No abstract

dedicated to professor jack k. hale on the occasion of his 70th birthday There are two objectives in this paper. First we develop a theory which is valid in the infinite dimensional setting and which shows that, under reasonable conditions, if M is a normally hyperbolic, compact, invariant manifold for a semiflow S 0 (t) generated by a given evolutionary equation on a Banach space W, then for every small perturbation G of the given evolutionary equation, there is a homeo-is a normally hyperbolic, compact, invariant manifold for the perturbed semiflow S G (t), and that h G converges to the identity mapping (on M), as G converges to 0. The second objective is to develop a methodology which is rich enough to show that this theory can be easily applied to a wide range of applications, including the Navier Stokes equations. It is noteworthy in this regard that, in order to be able to apply this theory in the analysis of numerical schemes used to study discretizations of partial differential equations, one needs to use a new measure or norm of the perturbation term G that arises in these schemes. Academic PressKey Words: approximation dynamics; Bubnov Galerkin approximations; Couette Taylor flow; evolutionary equations; exponential dichotomy; exponential trichotomy; ordinary differential equations; Navier Stokes equations; normal hyperbolicity; numerical schemes; partial differential equations; reaction diffusion equations; robustness.

We introduce the concept of a weakly, normally hyperbolic set for a system of ordinary differential equations. This concept includes the notion of a hyperbolic flow, as well as that of a normally hyperbolic invariant manifold. Moreover, it has the property that it is closed under finite set products. Consequently, the theory presented here can be used for the study of perturbations of the dynamics of coupled systems of weakly, normally hyperbolic sets. Our main objective is to show that under a small C 1 -perturbation, a weakly, normally hyperbolic set K is preserved by a homeomorphism, where the image K Y is a compact invariant set, with a related hyperbolic structure, for the perturbed equation. In addition, the homeomorphism is close to the identity in C 0, 1 and the perturbed dynamics on K Y are close to the original dynamics on K.1998 Academic Press

No abstract

No abstract