Global regular nonstationary flow for the Navier–Stokes equations in a cylindrical pipe
Global existence of regular solutions to the Navier–Stokes equations describing the motion of an incompressible viscous fluid in a cylindrical pipe with large inflow and outflow is shown. Global existence is proved in two steps. First, by the Leray–Schauder fixed point theorem we prove local existence with large existence time. Next, the local solution is prolonged step by step. The existence is proved without any restrictions on the magnitudes of the inflow, outflow, external force and initial velocity.