Center manifolds in equations from hydrodynamics
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 properties of these induced mappings are described by a series of lemmas and corollaries. With these concepts and results at hand it is then possible to discuss the so called reduced equations, which typically arise in connection with center manifold theory. The novelty in the setting under discussion is the systematic use of certain weighted Hölder spaces, in contrast to the more frequently used Sobolev spaces. Further items which are discussed are the regularity properties of the solutions of the reduced equations, and applications to concrete problems such as water waves.
elliptic problems on cylindrical domains
scientific article; zbMATH DE number 736829
scientific article