An $L^q(L^2)$-theory of the generalized Stokes resolvent system in infinite cylinders
Volume 178 / 2007
Studia Mathematica 178 (2007), 197-216
MSC: 35Q30, 76D07, 42A45, 46E40.
DOI: 10.4064/sm178-3-1
Abstract
Estimates of the generalized Stokes resolvent system, i.e. with prescribed divergence, in an infinite cylinder ${\mit\Omega}={\mit\Sigma}\times\mathbb R$ with ${\mit\Sigma}\subset \mathbb R^{n-1}$, a bounded domain of class $C^{1,1}$, are obtained in the space $L^q(\mathbb R;L^2({\mit\Sigma}))$, $q\in (1,\infty)$. As a preparation, spectral decompositions of vector-valued homogeneous Sobolev spaces are studied. The main theorem is proved using the techniques of Schauder decompositions, operator-valued multiplier functions and $R$-boundedness of operator families.
