The authors study the asymptotic behaviour of weak solutions of the Navier-Stokes equations
considered on a smooth bounded domain , and subject to homogeneous Dirichlet boundary conditions; here is the Stokes operator, while is the nonlinearity. Eq. (1) is studied in an -setting based on the familiar spaces , [R. Temam, Navier-Stokes equations. Theory and numerical analysis. Amsterdam etc.: North-Holland (1979; Zbl 0426.35003)]. Following these lines, a standard definition of weak solution is given. Starting point is an unproved assumption (H) imposed on the exterior force :
(H) Let ; there exists a function , continuous and nondecreasing in and nonincreasing in , such that for any there exists a globally defined weak solution such that and
It then follows that the weak solutions provided by assumption (H) satisfy
and are unique in the class of solutions which satisfy (3). The authors then prove a series of theorems, all expressing properties of weak solutions provided by (H). Based on these preparations, the authors then proceed to study a class of attractors associated with these weak solutions. They set in (1)–(3), assume that in (1) is autonomous and define a set valued mapping from into the set of all nonempty subsets of as follows:
(4) is the set of all , where is a globally defined weak solution subject to (3), and with It is proved that is a strict multivalued semiflow, i.e.
For such semiflows the notion of global attractor can be defined along established lines. The authors prove that the semiflow has a global attractor , which is minimal. The case of nonautonomous is more involved and requires the introduction of multivalued processes in the sense of A. V. Babin and M. I. Vishik [Attractors of evolution equations. Moscow: Nauka (1989; Zbl 0804.58003)]. The authors prove the existence of a global attractor in this more general situation. The paper concludes with a theory of attractors for multivalued processes on topological spaces, extending results obtained by V. S. Melnik and J. Valero [Set-Valued Anal. 8, No. 4, 375–403 (2000; Zbl 1063.35040)] for metric spaces.