Smoothness of inertial manifolds

Abstract

The theory of invariant manifolds plays an important role in the study of dynamics of nonlinear systems in finite dimensional or infinite dimensional spaces. Such invariant manifolds include stable manifolds, unstable manifolds, and center manifolds, see Pliss [36], for example. Recently the theory of inertial manifolds has been developed for some dissipative evolution equations. The inertial manifold introduced by Foias, Sell, and Ttmam [ 141 is a finite dimensional Lipschitz invariant manifold attracting solutions exponentially. One of the important properties of inertial manifolds is that they contain the global attractors. Thus, the study of dynamics of infinite dimensional nonlinear systems can be reduced to the study of dynamics of flows on the inertial manifold, which, in turn, is described by the dynamics of an ordinary differential equation. There are extensive works on this subject. See, for example, Constantin [3], Constantin, Foias, Nicolaenko,