Inertial manifolds and finite-dimensional reduction in dissipative PDEs

The aim of this talk is to review the classical and recent achievements in the theory of infinite-dimensional dissipative dynamical systems. The main attention will be payed to inertial manifolds (IMs) which are the most natural tools for justifying the finite dimensionality of the associated limit dynamics and for constructing the reduced system of ODEs describing this dynamics. Various methods of verifying the existence of IMs (including the classical spectral gap conditions, spatial and spatio-temporal averaging technique, etc.) will be discussed and illustrated on concrete classes of PDEs. The examples of natural dissipative PDEs which have global attractors with finite and small box-counting dimensions, but do not possess any finite dimensional IMs will be considered and the infinite dimensionality of the limit dynamics in these equations will be demonstrated. Finally, the recent results on smoothness of IMs will be also presented.