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

Fucai Li, Yichun Wang

TL;DR

The paper proves the global-in-time validity of the Hilbert expansion for the ionic Vlasov-Poisson-Boltzmann system in $\mathbb{R}^3$, deriving the compressible ionic Euler-Poisson system as the $\varepsilon\to0$ limit. It constructs a truncated expansion up to order $2k-1$ and analyzes the remainder via a coupled nonlinear Poisson equation for the potential and kinetic remainder dynamics, obtaining uniform-in-$\varepsilon$ estimates on a time interval $0\le t\le \varepsilon^{-m}$ with $0<m<\tfrac{1}{2}\tfrac{2k-3}{2k-2}$ (for $k\ge6$). A key advance is handling the exponential nonlinearity in the Poisson equation by developing a $L^2$--$L^\infty$ framework, refined elliptic bounds using a $(e^{\phi_0}-\Delta)^{-1}$ parametrix, and Faà di Bruno-type control of $H(\varepsilon)-T_{2k-1}(H(\varepsilon))$, which together close the hierarchy of coefficients and the remainder. The results provide a rigorous bridge from a kinetic VPB description of ions to the fluid Euler-Poisson dynamics with an irrotational flow, establishing the leading-order Euler-Poisson limit and the global stability of the Hilbert expansion under precise time-scale and regularity conditions.

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.

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

TL;DR

The paper proves the global-in-time validity of the Hilbert expansion for the ionic Vlasov-Poisson-Boltzmann system in , deriving the compressible ionic Euler-Poisson system as the limit. It constructs a truncated expansion up to order and analyzes the remainder via a coupled nonlinear Poisson equation for the potential and kinetic remainder dynamics, obtaining uniform-in- estimates on a time interval with (for ). A key advance is handling the exponential nonlinearity in the Poisson equation by developing a -- framework, refined elliptic bounds using a parametrix, and Faà di Bruno-type control of , which together close the hierarchy of coefficients and the remainder. The results provide a rigorous bridge from a kinetic VPB description of ions to the fluid Euler-Poisson dynamics with an irrotational flow, establishing the leading-order Euler-Poisson limit and the global stability of the Hilbert expansion under precise time-scale and regularity conditions.

Abstract

We justify the global-in-time validity of Hilbert expansion for the ionic Vlasov-Poisson-Boltzmann system in , 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 estimates for the potential-dependent terms.
Paper Structure (21 sections, 10 theorems, 239 equations)

This paper contains 21 sections, 10 theorems, 239 equations.

Table of Contents

  1. Introduction
  2. The model and our results
  3. Previous works
  4. Hilbert expansion
  5. Kinetic models for ions
  6. Compressible ionic Euler-Poisson system
  7. Main difficulties and our strategies in the proof
  8. Formations of the model
  9. $H^s$ estimates of the coefficients $\phi_n$ $(1\leq n\leq 2k-1)$
  10. The estimates of $H(\varepsilon)-T_{2k-1}(H(\varepsilon))$ and its first-order derivatives
  11. The interplay between $L^2$ and $W^{1,\infty}$ norms
  12. Outline of the paper
  13. Coefficients of the Hilbert Expansion
  14. The solution $\phi_R$ to the remainder equation \ref{['vpbr']}$_2$
  15. $L^2$ Estimates for the Remainders $R$ and $\phi_R$
  16. ...and 6 more sections

Key Result

Theorem 1.1

Suppose that the functions $[\rho_0(t,x),u_0(t,x),\phi_0(t,x)]$ are solutions to the compressible ionic Euler-Poisson system EPS constructed by Guo-ion-2011 provided that the smooth initial data $\rho_0^{\rm{in}}(x)-1$ and $u_0^{\rm{in}}(x)$ are sufficiently small and satisfies $\nabla\times u_0^{\r for all $0<m<\frac{1}{2}\frac{2k-3}{2k-2}$ and $\beta\geq \frac{7}{2}$.

Theorems & Definitions (21)

  • Remark 1.1
  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Proposition 2.1
  • proof
  • Proposition 3.1
  • Proposition 3.2
  • proof : Proof of Proposition \ref{['prop1']}
  • ...and 11 more