The authors consider the Navier-Stokes equations with delay in 2D. The equation in question is
Here, , ; the delay is encoded in , . is a smooth bounded domain. In order to cast (1) into an abstract form, a standard -setting is imposed on (1). E.g., is the -closure of , , while is the closure of with respect to the norm . In order to take care of the delay, spaces , , are introduced, where etc. The delay is taken into account by which is a mapping from to . The function is subject to a series of assumption involving measurability and two kinds of Lipschitz conditions. System (1) is then put into the abstract form
along conventional lines. The authors first recall a result obtained in an earlier paper, stating that (under some assumptions) (2) admits a unique global solution in some suitable function space. The authors then proceed to show that (2) admits a global attractor.
In this context, the usual notion of attractor does not suffice; it has to be replaced by a new notion, i.e. by a family called pullback attractor, related to a family of evolution operators associated with (2). After proving the existence of an absorbing set, the authors then show that under suitable assumptions on the data, a unique, uniformly bounded pullback attractor for (2) exists. The paper concludes with an application to the case where (2) is supplied by a forcing term.