Pullback attractors in V for non-autonomous 2D-Navier–Stokes equations and their tempered behaviour

In this paper the asymptotic behaviour of the solutions to a non-autonomous 2D-Navier–Stokes model is analyzed when the initial datum belongs to V, for two frameworks: the universe of fixed bounded sets, and also for another universe given by a tempered condition. The existence of pullback attractors in these different universes is established, and thanks to regularity properties, the relation between these several families of attractors and the corresponding in H is successfully studied. Finally, two results about the tempered behaviour in V and (H22(Ω)) of the pullback attractors, when time goes to −∞, are obtained.