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The compressible Navier-Stokes-Poisson system takes the form of usualNavier-Stokes equations coupled with the self-consistent Poisson equation,which is used to simulate the transport of charged particles under the electricfield of electrostatic potential force. In this paper, we focus on the largetime behavior of global strong solutions in the $L^{p}$ Besov spaces ofcritical regularity. By exploring the dissipative effect arising from Poissonpotential, we posed the new regularity assumption of low frequencies and thenestablish a \textit{sharp} time-weighted inequality, which leads to the optimaltime-decay estimates of the solution. Indeed, we see that the decay of densityis faster at the half rate than that of velocity, which is a differentingredient in comparison with the situation of usual Navier-Stokes equations.Our proof mainly depends on tricky and non classical Besov product estimateswith respect to various Sobolev embeddings.

A sharp time-weighted inequality for the compressible Navier-Stokes-Poisson system in the critical $L^{p}$ framework