Rate of convergence of random attractors towards deterministic singleton attractor for a class of non-Newtonian fluids of differential type
Kush Kinra
TL;DR
This paper addresses the long-time behavior of 2D and 3D third-grade non-Newtonian fluids by developing a rigorous variational framework and proving that, under sufficiently small external forcing, the deterministic global attractor is a singleton $\mathscr{A}=\{\mathbf{a}_*\}$. It then extends the analysis to stochastic perturbations in the form of infinite-dimensional additive white noise, establishing the existence of random attractors and quantifying their convergence to the deterministic singleton as the noise intensity $\varsigma$ vanishes; the convergence rate is shown to be $\dist_{\mathbb{H}}(\mathscr{A}_{\varsigma}(\omega), \mathscr{A}) \lesssim \varsigma^{\frac{2}{3}}$ under a small-forcing regime. The results rely on a careful functional-analytic setup, Helmholtz decompositions on (un)bounded domains, and energy estimates that control the nonlinear terms $\mathcal{B}$, $\mathcal{J}$, and $\mathcal{K}$ in the abstract equation $\frac{d\mathfrak{m}}{dt}+\nu\mathcal{A}\mathfrak{m}+\mathcal{B}(\mathfrak{m})+\alpha\mathcal{J}(\mathfrak{m})+\beta\mathcal{K}(\mathfrak{m})=\mathcal{P}\mathfrak{g}$. The study also contrasts the convergence rate with known results for Newtonian fluids, highlighting a slower decay in the non-Newtonian setting. Overall, the work advances understanding of how stochastic perturbations influence the asymptotic singleton structure and provides quantitative rates for the approach to determinism. $\mathscr{A}$, $\mathbf{a}_*$, and $\mathscr{A}_{\varsigma}(\omega)$ are central objects throughout the analysis, and the rate $\varsigma^{\frac{2}{3}}$ is a key quantitative takeaway with implications for stochastic fluid models in non-Newtonian rheology.
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.