The stationary and non‐stationary stokes system in exterior domains with non‐zero divergence and non‐zero boundary values

In an exterior domain Ω⊂ℝ n , n ⩾ 2, we consider the generalized Stokes resolvent problem in L q ‐space where the divergence g = div u and inhomogeneous boundary values u = ψ with zero flux ∫ ∂ Ωψ· N do = 0 may be prescribed. A crucial step in our approach is to find and to analyse the right space for the divergence g . We prove existence, uniqueness and a priori estimates of the solution and get new results for the divergence problem. Further, we consider the non‐stationary Stokes system with non‐homogeneous divergence and boundary values and prove estimates of the solution in L 5 (0, T ; L q (Ω)) for 1 < s, q < ∞.