\(H^{\infty}\)-calculus for the Stokes operator on unbounded domains
type: Item
instance of: scholarly article
title: \(H^{\infty}\)-calculus for the Stokes operator on unbounded domains
zbMATH Open document ID: 1160.47058
author: Peer Christian Kunstmann
published in: Archiv der Mathematik
publication date: 2008-09-12T00:00:00Z
review text: The Stokes operator, denoted \(A\), on an unbounded domain \(\Omega\) of uniform \(C^{1,1}\)-type is investigated. It is well-known that \(L^q\)-boundedness of the Helmholtz projection does not hold for general unbounded domains and for \(q\) different from \(2\), so there is no \(L^q\)-theory is this case. This lack has been overcome by \textit{R.\,Farwig, H.\,Kozono} and \textit{H.\,Sohr}, who proved in [``On the Stokes operator in general unbounded domains'' (Preprint No.\ 2498, TU Darmstadt (2007)], among other things, that \(-A\) generates an analytic semigroup in \(X^q(\Omega), \;1<q<\infty\), where \(X^q(\Omega)=L^q(\Omega)\cap L^2(\Omega)\) for \(q\geq 2\) and \(X^q(\Omega)=L^q(\Omega)+L^2(\Omega)\) for \(q\in (1,\infty )\). In the present paper, the author shows that \(\varepsilon + A\) has a bounded \(H^{\infty}\)-calculus in \(X^q(\Omega)\) for all \(q\in (1,\infty )\) and \(\varepsilon >0\). This allows to identify domains of fractional powers of \(A\).
Mathematics Subject Classification ID: 47N20
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 47A60
zbMATH DE Number: 5343877
zbMATH Keywords: Stokes operator
zbMATH Keywords: unbounded domain
zbMATH Keywords: Helmholtz decomposition
zbMATH Keywords: fractional powers
reviewed by: Gheorghe Moroşanu
MaRDI profile type: Publication
full work available at URL: s00013-008-2621-0
OpenAlex ID: W2141982487
DOI: 10.1007/S00013-008-2621-0
description: scientific article; zbMATH DE number 5343877