Rate of convergence of random attractors towards deterministic singleton attractor for a class of non-Newtonian fluids of differential type

TL;DR

This paper studies how the long-term behavior of certain non-Newtonian fluids simplifies to a single state under small external forces and analyzes how stochastic noise affects the convergence of their random attractors to this deterministic state.

Contribution

It establishes conditions for the deterministic singleton attractor and quantifies the convergence rate of random attractors under stochastic perturbations for third-grade fluids.

Findings

Deterministic attractor reduces to a point with small forcing.

Estimated the convergence rate of random attractors as noise diminishes.

Provided insights into stochastic effects on non-Newtonian fluid dynamics.

Abstract

In this article, we investigate the long-term dynamics of a class of two- and three-dimensional non-Newtonian fluids of differential type, known as third-grade fluids. We first show that when the external forcing is sufficiently small, the global attractor of the underlying system (which characterizes its asymptotic behavior) reduces to a single point. We then consider the system under stochastic perturbations, specifically infinite-dimensional additive white noise. In this random setting, we do not find conclusive evidence that the corresponding random attractor remains a single point, as in the deterministic case. However, we are able to estimate the rate at which the random attractor approaches the deterministic singleton attractor as the intensity of the stochastic noise tends to zero.