  In view of the possibility that the 3D Navier-Stokes equations (NSE) might
not always have regular solutions, we introduce an abstract framework for
studying the asymptotic behavior of multi-valued dissipative evolutionary
systems with respect to two topologies--weak and strong. Each such system
possesses a global attractor in the weak topology, but not necessarily in the
strong. In case the latter exists and is weakly closed, it coincides with the
weak global attractor. We give a sufficient condition for the existence of the
strong global attractor, which is verified for the 3D NSE when all solutions on
the weak global attractor are strongly continuous. We also introduce and study
a two-parameter family of models for the Navier-Stokes equations, with similar
properties and open problems. These models always possess weak global
attractors, but on some of them every solution blows up (in a norm stronger
than the standard energy one) in finite time.
