Bi-spatial random attractors, a stochastic Liouville type theorem and ergodicity for stochastic Navier-Stokes equations on the whole space

TL;DR

This paper studies the long-term behavior of 2D stochastic Navier-Stokes equations on the whole space, establishing the existence of bi-spatial random attractors, invariant measures, and a stochastic Liouville theorem, with novel results for unbounded domains.

Contribution

It introduces the first bi-spatial random attractors and stochastic Liouville theorem for 2D SNSE with linear multiplicative noise on unbounded domains.

Findings

Existence of bi-spatial pullback random attractors in  and  norms.

Existence of invariant sample measures satisfying a stochastic Liouville theorem.

Uniqueness of invariant measures for zero forcing and positive noise intensity.

Abstract

This article concerns the random dynamics and asymptotic analysis of the well known mathematical model, the Navier-Stokes equations. We consider the two-dimensional stochastic Navier-Stokes equations (SNSE) driven by a \textsl{linear multiplicative white noise of It\^o type} on the whole space $\mathbb{R}^2$. Firstly, we prove that the non-autonomous 2D SNSE generates a bi-spatial $(\mathbb{L}^2(\mathbb{R}^2),\mathbb{H}^1(\mathbb{R}^2))$-continuous random cocycle. Due to the bi-spatial continuity property of the random cocycle associated with SNSE, we show that if the initial data is in $\mathbb{L}^2(\mathbb{R}^2)$, then there exists a unique bi-spatial $(\mathbb{L}^2(\mathbb{R}^2),\mathbb{H}^1(\mathbb{R}^2))$-pullback random attractor for non-autonomous SNSE which is compact and attracting not only in $\mathbb{L}^2$-norm but also in $\mathbb{H}^1$-norm. Next, as a consequence of the existence of pullback random attractors, we prove the existence of a family of invariant sample measures for non-autonomous random dynamical system generated by 2D non-autonomous SNSE. Moreover, we show that the family of invariant sample measures satisfies a stochastic Liouville type theorem. Finally, we discuss the existence of an invariant measure for the random cocycle associated with 2D autonomous SNSE. We prove the uniqueness of invariant measures for $\boldsymbol{f}=\mathbf{0}$ and for any $\nu>0$ by using the linear multiplicative structure of the noise coefficient and exponential stability of solutions. The above results for SNSE defined on $\mathbb{R}^2$ are totally new, especially the results on bi-spatial random attractors and stochastic Liouville type theorem for 2D SNSE with linear multiplicative noise are obtained in any kind of domains for the first time.