Global Hilbert Expansion for the Vlasov-Poisson-Boltzmann System

Abstract

We study the Hilbert expansion for small Knudsen number $\varepsilon$ for the Vlasov-Boltzmann-Poisson system for an electron gas. The zeroth order term takes the form of local Maxwellian: $ F_{0}(t,x,v)=\frac{ρ_{0}(t,x)}{(2πθ_{0}(t,x))^{3/2}} e^{-|v-u_{0}(t,x)|^{2}/2θ_{0}(t,x)},\text{\ }θ_{0}(t,x)=Kρ_{0}^{2/3}(t,x).$ Our main result states that if the Hilbert expansion is valid at $t=0$ for well-prepared small initial data with irrotational velocity $u_0$, then it is valid for $0\leq t\leq \varepsilon ^{-{1/2}\frac{2k-3}{2k-2}},$ where $ρ_{0}(t,x)$ and $ u_{0}(t,x)$ satisfy the Euler-Poisson system for monatomic gas $γ=5/3$.