Abstract
The object of this article is to survey some recent developments in the theory of infinite dimensional dynamical systems. We shall successively consider the derivation of optimal bounds for the dimension of the attractor for the Navier-Stokes equations in space dimension three; new developments in the theory of inertial manifolds and their connection to the concept of slow manifolds broadly used in meteorology; and the approximation of inertial manifolds using finite differences in connection with multigrid methods and wavelets.
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