes equations system on a torus with a degenerate, one dimensional noise. In particular, for some initial data and noises we identify the invariant probability measure for the system and give a sufficient condition under which it is unique and stochastically stable. In the second part of the note, we consider a simple example of a finite-dimensional system of stochastic differential equations driven by a one dimensional Wiener process with a drift, that displays some similarity with the stochastic N.S.E., and investigate its ergodic properties depending on the strength of the drift. If the latter is sufficiently small and lies below a critical threshold, then the system admits a unique invariant probability measure which is Gaussian. If, on the other hand, the strength of the noise drift is larger than the threshold, then in addition to a Gaussian invariant probability measure, there exist another one. In particular, the generator of the system is not hypoelliptic.