On the Stokes and Navier‐Stokes System for Domains with Noncompact Boundary in Lq‐spaces

Abstract

AbstractWe consider the Lq‐theory of weak solutions of the Stokes and Navier‐Stokes equations in two classes of unbounded domains with noncompact boundary, namely in perturbed half spaces which are obtained by a perturbation of the half space IRn, and in aperture domains consisting of two disjoint half spaces separated by a wall but connected by a hole (aperture) through this wall. The proofs rest on the cut‐off procedure and a new multiplier approach to the half space problem. In an aperture domain we additionally prescribe either the flux through the wall or the pressure drop at infinity to single out a unique solution. The nonlinear problem is solved for sufficiently small data and requires q =n/2, n ≥ 3, to estimate the nonlinearity.