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 [García-Luengo, Marín-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 norm and the 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.