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Global regular solutions to the Navier-Stokes equations with large flux
We consider the Navier-Stokes motion in a bounded cylinder with
boundary slip conditions. We assume an inflow and an outflow of the fluid through the bottom and the top of the cylinder where the magnitude of the flux is not restricted. We require that the derivatives of the initial velocity and the external
force with respect to the variable along the axis of the cylinder are sufficiently small. Under these conditions we are able to prove global existence of regular solutions. Since we are interested in nonvanishing in time flux we need to use the Hopf function to derive global energy estimate.