Long time asymptotic equivalence of the Vlasov-Poisson-Boltzmann equations at the level of the compressible Navier-Stokes-Poisson equations

In this paper, we are concerned with long time asymptotic equivalence between the Vlasov-Poisson-Boltzmann (VPB) system and the compressible Navier-Stokes-Poisson (NSP) system. The density of the VPB system is proved to be asymptotically equivalent (mod decay rate $ (1+t)^{-1} $) as $ t\rightarrow \infty $ to that of the compressible Navier-Stokes-Poisson equations for the corresponding initial data. Regarding the momentum difference, we separately analyze the real and imaginary parts of its Fourier transform. It is shown that the real part decays at a rate of $ (1+t)^{-\frac{3}{4}}\ln(1+t) $, while the imaginary part decays at an optimal rate of $ (1+t)^{-\frac{1}{4}} $.