H^∞-calculus for Stokes operators on rough and on unbounded domains

In this article, we give an overview on known as well as new results on the boundedness of the H∞-calculus of the Stokes operator in rough as well as in unbounded (smoother) domains. We present a special case of an abstract comparison principle due to Kunstmann and Weis ({KuW:Hinfty-Stokes}) that serves as the basis for all considerations. Subsequently, we show how this result can be applied to arrive at a bounded H∞-calculus for the Stokes operator. We sketch the proof for no slip boundary conditions in bounded Lipschitz domains which was given in~{KuW:Hinfty-Stokes}. For unbounded domains this approach yields a shorter proof compared to previous arguments. Moreover, we further establish the boundedness of the H∞-calculus for the Stokes operator with Neumann type boundary conditions in bounded convex domains which is entirely new.