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

We consider the L q ‐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 IR n , 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.