Random dynamics and invariant measures for a class of non-Newtonian fluids of differential type on 2D and 3D Poincaré domains

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 $\Psi$. 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 $\Psi$. 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 $\Psi$ 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.

References in corpus (1)

• Invariant measures for a class of stochastic third grade fluid equations in $2D$ and $3D$ bounded domains