We consider the motion of a viscous incompressible fluid described by
the Navier-Stokes equations in a bounded cylinder with slip boundary
conditions. Assuming that $L_2$ norms of the derivative of the initial
velocity and the external force with respect to the variable along the
axis of the cylinder are sufficiently small we are able to prove long
time existence of regular solutions. By the regular solutions we mean
that velocity belongs to $W^{2,1}_2 (Dx(0,T))$ and gradient of pressure
to $L_2(Dx(0,T))$. To show global existence we prolong the local solution
with sufficiently large T step by step in time up to infinity. For this purpose
we need that $L_2(D)$ norms of the external force and derivative
of the external force in the direction along the axis of the cylinder
vanish with time exponentially.
Next we consider the inflow-outflow problem. We assume that the normal
component of velocity is nonvanishing on the parts of the boundary which
are perpendicular to the axis of the cylinder. We obtain the energy type
estimate by using the Hopf function. Next the existence of weak solutions is
proved.