Electronic Journal of Differential Equations,
Vol. 2019 (2019), No. 61, pp. 1-18.

Title: Decay rate of strong solutions to compressible Navier-Stokes-Poisson equations
       with external force

Authors: Yeping Li     (East China Univ. of Science and Tech., Shanghai, China)
         Nengqiu Zhang (East China Univ. of Science and Tech., Shanghai, China)

Abstract:
 In this article, we consider the three dimensional compressible
 Navier-Stokes-Poisson equations with the effect of external
 potential force. First, the stationary solution is established by
 solving a nonlinear elliptic system. Next, we show global well-posedness
 of the strong solutions for the initial value problem to the three
 dimensional compressible Navier-Stokes-Poisson equations when the
 initial data are close to the stationary solution in $H^2(\mathbb{R}^3)$.
 Moreover, if the $L^1(\mathbb{R}^3)$-norm of initial perturbation is finite,
 we prove the optimal $L^p(\mathbb{R}^3)$ $(2\leq p\leq6)$ decay rates for such
 strong solution and $L^2(\mathbb{R}^3)$ decay rate of its first-order
 spatial derivatives via a low frequency and high frequency decomposition.

Submitted October 7, 2018. Published May 07, 2019.
Math Subject Classifications: 35M20, 35Q35, 76W05.
Key Words: Navier-Stokes-Poisson equation; stationary solution;
           strong solution; energy estimate; optimal decay rate.