TY - JOUR

T1 - Perturbations of normally hyperbolic manifolds with applications to the navier-stokes equations

AU - Pliss, Victor A.

AU - Sell, George R.

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2001/1/20

Y1 - 2001/1/20

N2 - 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 S0(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 hG:M→W such that MG=hG(M) is a normally hyperbolic, compact, invariant manifold for the perturbed semiflow SG(t), and that hG 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.

AB - 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 S0(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 hG:M→W such that MG=hG(M) is a normally hyperbolic, compact, invariant manifold for the perturbed semiflow SG(t), and that hG 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.

KW - Approximation dynamics

KW - Bubnov-Galerkin approximations

KW - Couette-Taylor flow

KW - Evolutionary equations

KW - Exponential dichotomy

KW - Exponential trichotomy

KW - Navier-Stokes equations

KW - Ordinary differential equations

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U2 - 10.1006/jdeq.2000.3905

DO - 10.1006/jdeq.2000.3905

M3 - Article

AN - SCOPUS:0035915837

VL - 169

SP - 396

EP - 492

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 2

ER -