A sharp time-weighted inequality for the compressible Navier–Stokes–Poisson system in the critical Lp framework

Abstract

The compressible Navier–Stokes–Poisson system takes the form of usual Navier–Stokes equations coupled with the self-consistent Poisson equation, which is used to simulate the transport of charged particles under the electrostatic potential force. In this paper, we focus on the large-time behavior of global strong solutions in the Lp critical Besov spaces. Inspired by the dissipative effect arising from Poisson potential, we formulate a new regularity assumption of low frequencies and then establish the sharp time-weighted inequality, which leads to the optimal time-decay estimates of strong solutions. Indeed, we see that the decay of density is faster at the half rate than that of velocity, which is a different ingredient in comparison with the situation of compressible Navier–Stokes equations. Our proof mainly depends on tricky and non classical Besov product estimates with respect to various Sobolev embeddings.