Uniform Structural stability of Hagen-Poiseuille flows in a pipe

TL;DR

This paper proves the uniform nonlinear structural stability of Hagen-Poiseuille flows with large fluxes in pipes, providing foundational estimates crucial for solving steady Navier-Stokes problems in complex geometries.

Contribution

It establishes the first uniform nonlinear stability result for Hagen-Poiseuille flows with large fluxes, advancing understanding of steady Navier-Stokes solutions in pipes.

Findings

Linear stability is proved via Fourier analysis of the stream function.

Uniform estimates are achieved for flows with different fluxes and frequencies.

Boundary layer analysis is key for large flux and intermediate frequency cases.

Abstract

In this paper, we prove the uniform nonlinear structural stability of Hagen-Poiseuille flows with arbitrary large fluxes in the axisymmetric case. This uniform nonlinear structural stability is the first step to study Liouville type theorem for steady solution of Navier-Stokes system in a pipe, which may play an important role in proving the existence of solutions for the Leray's problem, the existence of solutions of steady Navier-Stokes system with arbitrary flux in a general nozzle. A key step to establish nonlinear structural stability is the a priori estimate for the associated linearized problem for Navier-Stokes system around Hagen-Poiseuille flows. The linear structural stability is established as a consequence of elaborate analysis for the governing equation for the partial Fourier transform of the stream function. The uniform estimates are obtained based on the analysis for the solutions with different fluxes and frequencies. One of the most involved cases is to analyze the solutions with large flux and intermediate frequency, where the boundary layer analysis for the solutions plays a crucial role.