  For the one-species Vlasov-Poisson-Boltzmann (VPB) system in the scaling
under which the moments of the fluctuations formally converge to the
incompressible Navier-Stokes-Fourier-Poisson (NSFP) system, we prove the
uniform estimates with respect to the Knudsen number $\epsilon$ for the
fluctuations. As a consequence, the existence of the global-in-time classical
solutions of VPB with all $\epsilon \in (0,1]$ is established in whole space
under small size of initial data, and the convergence to incompressible NSFP as
$\epsilon$ go to 0 is rigorously justified.

Analysis of PDEs