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

Optimal time-decay rates of the compressible Navier–Stokes–Poisson system in <inline-formula><tex-math id="M1">$ \mathbb R^3 $</tex-math></inline-formula>

“…Wang [25] obtained the optimal asymptotic decay of solutions just by pure energy estimates, and particularly he proved that the density of the compressible NSP system decays at the L 2 -rate (1 + t) − 5 4 , which is faster than the L 2 -rate (1+t) − 3 4 for the NS system due to the effect of the electric field. Hao-Li [9], Tan-Wu [22], Chikami-Danchin [4], Bie-Wang-Yao [2] and Shi-Xu [21] also established the unique global solvability and the optimal decay rates in critical spaces. We mention that there are many results on the existence and long time behavior of the weak solutions or non-constant stationary solutions, see, for example [1,5,8,28] and the references therein.…”

Large time behavior of solutions to wave equations arising from the linearized compressible Navier-Stokes-Poisson system

“…Bie, Wang and Yao [2] established the optimal decay rate for the CNSP system in critical Besov spaces. Their results were improved by Shi and Xu in [14] later. If the doping profile are functions respect to space variable, Tan, Wang and Wang [12] proved the time decay rates for the solution provided that the initial perturbation in L p with 1 ≤ p < 3 2 .…”

“…As the CNSP system has strong physical background, which attract mathematicians and physicians to study it and many results have been established, here we mainly pay attention to the results related to the asymptotic behavior. On one hand, some researchers focused on establishing the optimal time decay estimates of strong solutions to the CNSP system, see [2,4,10,11,12,13,14,16,19]. On the other hand, some researchers investigated the quasineutral limit and inviscid limit of CNSP system, see [3,7,8,9,15].…”

Optimal time-decay estimates for the compressible Navier-Stokes-Poisson equations without additional smallness assumptions

“…Recently, Chikami and Danchin [6] proposed the description of the timedecay which allows one to handle dimension d ≥ 2 in the L 2 critical Besov space. In [26], the author & Xu developed a new regularity assumption of low frequencies, where the regularity s 1 belongs to (1 − d 2 , s 0 ] with s 0 2d p − d 2 , and established the sharp time-weighted inequality, which led to the optimal time-decay rates of strong solutions. These recent works (see for example [2,6,20,21,26,28,29] and references therein) mainly relies on the refined time-weighted energy approach in the Fourier semi-group framework, so the smallness assumption of low frequencies of initial data plays a key role.…”

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The present paper is dedicated to the large time asymptotic behavior of global strong solutions near constant equilibrium (away from vacuum) to the compressible Navier-Stokes-Poisson equations. Precisely, we present that under the same regularity assumptions as in [26], a different time-decay framework of the Ḃs p,1 norm of the critical global solutions is established. The proof mainly depends on the pure energy argument without the spectral analysis, which allows us to remove the usual smallness assumption of low frequencies of initial data.

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