Uniqueness and uniform structural stability of Poiseuille flows in an infinitely long pipe with Navier boundary conditions

TL;DR

This paper proves the uniqueness and stability of Poiseuille flows in an infinite pipe with Navier boundary conditions, showing these properties hold uniformly for various flow fluxes and slip coefficients.

Contribution

It establishes uniform estimates for Poiseuille flows' uniqueness and stability under Navier boundary conditions, even with large external forces and fluxes.

Findings

Uniqueness and stability are uniform with respect to flux and slip coefficient.

Existence of steady solutions is guaranteed for large fluxes and forces.

Special decomposition techniques are key to deriving the estimates.

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. 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.