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{\rho_{0}(t,x)}{(2\pi \theta_{0}(t,x))^{3/2}} e^{-|v-u_{0}(t,x)|^{2}/2\theta_{0}(t,x)},\text{\ }\theta_{0}(t,x)=K\rho_{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$\rho_{0}(t,x)$and$ u_{0}(t,x)$satisfy the Euler-Poisson system for monatomic gas$\gamma=5/3$