Weak and strong attractors for the 3D Navier-Stokes system

We study in this paper the asymptotic behaviour of the weak solutions of the three-dimensional Navier-Stokes equations. On the one hand, using the weak topology of the usual phase space H (of square integrable divergence free functions) we prove the existence of a weak attractor in both autonomous and nonautonomous cases. On the other, we obtain a conditional result about the existence of the strong attractor, which is valid under an unproved hypothesis. Also, with this hypothesis we obtain continuous weak solutions with respect to the strong topology of H. (c) 2007 Elsevier Inc. All rights reserved.