Random dynamics of two-dimensional stochastic second grade fluids

Abstract

In this paper, we consider a stochastic model of incompressible non-Newtonian fluids of second grade on a bounded domain of ${\mathbb{R}}^2$ with multiplicative noise. We first show that the solutions to the stochastic equations of second grade fluids generate a continuous random dynamical system. Second, we investigate the Fréchet differentiability of the random dynamical system. Finally, we establish the asymptotic compactness of the random dynamical system, and the existence of random attractors for the random dynamical system, we also obtain the upper semi-continuity of the perturbed random attractors when the noise intensity approaches zero.

Keywords:

• Second grade fluids
• non-Newtonian fluids
• stochastic partial differential equations
• Fréchet differentiability
• asymptotically compact random dynamical systems
• random attractors

Acknowledgements

The author sincerely thanks Professor Tusheng Zhang and Jianliang Zhai for their instructions and many invaluable suggestions.

Additional information

Funding

This work is partly supported by National Natural Science Foundation of China (Nos. 11671372, 11431014, and 11721101) and the Fundamental Research Funds for the Central Universities (No. WK0010000057).