Optimal Decay Rate of the Compressible Navier–Stokes–Poisson System in ${\mathbb {R}^3}$

Abstract

The compressible Navier–Stokes–Poisson (NSP) system is considered in ${\mathbb {R}^3}$ in the present paper, and the influences of the electric field of the internal electrostatic potential force governed by the self-consistent Poisson equation on the qualitative behaviors of solutions is analyzed. It is observed that the rotating effect of electric field affects the dispersion of fluids and reduces the time decay rate of solutions. Indeed, we show that the density of the NSP system converges to its equilibrium state at the same L ²-rate ${(1+t)^{-\frac {3}{4}}}$ or L ^∞-rate (1 + t)^−3/2 respectively as the compressible Navier–Stokes system, but the momentum of the NSP system decays at the L ²-rate ${(1+t)^{-\frac {1}{4}}}$ or L ^∞-rate (1 + t)⁻¹ respectively, which is slower than the L ²-rate ${(1+t)^{-\frac {3}{4}}}$ or L ^∞-rate (1 + t)^−3/2 for compressible Navier–Stokes system [Duan et al., in Math Models Methods Appl Sci 17:737–758, 2007; Liu and Wang, in Comm Math Phys 196:145–173, 1998; Matsumura and Nishida, in J Math Kyoto Univ 20:67–104, 1980] and the L ^∞-rate (1 + t)^−p with ${p \in (1, 3/2)}$ for irrotational Euler–Poisson system [Guo, in Comm Math Phys 195:249–265, 1998]. These convergence rates are shown to be optimal for the compressible NSP system.