  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.
