Global in time L p ‐estimates for the instationary Stokes and Navier–Stokes flow through an aperture

Using a characterisation of maximal L^p‐regularity by ℛ‐bounded operator families we prove global in time estimates in $L^{p}({\mathbb{R}}_{+};L^{q}(\varOmega))$ , 1<p,q<∞, for solutions of the instationary Stokes system in an aperture domain $\varOmega\subset{\mathbb{R}} ^{n}$ , n≥3, with $\curpartial\varOmega\in C^{1,1}.$ The results are applied to obtain new global in time estimates for weak solutions of the Navier–Stokes equations with nonvanishing flux through the aperture.