\(H^{\infty}\)-calculus for the Stokes operator on unbounded domains
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 \(A\).
scientific article; zbMATH DE number 5343877