Large time behaviour for a class of 2D and 3D stochastic non-Newtonian fluids of differential types: Attractors and invariant measures

TL;DR

This work analyzes the long-time behavior of stochastic non-Newtonian flows of differential type, specifically stochastic third-grade fluids on general (bounded and unbounded) domains driven by linear multiplicative Itô noise. By applying a Doss–Sussman transformation and reformulating the problem as a non-autonomous random dynamical system, the authors obtain a comprehensive long-time description: existence of a pullback random absorbing set, and both bounded- and unbounded-domain pullback random attractors, along with the existence of invariant measures and ergodicity results. Key contributions include establishing pullback attractors on general unbounded domains and proving uniqueness of the invariant measure in the zero-forcing case via exponential stability, thereby advancing the theory of random attractors and ergodicity for stochastic third-grade fluids in broad geometries. The analysis highlights the role of Itô noise in domains lacking Poincaré inequality and provides a rigorous framework for understanding the stochastic long-time dynamics of highly nonlinear non-Newtonian fluids.

Abstract

This study investigates a stochastic version of a class of non-Newtonian fluids governed by third-grade fluid equations, which exhibit complex and highly nonlinear dynamics. In particular, we address the random dynamics and asymptotic behavior of stochastic third-grade fluid equations (STGFEs) driven by a \emph{linear multiplicative Itô-type white noise} on general domains $\mathbb{Q}\subseteq\mathbb{R}^d$, $d\in\{2,3\}$. We first prove that the non-autonomous STGFEs generate a continuous non-autonomous random dynamical system $Φ$, and we establish the existence of a pullback absorbing set. Using compact Sobolev embeddings on bounded domains and uniform tail estimates on unbounded domains, we show the pullback asymptotic compactness of $Φ$, which leads to the existence of pullback random attractors that are compact and attracting in $\mathbb{L}^2(\mathbb{Q})$. As a consequence, we demonstrate the existence of an invariant measure associated with the STGFEs and, exploiting the linear multiplicative structure of the noise along with the exponential stability of solutions, we prove uniqueness of the invariant measure in the case of zero external forcing. These results are entirely new for STGFEs on general domains, and, in particular, the existence of pullback random attractors with linear multiplicative noise is obtained here for the first time. We further note that, unlike Stratonovich noise, which is widely used in the literature to study random attractors, Itô noise is more appropriate for domains that do not satisfy the Poincaré inequality. Overall, this work resolves several open problems regarding random attractors, invariant measures, and ergodicity for stochastic third-grade fluids on general unbounded domains $\mathbb{Q}\subseteq\mathbb{R}^d$, $d\in\{2,3\}$.

Theorems & Definitions (52)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Lemma 2.1
• Lemma 2.2: Kunstmann_2010
• Remark 2.3
• Remark 2.4
• Definition 2.5
• Lemma 2.6
• proof