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# Communications in Mathematical Sciences

## Volume 16 (2018)

### Number 1

### Ergodicity and dynamics for the stochastic 3D Navier–Stokes equations with damping

Pages: 97 – 122

DOI: http://dx.doi.org/10.4310/CMS.2018.v16.n1.a5

#### Authors

#### Abstract

The stochastic 3D Navier–Stokes equation with damping driven by a multiplicative noise is considered in this paper. The existence of invariant measures is proved for $3 \lt \beta \leq 5$ with any $\alpha \gt 0$ and $\alpha \geq \frac{1}{2}$ as $\beta = 3$. Using asymptotic strong Feller property, the uniqueness of invariant measures is obtained for the degenerate additive noise. The existence of a random attractor for the random dynamical systems generated by the solution of stochastic 3D Navier–Stokes equations with damping is proved for $\beta \gt 3$ with any $\alpha \gt 0$ and $\alpha \geq \frac{1}{2}$ as $\beta = 3$.

#### Keywords

stochastic Navier–Stokes equations, existence, uniqueness, ergodicity, random attractor

#### 2010 Mathematics Subject Classification

35B41, 35Q30, 37A25, 37L40, 76D05

The work of Gao is partially supported by a NSFC Grant No. 11531006, PAPD of Jiangsu Higher Education Institutions, and Jiangsu Center for Collaborative Innovation in Geographical Information Resource Development and Application.

Received 5 November 2016

Accepted 12 September 2017

Published 29 March 2018