Global Hilbert expansion for the ionic Vlasov-Poisson-Boltzmann system

TL;DR

This paper rigorously proves the validity of the Hilbert expansion for the ionic Vlasov-Poisson-Boltzmann system, showing convergence to the compressible Euler-Poisson system in the zero Knudsen number limit.

Contribution

It provides a rigorous mathematical justification for the global-in-time Hilbert expansion in a fundamental plasma model, including detailed analysis of the multi-layered structure and nonlinear elliptic estimates.

Findings

Validated the global-in-time Hilbert expansion for the system.

Derived the compressible Euler-Poisson system as the zero Knudsen limit.

Developed new elliptic estimates for nonlinear potential terms.

Abstract

We justify the global-in-time validity of Hilbert expansion for the ionic Vlasov-Poisson-Boltzmann system in $\mathbb{R}^3$, a fundamental model describing ion dynamics in dilute collisional plasmas. As the Knudsen number approaches zero, we rigorously derive the compressible Euler-Poisson system governing global smooth irrotational ion flows. The truncated Hilbert expansion exhibits a multi-layered mathematical structure: the expansion coefficients satisfy linear hyperbolic systems, while the remainder equation couples with a nonlinear Poisson equation for the electrostatic potential. This requires refined elliptic estimates addressing the exponential nonlinearities and some new enclosed $L^2\cap W^{1,\infty}$ estimates for the potential-dependent terms.