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In this paper, we investigate the optimal decay rate for the higher orderspatial derivative of global solution to the compressible Navier-Stokes (CNS)equations with or without potential force in three-dimensional whole space.First of all, it has been shown in \cite{guo2012} that the $N$-th order spatialderivative of global small solution of the CNS equations without potentialforce tends to zero with the $L^2-$rate $(1+t)^{-(s+N-1)}$ when the initialperturbation around the constant equilibrium state belongs to$H^N(\mathbb{R}^3)\cap \dot H^{-s}(\mathbb{R}^3)(N \ge 3 \text{~and~} s\in [0,\frac32))$. Thus, our first result improves this decay rate to$(1+t)^{-(s+N)}$. Secondly, we establish the optimal decay rate for the globalsmall solution of the CNS equations with potential force as time tends toinfinity. These decay rates for the solution itself and its spatial derivativesare really optimal since the upper bounds of decay rates coincide with thelower ones.

Optimal decay of compressible Navier-Stokes equations with or without potential force