Uniqueness and uniform structural stability of Poiseuille flows in a periodic pipe with Navier boundary conditions

TL;DR

This paper establishes the uniqueness and uniform structural stability of Poiseuille flows in a periodic pipe with Navier boundary conditions, analyzing their behavior under varying flux and slip coefficients.

Contribution

It proves the uniform linear structural stability of axisymmetric Poiseuille flows with Navier boundary conditions, including boundary layer analysis across different regimes.

Findings

Stability is uniform with respect to flux and slip coefficient.

Non-zero frequency velocity component bounded by a power function of flux.

Proved the uniqueness of Poiseuille flows under specified conditions.

Abstract

In this paper, we prove the uniqueness and structural stability of Poiseuille flows for axisymmetric solutions of steady Navier-Stokes system supplemented with Navier boundary conditions in a periodic pipe. Moreover, the stability is uniform with respect to both the flux and the slip coefficient of Navier boundary conditions. It is also showed that the non-zero frequency part of the velocity is bounded by a power function of the flux with negative power as long as the flux is suitably large. One of the key ingredients of the analysis is to prove the uniform linear structural stability, where the analysis for the boundary layers and the swirl velocity corresponding to the flux and the slip coefficients in different regimes plays a crucial role.