Global Hilbert Expansion for the Vlasov-Poisson-Boltzmann System

TL;DR

This paper proves the validity of the Hilbert expansion for the Vlasov-Poisson-Boltzmann system with small Knudsen number, linking kinetic and fluid models for electron gases under specific initial conditions.

Contribution

It establishes the global-in-time validity of the Hilbert expansion for the system, connecting kinetic theory to Euler-Poisson fluid dynamics for the first time.

Findings

Validity of Hilbert expansion for small initial data

Connection between kinetic and fluid models

Time interval of expansion validity depends on initial conditions

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 $\gamma=5/3$.