Uniform Structural Stability of Hagen–Poiseuille Flows in a Pipe

In this paper, we prove the uniform nonlinear structural stability of Hagen–Poiseuille flows with arbitrary fluxes in the axisymmetric case in an infinitely long pipe. This uniform nonlinear structural stability is the first step to study Liouville type property for steady solutions of Navier–Stokes system in a pipe, which may play an important role in proving the existence of solutions with arbitrary flux to steady Navier–Stokes system in a nozzle with Poiseuille flows as far field asymptotic states (Leray’s problem). A key step 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 on 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 plays a crucial role.