Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 64, pp. 1-18.

Title: Optimal decay rates for higher-order derivatives of solutions to 
  3D compressible Navier-Stokes-Poisson equations with external force
       
Authors: Liuna Qin     (Guangxi Normal Univ., Guilin, Guangxi, China)
         Changguo Xiao (Guangxi Normal Univ., Guilin, Guangxi, China)
         Yinghui Zhang (Guangxi Normal Univ., Guilin, Guangxi, China)


Abstract: 
We investigate optimal decay rates for higher-order spatial
derivatives of solutions to the 3D compressible
 Navier-Stokes-Poisson equations with external force.
The main novelty of this article is twofold:
First, we prove the first and second order spatial derivatives of
the solutions converge to zero at the L<sup>2</sup>-rate
(1+t)<sup>-5/4</sup>, which is faster than the L<sup>2</sup>-rate
(1+t)<sup>^-3/4</sup> in Li-Zhang [15].
Second, for well-chosen initial data, we  show the lower optimal decay rates
of the first order spatial derivative of the solutions.
Therefore, our decay rates are optimal in this sense.

Submitted August 1, 2021. Published September 07, 2022.
Math Subject Classifications: 76B03, 35Q35.
Key Words: Compressible Navier-Stokes-Poisson system;  external force; higher-order derivative.