On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems

For an abstract dynamical system, we establish, under minimal assumptions, the existence of DD-attractor, i.e. a pullback attractor for a given class DD of families of time varying subsets of the phase space. We relate this concept with the usual attractor of fixed bounded sets, pointing out its usefulness in order to ensure the existence of this last attractor in particular situations. Moreover, we prove that under a simple assumption these two notions of attractors generate, in fact, the same object. This is then applied to a Navier–Stokes model, improving some previous results on attractor theory.