Global existence for the inflow-outflow problem for the Navier–Stokes equations in a cylinder
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. To prove the long time existence we need smallness of derivatives, with respect to the variable along the axis of the cylinder, of the external force and of the initial velocity in $L_2$-norms. Moreover, we need smallness of derivatives of inflow and outflow with respect to tangent directions to the boundary and with respect to time in some norms. The global existence is proved step by step using the existence on the time interval $[0,T]$, with $T$ sufficiently large.