Optimal decay rate of the compressible Navier-Stokes-Poisson system in R^3

TL;DR

This paper investigates the decay rates of solutions to the compressible Navier-Stokes-Poisson system in three-dimensional space, revealing how electric fields influence fluid dispersion and decay behavior, with results showing optimal decay rates for different solution components.

Contribution

It establishes the optimal decay rates of solutions to the compressible NSP system in R^3, highlighting the impact of electric fields on the decay behavior compared to the classical Navier-Stokes system.

Findings

Density decays at the same rate as Navier-Stokes system.

Momentum decays slower due to electric field effects.

Decay rates are proven to be optimal.

Abstract

The compressible Navier-Stokes-Poisson (NSP) system is considered in $R^3$ 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 $(1+t)^{-\frac34}$ or $L^\infty$-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 $(1+t)^{-\frac14}$ or $L^\infty$-rate $(1+t)^{-1}$ respectively, which is slower than the $L^2$-rate $(1+t)^{-\frac34}$ or $L^\infty$-rate $(1+t)^{-3/2}$ for the compressible Navier-Stokes system. These convergence rates are also shown to be optimal for the compressible NSP system.