We study the initial value problems for the Stokes and Navier-Stokes equations in an aperture domain , which consists of two disjoint half spaces separated by a wall but connected by an aperture in the wall. This class of unbounded domains with noncompact boundaries is interesting because of the following remarkable feature (Heywood, 1976): either a prescribed flux of the velocity field through the aperture or a prescribed pressure drop at infinity may be required as an additional boundary condition in order to get a unique solution. We consider the Stokes equation with zero flux through the aperture, which generates a bounded analytic semigroup in (Farwig and Sohr, 1996), and derive some decay estimates (- estimates) of the semigroup:
E-Mail: | hishida@mathematik.tu-darmstadt.de |