Perturbations of normally hyperbolic manifolds with applications to the Navier-Stokes equations

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:169

Article

Pliss, VA ; Sell, GR

关键词: 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;   

DOI  :  10.1006/jdeq.2000.3905

来源: Elsevier

【 摘 要 】

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-o(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 homeomorphism h(G): M --> W such that M-G = h(G)(M) 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 secund 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. (C) 2001 Academic Press.

【 授权许可】

Free