Center manifolds in equations from hydrodynamics
type: Item
instance of: scholarly article
title: Center manifolds in equations from hydrodynamics
zbMATH Open document ID: 0822.35042
DOI: 10.1007/BF01194039
author: Robert E. L. Turner
author: Charles J. Amick
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 1995-10-09T00:00:00Z
review text: In the paper under discussion the authors describe a new setting in order to handle center manifold constructions, as they arise in connection with elliptic problems on cylindrical domains. We have to content us with a few indications. Starting point is a cylindrical domain \(\Omega= \mathbb{R}\times I\), where \(I\subseteq \mathbb{R}^ n\) is a smooth bounded domain. Such domains serve as carriers for elliptic problems, in which the unbounded variable \(x\in \mathbb{R}\) is treated like the dynamical time variable \(t\). Center manifold constructions within this context originate in the work of K. Kirchgässner. The authors introduce Hölder spaces \(C_ \mu^{j\alpha} (\Omega)\) \((j\in \mathbb{N}\), \(\alpha\in (0,1))\) which differ from the usual ones only in that the distinguished variable \(x\) is affected by a weighting factor \(\exp (-\mu | x|)\). The definition of \(C_ \mu^{j\alpha} (\Omega)\) extends in an obvious way to vector valued functions \(u(x,y)\in \mathbb{R}^ n\), \((x\in \mathbb{R}\), \(y\in \mathbb{R}^ n)\). In a next step, certain subsets \(C^{j\alpha}_{\mu d} \subseteq C^{j\alpha}_{\mu d}\) are defined, where \(d= (d_ 0, \dots, d_ j, d_{j+\alpha})\in \mathbb{R}^{j+2}\) has positive components. By definition \(u\in C^{j\alpha}_{\mu d}\) iff \(u\in C_ \mu^{j\alpha}\) and \(| \partial^ \beta u|\leq d_ k\) for \(| \beta| =k\) and \(k\leq j\), and a similar condition for the Hölder norms of \(\partial^ \beta u\), \(| \beta|=j\), involving \(d_{j+\alpha}\). With the Hölder spaces and the subsets \(C^{j\alpha}_{\mu d}\) at disposal, mappings \(g\) from \(\mathbb{R}^ n\times \mathbb{R}^ m\) into \(\mathbb{R}^ n\) are introduced, which satisfy certain smoothness conditions and where \(\mathbb{R}^ m\) is a space of parameters. It is then shown that such functions \(g\) induce via composition well behaved mappings from \(C^{j\alpha}_{\mu d}\) into \(C^{j\alpha}_{\mu e}\) where \(e\) depends in a rather complicated way on \(j\), \(\alpha\), \(\mu\), \(d\). The…
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 58D25
zbMATH DE Number: 736829
zbMATH Keywords: center manifold constructions
zbMATH Keywords: elliptic problems on cylindrical domains
zbMATH Keywords: weighted Hölder spaces
reviewed by: Bruno Scarpellini
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 736829