For certain dissipative partial differential equations unique or regular solutions in a given state space and for all time are not available, either because solutions cease to exist in the relevant topology or because it is not known whether such solutions exist in the first place. A famous example is furnished by strong solutions of the three-dimensional Navier-Stokes equations. In this situation the concept of a global attractor, the minimal closed set in the state space that uniformly attracts (in the topology of the state space) the trajectories starting from any bounded subset of initial states, is not suitable to describe the asymptotic behavior of solutions.
The work under review introduces an abstract framework which allows studying the long-term dynamics of possibly multi-valued evolution systems with respect to two related metric topologies of the state space, referred to as the weak and strong topology. While in this context the dissipative system always possesses a global attractor with respect to the weak topology (“weak global attractor”), this need not hold true in the strong topology (“strong global attractor”). However, if the strong global attractor exists and is closed in the weak topology, then it coincides with the weak global attractor.
The authors give a sufficient condition for the existence of the strong global attractor and demonstrate that this condition is satisfied for the three-dimensional space-periodic Navier-Stokes equations when all solutions on the weak global attractor are continuous in the strong topology. They extend their analysis to so-called tridiagonal models for the Navier-Stokes equations: a two-parameter family of simple models for the Navier-Stokes equations with a similar nonlinearity. These model equations fall within the framework introduced here and have weak global attractors. Solutions for certain parameter values, however, are shown to blow up in finite time (with respect to the appropriate norm). Some open questions on blow-up of solutions and strong global attractors for the tridiagonal models of the Navier-Stokes are posed and corresponding results for tridiagonal models of the Euler equations are discussed.