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 a fluid in a cylindrical pipe with large inflow and outflow in shown. The global existence is proved under the following conditions: \roster \item"(1)" small variations of velocity and pressure with respect to the variable along the pipe, \item"(2)" inflow and outflow are very close to homogeneous and decay exponentially with time, \item"(3)" the external force decays exponentially with time. \endroster Global existence is proved in two steps. First by the Leray-Schauder fixed point theorem we prove local existence with large existence time which is inversely proportional to the above smallness restrictions. Next the local solution is prolonged step by step. The existence is proved for a solution without any restrictions on the magnitudes of inflow, outflow, external force and the initial velocity