The squeezing property and the decay of small wavelengths in dissipative dynamical systems

The concepts of inertial manifolds and approximate inertial manifolds for dissipative dynamical systems are based upon a decomposition of the underlying phase space into a finitedimensional, inertial subspace, and an infinite-dimensional,highly dissipated, orthogonal complement. We present here a more detailed squeezing property which shows that, under the same conditions as the usual queering property, there exists a small parameter r > 0 such that the percentage p of the total energy contained in the highly dissipated modes is either dose to unity, and the entire system is rapidly dissipatingto rero, or p passes a threshold value 1 e end subsequently decays exponentially with a large rate to a value inferior to E, and remains there for dl forward time.