  We consider Leray's problem on stationary Navier-Stokes flows with arbitrary
large fluxes in an unbounded cylinder with several exits to infinity. For a
stationary Navier-Stokes flow with large fluxes in the unbounded cylinder we
prove that, if the difference between the pressure of the main flow and the
pressure of the Poiseuille flow with the same flux in a branch of the cylinder
satisfies some asymptotic boundedness condition at $|x|\rightarrow \infty$ then
the flow behaves at infinity of the branch like the Poiseuille flow.
