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

TL;DR

The paper proves ergodicity and exponential mixing for the 2D Navier–Stokes equations on ${\mathbb T}^2$ driven by a highly degenerate multiplicative Gaussian noise that acts in a finite set of modes and depends on the current state. It develops a Malliavin calculus framework tailored to multiplicative, state-dependent noise and establishes the invertibility of the Malliavin matrix, enabling the derivation of an asymptotically strong Feller property. The main results guarantee a unique invariant measure $\mu_*$ and exponential convergence of the semigroup to $\mu_*$ in a suitable weighted topology, under controllability and regularity conditions on the noise functions $q_k$. The methods extend ergodicity results beyond additive or highly-activated noise to cases with very limited directions of forcing, providing a robust approach to degenerate stochastic PDEs in fluid dynamics.

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.

Theorems & Definitions (64)

• Remark 1.3
• Theorem 1.4
• proof
• Proposition 1.5
• Example 1.6
• Example 1.7
• proof
• Lemma 2.1
• proof
• Lemma 2.2