Random dynamics of two-dimensional stochastic second grade fluids

TL;DR

This paper studies the behavior of two-dimensional stochastic second grade fluids with multiplicative noise, establishing the existence of random attractors and analyzing the system's differentiability and asymptotic properties.

Contribution

It introduces a new stochastic model for second grade fluids, proves the existence of a continuous random dynamical system, and demonstrates the presence of random attractors and their upper semi-continuity.

Findings

Solutions generate a continuous random dynamical system

Existence of random attractors established

Upper semi-continuity of attractors as noise diminishes

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 Fr\'{e}chet 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.