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Decay of the Navier-Stokes-Poisson equations. (English) Zbl 1239.35117
Summary: We establish the time decay rates of the solution to the Cauchy problem for the compressible Navier-Stokes-Poisson system via a refined pure energy method. In particular, the optimal decay rates of the higher-order spatial derivatives of the solution are obtained. The $$\dot H^{-s} (0\leqslant s <3/2)$$ negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates. As a corollary, we also obtain the usual $$L^{p}$$-$$L^{2}$$ $$(1<p \leq 2)$$ type of the optimal decay rates. Compared to the compressible Navier-Stokes system and the compressible irrotational Euler-Poisson system, our results imply that both the dispersion effect of the electric field and the viscous dissipation contribute to enhance the decay rate of the density. Our proof is based on a family of scaled energy estimates with minimum derivative counts and interpolations among them without linear decay analysis.

##### MSC:
 35Q30 Navier-Stokes equations 76N15 Gas dynamics, general 35Q31 Euler equations 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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