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

Victor A. Pliss; George R. Sell

doi:10.1006/jdeq.2000.3905

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

Victor A. Pliss, George R. Sell

School of Mathematics

Research output: Contribution to journal › Article › peer-review

28 Scopus citations

Abstract

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 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 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.

Original language	English (US)
Pages (from-to)	396-492
Number of pages	97
Journal	Journal of Differential Equations
Volume	169
Issue number	2
DOIs	https://doi.org/10.1006/jdeq.2000.3905
State	Published - Jan 20 2001

Bibliographical note

Funding Information:
1 This research was supported in part by grants from the Russian Foundation for Fundamental Studies. Both authors express appreciation to the Faculty of Mathematics and Mechanics, in St. Petersburg; and to the Ordway Visiting Professorship program in the School of Mathematics, the Institute for Mathematics and its Applications, and the Minnesota Supercomputer Institute, in Minneapolis, for their help in sponsoring this project. We extend our sincere appreciation to both Yuncheng You and the referee for their very careful readings of our manuscript. We are very grateful for their many helpful suggestions and comments.

Keywords

Approximation dynamics
Bubnov-Galerkin approximations
Couette-Taylor flow
Evolutionary equations
Exponential dichotomy
Exponential trichotomy
Navier-Stokes equations
Ordinary differential equations

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abstract = "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.",

keywords = "Approximation dynamics, Bubnov-Galerkin approximations, Couette-Taylor flow, Evolutionary equations, Exponential dichotomy, Exponential trichotomy, Navier-Stokes equations, Ordinary differential equations",

author = "Pliss, {Victor A.} and Sell, {George R.}",

note = "Funding Information: 1 This research was supported in part by grants from the Russian Foundation for Fundamental Studies. Both authors express appreciation to the Faculty of Mathematics and Mechanics, in St. Petersburg; and to the Ordway Visiting Professorship program in the School of Mathematics, the Institute for Mathematics and its Applications, and the Minnesota Supercomputer Institute, in Minneapolis, for their help in sponsoring this project. We extend our sincere appreciation to both Yuncheng You and the referee for their very careful readings of our manuscript. We are very grateful for their many helpful suggestions and comments.",

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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.

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

Abstract

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Keywords

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