This is a very interesting paper, concerning, the “abstract” solutions of a problem with “initial” data, such as the unsteady Navier-Stokes problem in 3D. A generalized semiflow is a family of “solutions” \(f\) from \((0, \infty)\) to a suitable space \(X\). \(f(0)\) is prescribed.
A condition is given under which the elements of a generalized semiflow \(G\) are continuous from \((0,\infty) \) to \(X\). Another condition is specified under which \(G\) has a global attractor. A characterization of \(G\) is given by using the method of [O. Ladyzhenskaya, Attractors for semigroups and evolution equations, Cambridge Univ. Press (1991; Zbl 0755.47049)]. In the final part, the set \(G_{\text{NS}}\) of all weak solutions to Navier-Stokes (NS) equations is considered; \(G_{\text{NS}}\) has at least one element, obtained by the Galerkin method. It is proved that \(G_{\text{NS}}\) is a generalized semiflow on a suitable space \(H\) iff each weak solution is continuous from \((0,\infty)\) to \(H\).
The main result is as follows: If \(G_{\text{NS}}\) is a generalized semiflow, then there exists a global attractor for \(G_{\text{NS}}\). Then the existence of a global attractor for the NS equations is related with the property that all weak solutions are continuous from \((0,\infty)\) to \(L^2\).
For the entire collection see [Zbl 0933.00017].