The Stokes System in Domains with Outlets of Bounded and Connected Cross-Sections

The Stokes system with prescribed fluxes is investigated. By smoothness assumptions on the boundary and by the boundedness of the diameters of the outlets it is ensured that the divergence equation in each bounded subdomain is solvable, the Poincaré inequality is valid and the constants in all the corresponding estimates are bounded independently of the location. We derive existence, uniqueness and regularity results in two different frameworks: On one hand we use weighted function spaces generated by L9 -norms, 1 < q < 00, where the weight is of exponential type and apply a technique of Maz'ya and Plamenevskii. On the other hand we use local spaces, since in order to solve the problem with non-zero flux it seems to us that to formulate results in local spaces is more adequate and physical senseful.