$L_q-L_r$-Estimates for Non-Stationary Stokes Equations in an Aperture Domain

Abstract

This article deals with asymptotic estimates of strong solutions of Stokes equations in aperture domains. An aperture domain is a domain, which outside a bounded set is identical to two half spaces separated by a wall and connected inside the bounded set by one or more holes in the wall. It is known that the corresponding Stokes operator generates a bounded analytic semigroup in the closed subspace $J_q(\Omega )$ of divergence free vector fields of $L_q(\Omega {)}^n$. We deal with $L_q-L_r$-estimates for the semigroup, which are known for ${\mathbb{R}}^n$, the half space and exterior domains.