Long-time dynamics for time-nonlocal generalized Rayleigh-Stokes equations

TL;DR

This paper studies the long-term behavior of solutions to time-nonlocal Rayleigh-Stokes equations modeling non-Newtonian fluids, establishing well-posedness, semi-dynamical systems, and attractors.

Contribution

It introduces a framework for analyzing the global dynamics of time-nonlocal Rayleigh-Stokes equations, including existence of attractors under dissipativity and Lipschitz conditions.

Findings

Proved global well-posedness of solutions.

Constructed a semi-dynamical system with a semi-group structure.

Established the existence of generalized attractors.

Abstract

In this paper, we consider an autonomous semi-dynamical system driven by semilinear time-nonlocal evolution equations, these type equations are used to describe the Rayleigh-Stokes problem for a non-Newtonain fluid to a generalized second grade fluid. We first investigate the global well-posedness of solutions consisting of global Lipschitz condition by a weighted space $\mathcal C$. Utilizing the topology convergence on compact subsets of $\mathcal C$, we construct a semi-dynamical system that satisfies the semi-group structure. It also is shown that this semi-dynamical system has an attracting set when the vector field function satisfies a dissipativity condition and a local Lipschitz condition. With the asymptotic compactness, we also establish the existence of generalized attractors in $\mathcal C_\alpha$ of subspace of $\mathcal C$ the weighted norm.