Hagen-Poietille管流的一致结构稳定性
Uniform Structural stability of Hagen-Poiseuille flows in a pipe
摘要：
本文证明了轴对称情形下任意大通量Hagen-Poietille流的一致非线性结构稳定性。这种一致的非线性结构稳定性是研究管道中Navier-Stokes系统定常解的Liouville型定理的第一步，它对证明Leray问题解的存在性，以及一般喷管中任意流量定常Navier-Stokes系统解的存在性都有重要作用。建立非线性结构稳定性的一个关键步骤是对Hagen-Poietille流的Navier-Stokes方程组的线性化问题进行先验估计。线性结构稳定性是通过对流函数的部分傅里叶变换的控制方程进行详细分析而建立的。通过对不同通量和频率下解的分析，得到了一致估计。其中，大流量、中频情况下的边界层分析是一个重要的问题。
英文摘要：
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.
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