Reduced and Generalized Stokes Resolvent Equations in Asymptotically Flat Layers, Part II: H_∞-Calculus

Abstract.

We study the generalized Stokes equations in asymptotically flat layers, which can be considered as compact perturbations of an infinite (flat) layer $\Omega _0 = \mathbb{R}^{n - 1} \times ( - 1,1).$ Besides standard non-slip boundary conditions, we consider a mixture of slip and non-slip boundary conditions on the upper and lower boundary, respectively. In this second part, we use pseudodifferential operator techniques to construct a parametrix to the reduced Stokes equations, which solves the system in L^q-Sobolev spaces, 1 < q < ∞, modulo terms which get arbitrary small for large resolvent parameters λ. This parametrix can be analyzed to prove the existence of a bounded H_∞-calculus of the (reduced) Stokes operator.