The instationary Stokes and Navier−Stokes equations are considered in a simultaneously space-timevariational saddle point formulation, so involving both velocities u and pressure p. For the instationaryStokes problem, it is shown that the corresponding operator is a boundedlyinvertible linear mapping between H1 and H'2, both Hilbertspaces H1 and H2 beingCartesian products of (intersections of) Bochner spaces, or duals of those. Based on theseresults, the operator that corresponds to the Navier−Stokes equations is shown to mapH1 into H'2, with a Fréchetderivative that, at any (u,p) ∈H1, is boundedly invertible. These resultsare essential for the numerical solution of the combined pair of velocities and pressureas function of simultaneously space and time. Such a numerical approach allows for theapplication of (adaptive) approximation from tensor products of spatial and temporal trialspaces, with which the instationary problem can be solved at a computational complexitythat is of the order as for a corresponding stationary problem.
