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Global compressible Euler-Poisson limit of the ionic Vlasov-Poisson-Boltzmann system for all cutoff potentials

Qin Ye, Fujun Zhou, Weijun Wu

TL;DR

This work rigorously justifies the compressible ionic Euler–Poisson limit as the Knudsen number $\varepsilon\to0$ for the ionic Vlasov–Poisson–Boltzmann system across the full cutoff potential range $-3<\gamma\le1$. The authors combine a truncated Hilbert expansion with a novel weighted $H^1_{x,v}$–$W^{1,\infty}_{x,v}$ analysis, introduce a global Maxwellian and velocity weights to handle high-velocity tails, and perform a macro–micro decomposition around a local Maxwellian to control both macroscopic and microscopic components. Central to their approach are uniform lemmas for hard and soft potentials, soft-potential weighted $H^1$ energy estimates, and $W^{1,\infty}_{x,v}$ bounds obtained via a carefully designed kernel decomposition and Dynkin-type Duhamel representation. The results yield global-in-time convergence with rate $O(\varepsilon)$ on $0\le t\le \varepsilon^{-1/2}$, significantly advancing the mathematical understanding of kinetic-to-fluid limits in plasmas and providing a unified treatment for both hard and soft collision kernels. This has implications for the rigorous derivation of macroscopic plasma models from kinetic theory under realistic interaction potentials.

Abstract

The ionic Vlasov-Poisson-Boltzmann system is a fundamental model in dilute collisional plasmas. In this work, we study the compressible ionic Euler-Poisson limit of the ionic Vlasov-Poisson-Boltzmann system for the full range of cutoff potentials $-3 < γ\leq 1$. By employing a truncated Hilbert expansion together with a novel weighted $H^1_{x,v}$-$W^{1,\infty}_{x,v}$ framework, we prove that the solution of the ionic Vlasov-Poisson-Boltzmann converges globally in time to the smooth global solution of the compressible ionic Euler-Poisson system.

Global compressible Euler-Poisson limit of the ionic Vlasov-Poisson-Boltzmann system for all cutoff potentials

TL;DR

This work rigorously justifies the compressible ionic Euler–Poisson limit as the Knudsen number for the ionic Vlasov–Poisson–Boltzmann system across the full cutoff potential range . The authors combine a truncated Hilbert expansion with a novel weighted – analysis, introduce a global Maxwellian and velocity weights to handle high-velocity tails, and perform a macro–micro decomposition around a local Maxwellian to control both macroscopic and microscopic components. Central to their approach are uniform lemmas for hard and soft potentials, soft-potential weighted energy estimates, and bounds obtained via a carefully designed kernel decomposition and Dynkin-type Duhamel representation. The results yield global-in-time convergence with rate on , significantly advancing the mathematical understanding of kinetic-to-fluid limits in plasmas and providing a unified treatment for both hard and soft collision kernels. This has implications for the rigorous derivation of macroscopic plasma models from kinetic theory under realistic interaction potentials.

Abstract

The ionic Vlasov-Poisson-Boltzmann system is a fundamental model in dilute collisional plasmas. In this work, we study the compressible ionic Euler-Poisson limit of the ionic Vlasov-Poisson-Boltzmann system for the full range of cutoff potentials . By employing a truncated Hilbert expansion together with a novel weighted - framework, we prove that the solution of the ionic Vlasov-Poisson-Boltzmann converges globally in time to the smooth global solution of the compressible ionic Euler-Poisson system.
Paper Structure (13 sections, 17 theorems, 287 equations)

This paper contains 13 sections, 17 theorems, 287 equations.

Table of Contents

  1. Introduction
  2. Problem Setup and Main Results
  3. Difficulties and Innovations
  4. Time Growth of $F_n$ and $\phi_n$
  5. Growth in $|v|$
  6. Local Maxwellian background $L(x, v)$
  7. Relevant literature
  8. New Lemmas for Hard and Soft Potentials
  9. Weighted H1 Estimates for the Remainder with Soft Potentials
  10. W Estimates for the Remainder with Soft Potentials
  11. Proof of the Main Results for the Remainder
  12. Estimates on the Time-Dependent Coefficients
  13. Other Preliminary Lemmas

Key Result

Theorem 1.1

Let $F_0 =\mu$ be as in LM. Fix an integer $s_1\ge 5$ and a constant $n_0>0$. Assume the initial perturbation $(\rho_0^{\mathrm{in}}, u_0^{\mathrm{in}})$ satisfies $\nabla_x\times u_0^{\mathrm{in}}=0$ and, for the constant $0 < \delta_0 \ll 1$, so that the existence of a unique global solution $(\rho_0(t,x), u_0(t,x), \phi_0(t,x))$ to the Euler--Poisson system ep is guaranteed by GuoCMP2011. Then

Theorems & Definitions (35)

  • Theorem 1.1: Soft potentials $-3< \gamma <0$
  • Theorem 1.2: Hard potentials $0\le \gamma \le 1$
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 25 more