Abstract
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. This is a key step to study general Leray problem for flows in a general infinitely long nozzle. The crucial point is that the estimate is uniform with respect to both the flux of flows and friction 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 are large. The delicate partition of the two dimensional plane for parameters, friction coefficient and Fourier variable associated with the axial coordinate, plays a key role to achieve these estimates.
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