Random dynamics and invariant measures for a class of non-Newtonian fluids of differential type on 2D and 3D Poincaré domains
Kush Kinra, Fernanda Cipriano
TL;DR
This work studies a stochastic incompressible non-Newtonian (third-grade) fluid model on $2$-D and $3$-D Poincaré domains, establishing well-posedness and a rigorous random dynamical framework via a Doss–Sussmann transformation. It proves the existence of a unique weak analytic solution and constructs a random dynamical system, then demonstrates the existence of random attractors both on bounded domains (using compact Sobolev embeddings) and on unbounded domains (via uniform-tail estimates), culminating in the existence of invariant measures. The results extend stochastic analysis for non-Newtonian fluids to unbounded geometries and connect long-time dynamics to invariant measures, with the caveat that uniqueness of invariant measures remains open. The methodology combines monotone operator techniques, Ornstein–Uhlenbeck processes, and tail-control strategies to overcome the lack of compactness in unbounded settings.
Abstract
In this article, we consider a class of incompressible stochastic third-grade fluids (non-Newtonian fluids) equations on two- as well as three-dimensional Poincaré domains $\mathcal{O}$ (which may be bounded or unbounded). Our aims are to study the well-posedness and asymptotic analysis for the solutions of the underlying system. Firstly, we prove that the underlying system defined on $\mathcal{O}$ has a unique weak solution (in the analytic sense) under Dirichlet boundary condition and it also generates random dynamical system $Ψ$. Secondly, we consider the underlying system on bounded domains. Using the compact Sobolev embedding $\mathbb{H}^1(\mathcal{O}) \hookrightarrow\mathbb{L}^2(\mathcal{O})$, we prove the existence of a unique random attractor for the underlying system on bounded domains with external forcing in $\mathbb{H}^{-1}(\mathcal{O})+\mathbb{W}^{-1,\frac{4}{3}}(\mathcal{O})$. Thirdly, we consider the underlying system on unbounded Poincaré domains with external forcing in $\mathbb{L}^{2}(\mathcal{O})$ and show the existence of a unique random attractor. In order to obtain the existence of a unique random attractor on unbounded domains, due to the lack of compact Sobolev embedding $\mathbb{H}^1(\mathcal{O}) \hookrightarrow\mathbb{L}^2(\mathcal{O})$, we use the uniform-tail estimates method which helps us to demonstrate the asymptotic compactness of $Ψ$. Note that due to the presence of several nonlinear terms in the underlying system, we are not able to use the energy equality method to obtain the asymptotic compactness of $Ψ$ in unbounded domains, which makes the analysis of this work in unbounded domains more difficult and interesting. Finally, as a consequence of the existence of random attractors, we address the existence of invariant measures for underlying system.