In an infinite cylinder Ω = Σ × R, where Σ ⊂ R n−1 , n ≥ 3, is a bounded domain of C 1,1 class, we study the unique solvability of Stokes resolvent systems in L q (R; L 2 (Σ)) for 1 < q < ∞ and in vector-valuedBy a partial Fourier transform along the axis of the cylinder Ω the given system is reduced to a parametrized system on Σ, for which parameter independent estimates are proved. For further applications we obtain even parameter independent estimates in L r (Σ), 1 < r < ∞, in the non-solenoidal case prescribing an arbitrary divergence g = div u. Then operator-valued multiplier theorems are used for the final estimates of the Stokes resolvent systems in Ω.

SUMMARYWe prove existence, uniqueness and exponential stability of stationary Navier-Stokes flows with prescribed flux in an unbounded cylinder of R n , n 3, with several exits to infinity provided the total flux and external force are sufficiently small. The proofs are based on analytic semigroup theory, perturbation theory and L r − L q -estimates of a perturbation of the Stokes operator in L q -spaces.