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

2025/06/05 by Kush Kinra, Kinra, Kush, Fernanda Cipriano +1
Engineering · Mathematics · #35B40 #35Q35 #35R60 #37L30 #76A05 #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Thin Films #Navier-Stokes equation solutions #Probability (math.PR) #Stability and Controllability of Differential Equations

paper · doi:10.48550/arxiv.2506.04801

openalex publication_date 2025/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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 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 ℍ¹(O) \hookrightarrow\mathbbL²(O), we prove the existence of a unique random attractor for the underlying system on bounded domains with external forcing in ℍ^-1(O)+\mathbbW^-1,(4)/(3)(O). Thirdly, we consider the underlying system on unbounded Poincaré domains with external forcing in \mathbbL²(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 ℍ¹(O) \hookrightarrow\mathbbL²(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.

Citations

• Random dynamics of solutions for three-dimensional stochastic globally modified Navier-Stokes equations on unbounded Poincaré domains
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• Random Attractors for the Stochastic Benjamin-Bona-Mahony Equation on Unbounded Domains
• Boundary-value problems for the system of elasticity theory in unbounded domains. Korn's inequalities

Related

• Random dynamics of two-dimensional stochastic second grade fluids
• Random dynamics of solutions for three-dimensional stochastic globally modified Navier-Stokes equations on unbounded Poincaré domains
• Asymptotically autonomous robustness in probability of random attractors for stochastic Navier-Stokes equations on unbounded Poincaré domains
• Random attractors for 2D and 3D stochastic convective\n Brinkman-Forchheimer equations in some unbounded domains
• Existence and asymptotic autonomous robustness of random attractors for three-dimensional stochastic globally modified Navier-Stokes equations on unbounded domains