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

Global existence of weak solutions to the Navier-Stokes equation in a cylindrical domain under the slip boundary conditions and with inflow and outflow was proved. To prove the energy estimate, crucial for the proof, we use the Hopf function. This makes us possible to derive such estimate that the inflow and outflow must not vanish as t converges to infinity. The proof requires estimates in weighted Sobolev spaces for solutions to the Poisson equation. Finally, the paper is the first step to prove the existence of global regular special solutions to the Navier-Stokes equations with inflow and outflow.