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

Hui Liu (Jiangsu Provincial Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, China; and School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong, China)

Hongjun Gao (Jiangsu Provincial Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, China)

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

Full Text (PDF format)

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