The first part of the reviewing article contains an overview of recent results on dynamical systems associated with dissipative evolution equations (attractors; inertial manifolds; approximate inertial manifolds). On the base of the work of J. L. Lions, the author and S. Wang [Nonlinearity 5, No. 2, 237-288 (1992; Zbl 0746.76019)] are indicated the possibilities of applications to the equations of meteorology, oceanography and climatology, for long time wheather prediction when viscous effects are considered.
In the second part of the article the results of A. Debussche and the author on the mathematical theory of slow manifolds are presented [Appl. Math. Lett. 4, No. 4, 73-76 (1991; Zbl 0723.58006) and Differ. Integral Equ. 4, No. 5, 897-931 (1991; Zbl 0753.35043)]. In particular the connections of approximate inertial manifolds with the method of nonlinear initialization in meteorology are given.
For the entire collection see [Zbl 0782.00023].