Stokes resolvent systems in an infinite cylinder

Abstract

Abstract 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‐valued homogeneous Besov spaces (ℝ; L r (Σ)) for 1 ≤ p , q ≤ ∞, s ∈ ℝ, 1 < r < ∞. By 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. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)