Ergodic and mixing properties of the 2D Navier-Stokes equations with a degenerate multiplicative Gaussian noise

TL;DR

This paper proves that the 2D Navier-Stokes equations with highly degenerate multiplicative Gaussian noise exhibit ergodic and mixing behavior, using Malliavin calculus to handle the complex invertibility of the Malliavin matrix.

Contribution

It extends ergodic and mixing results to the case of degenerate multiplicative noise, which was not previously addressed.

Findings

Established ergodicity of 2D Navier-Stokes with degenerate noise

Proved mixing properties under multiplicative noise

Developed new techniques for Malliavin matrix invertibility

Abstract

In this paper, we establish ergodic and mixing properties of stochastic 2D Navier-Stokes equations driven by a highly degenerate multiplicative Gaussian noise. The noise could appear in as few as four directions and the intensity of the noise depends on the solution. The case of additive Gaussian noise was treated in Hairer and Mattingly [\emph{Ann. of Math.}, 164(3):993--1032, 2006]. To obtain ergodic and mixing properties, we use Malliavin calculus to establish the asymptotically strong Feller property. The main difficulty lies in the proof of the "invertibility" of Malliavin matrix which is totally different from the additive case.