Ergodicity of 2D singular stochastic Navier-Stokes equations

TL;DR

This paper investigates the ergodic properties of 2D stochastic Navier-Stokes equations driven by irregular noise, establishing conditions under which the system admits an invariant measure with exponential tail bounds.

Contribution

It demonstrates the existence of an invariant measure for the 2D stochastic Navier-Stokes equations under weak high-frequency noise conditions, extending understanding of their long-term behavior.

Findings

Existence of invariant measure under certain noise conditions

Uniform bounds in time for the solutions

Stretched exponential tail bounds for the invariant measure

Abstract

We consider the 2D stochastic Navier-Stokes equations driven by noise that has the regularity of space-time white noise but doesn't exactly coincide with it. We show that, provided that the intensity of the noise is sufficiently weak at high frequencies, this systems admits uniform bounds in time, so that it has an invariant measure, for which we obtain stretched exponential tail bounds.