Optimal decay rates for higher-order derivatives of solutions to 3D compressible Navier-Stokes-Poisson equations with external force

We investigate optimal decay rates for higher-order spatial derivatives of solutions to the 3D compressibleNavier-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 \(L^2\)-rate \((1+t)^{-5/4}\), which is faster than the \(L^2\)-rate \((1+t)^{-3/4}\) 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.