Existence of global weak solutions for Navier-Stokes equations with large flux

Global existence of weak solutions to the Navier-Stokes equations in a cylindrical domain under boundary slip conditions and with inflow and outflow is proved. To prove the energy estimate, crucial for the proof, we use the Hopf function. This makes it possible to derive an estimate such that the inflow and outflow need not vanish as t→∞. The proof requires estimates in weighted Sobolev spaces for solutions to the Poisson equation. Our result is the first step towards proving the existence of global regular special solutions to the Navier-Stokes equations with inflow and outflow.