Pullback attractors for 2D Navier–Stokes equations with delays and the flattening property

Abstract

This paper treats the existence of pullback attractors for a 2D Navier–Stokes model with finite delay formulated in [Caraballo and Real, J. Differential Equations 205 (2004), 271–297]. Actually, we carry out our study under less restrictive assumptions than in the previous reference. More precisely, we remove a condition on square integrable control of the memory terms, which allows us to consider a bigger class of delay terms. Here we show that the asymptotic compactness of the corresponding processes required to establish the existence of pullback attractors, obtained in [Garcia-Luengo, Marin-Rubio and Real, Adv. Nonlinear Stud. 13 (2013), 331–357] by using an energy method, can be also proved by verifying the flattening property – also known as Condition (C). We deal with dynamical systems in suitable phase spaces within two metrics, the \begin{document}$ L^2 $\end{document} norm and the \begin{document}$ H^1 $\end{document} norm. Moreover, we provide results on the existence of pullback attractors for two possible choices of the attracted universes, namely, the standard one of fixed bounded sets, and secondly, one given by a tempered condition.