Electronic Journal of Differential 
Equations, Vol. 2020 (2020), No. 102, pp. 1-25.

Title: Optimal time decay rates for the compressible Navier-Stokes
       system with and without Yukawa-type potential
       
Authors: Qing Chen (Xiamen Univ. of Tech., Xiamen, Fujian, China)
         Guochun Wu (Huaqiao Univ., Quanzhou, China)
         Yinghui Zhang (Guangxi Normal Univ., Guilin, Guangxi, China)
         Lan Zou (Huaqiao Univ., Quanzhou, China)

Abstract:
 We consider the time decay rates of smooth solutions to the Cauchy
 problem for the compressible Navier-Stokes system with and without
 a Yukawa-type potential.
 We prove the existence and uniqueness of global solutions by the standard energy
 method under small initial data assumptions.
 Furthermore, if the initial data belong to $L^1(\mathbb R^3)$, we establish the optimal
 time decay rates of the solution as well as its higher-order spatial derivatives.
 In particular, we obtain the optimal decay rates of the highest-order spatial
 derivatives of the velocity. Finally, we derive the lower bound time decay rates for
 the solution and its spacial derivatives.
 
Submitted February 2, 2020. Published September 29, 2020.
Math Subject Classifications: 35Q30, 76N15
Key Words: Compressible flow; energy method; optimal decay rates.