The main concern of this paper is to investigate the asymptotic stability of stationary solution to the compressible Navier–Stokes–Poisson equations with the classical Boltzmann relation in a half line. We first show the unique existence of stationary solution with the aid of the stable manifold theory, and then prove that the stationary solution is time asymptotically stable under the small initial perturbation by the elementary energy method. Finally, we discuss the convergence rate of the time-dependent solution towards the stationary solution, and give a new condition to ensure an algebraic decay or an exponential decay. The proof is based on a time and space weighted energy method by fully utilizing the self-consistent Poisson equation.

本文的主要关注点是研究具有经典Boltzmann关系的可压缩Navier-Stokes-Poisson方程在半直线上的平稳解的渐近稳定性。我们首先借助稳定流形理论证明了平稳解的唯一性，然后通过基本能量方法证明了该平稳解在较小的初始扰动下是时间渐近稳定的。最后，我们讨论了时间相关解向平稳解的收敛速度，并给出了确保代数衰减或指数衰减的新条件。该证明是基于时空加权能量方法的，它充分利用了自洽Poisson方程。