Generalized Stokes Problems

Abstract

This chapter is devoted to maximal L _p -regularity of one-phase linear generalized Stokes problems on domains ${\Omega} \subset {\mathbb{R}}^{n}$ which are either ${\mathbb{R}}^{n}$, ${\mathbb{R}}_{ + }^{n}$, or domains with compact boundary $\partial {\Omega}$ of class C ³, i.e., interior or exterior domains. Here we only consider the physically natural boundary conditions no-slip, pure slip, outflow, and free. As in Chap. 6, our approach is based on vector-valued Fourier multiplier theory, perturbation, and localization. It turns out that due to the divergence condition (and the pressure), the analysis for the half-space as well as the localization procedure are much more involved than in the previous chapter. Nevertheless, besides some extra compatibility condition which comes from the divergence condition, the main results will parallel those in Chap. 6.