Inertial manifolds and normal hyperbolicity
type: Item
instance of: scholarly article
title: Inertial manifolds and normal hyperbolicity
zbMATH Open document ID: 0863.35014
DOI: 10.1007/BF00047882
author: R. Temam
author: Ricardo M. S. Rosa
published in: Acta Applicandae Mathematicae
publication date: 1997-06-03T00:00:00Z
review text: This is a rather long technical digression on inertial manifolds of evolution equations of the form: \[ u_t+Au= f(u). \] Here \(-A\) generates a strongly continuous semigroup \(V_t\), \(t\geq 0\) on an underlying Banach space \(E_0\). There are two further Banach spaces \(E\), \(F\) related to \(E_0\) by the continuous embeddings \(E\subset F\subset E_0\). Various hypothesis are imposed on \(f\), \(A\) so as to secure the existence of inertial manifolds. In a first setting, \(f\) is globally Lipschitz from \(E\) to \(F:|f(u)-f(v)|_F\leq M_1|u-v|_E\). Furthermore, one is given numbers \(0<\lambda_n\leq\Lambda_n\), \(n\geq 1\), finite-dimensional projection operators \(P_n\), \(n\geq 1\), a number \(\alpha\in(0,1)\) and constants \(K_1\), \(K_2\) such that the following holds (where \(Q_n=1-P_n\)): (1) \(P_nE_0\) is invariant under \(V_t\), \(t\geq 0\), and \(V_tP_n\), \(t\geq 0\) extends into a group \(V_tP_n\), \(t\in\mathbb{R}\) on \(P_nE_0\) such that \(|V_tP_n|_{L(E)}\leq K_1e^{-\lambda_nt}\), \(t\leq 0\) and \(|V_tP_n|_{L(F,E)}\leq K_2\lambda^\alpha_ne^{-\lambda_nt}\), \(t\leq 0\), (2) \(Q_nE\) is invariant under \(V_t\), \(t\geq 0\) with \(|V_tQ_n|_{L(E)}\leq K_2e^{-\Lambda_nt}\), \(t\geq 0\), \(|V_tQ_n|_{L(E,F)}\leq K_2(t^{-\alpha}+\Lambda^\alpha_n)e^{-\Lambda_nt}\), \(t\geq 0\). The assumption \(V_tF\subset E\), \(t>0\) is incorporated in (1), (2). Based on this framework, the existence of an inertial manifold can be proved under the assumption that a spectral gap condition is satisfied. A number of properties of the inertial manifolds are proved, among others asymptotic completeness. At a later stage, the assumption that \(f\) is globally Lipschitz is removed by a well-known cut off procedure. A series of further properties such as smoothness are discussed. For reasons of space we cannot do justice to this rather long paper. However, we have to point out that the paper can be highly recommended to everybody interested in inertial manifolds and the techniques…
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35G10
Mathematics Subject Classification ID: 35K25
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 943757
zbMATH Keywords: spectral gap condition
zbMATH Keywords: asymptotic completeness
reviewed by: Bruno Scarpellini
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00047882
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description: scientific article; zbMATH DE number 943757