  In this paper, uniqueness and uniform structural stability of Poiseuille
flows in an infinitely long pipe with Navier boundary conditions are
established for axisymmetric solutions of steady Navier-Stokes system. The
crucial point is that the estimate is uniform with respect both the flux of
flows and slip coefficient which appeared in Navier boundary conditions. With
the aid of special structure of Navier-Stokes system and the refined estimate
for some quantities such as radial velocity, the uniqueness and existence of
steady solutions of Navier-Stokes system can be obtained even when the external
forces are large as long as the fluxes of flows are large. The delicate
decomposition in the two dimensional plane for slip coefficient and frequency
corresponding to Fourier variable in the axial direction plays a key role to
achieve these estimates.
