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The non-cutoff Vlasov-Poisson-Boltzmann system with weak collisions

Yuanjie Lei, Shuangqian Liu, Qinghua Xiao, Huijiang Zhao

Abstract

We prove global existence of smooth solutions near Maxwellians for the non-cutoff Vlasov-Poisson-Boltzmann system in the weakly collisional regime. To address the weak dissipation of the non-cutoff linearized Boltzmann operator, we develop a refined velocity-weighted energy framework combined with vector-field techniques to control the transport term, nonlinear collisions, and the self-consistent electric field. This approach yields uniform-in-time bounds, captures enhanced dissipation of the solution, and establishes Landau damping for both the density and electric field, providing the first global-in-time result of this type for the non-cutoff Vlasov-Poisson-Boltzmann system. Our approach is inspired by the recent work of Chaturvedi-Luk-Nguyen ({\it J. Amer. Math. Soc.} {\bf 36} (2023), no. 4, 1103--1189.)

The non-cutoff Vlasov-Poisson-Boltzmann system with weak collisions

Abstract

We prove global existence of smooth solutions near Maxwellians for the non-cutoff Vlasov-Poisson-Boltzmann system in the weakly collisional regime. To address the weak dissipation of the non-cutoff linearized Boltzmann operator, we develop a refined velocity-weighted energy framework combined with vector-field techniques to control the transport term, nonlinear collisions, and the self-consistent electric field. This approach yields uniform-in-time bounds, captures enhanced dissipation of the solution, and establishes Landau damping for both the density and electric field, providing the first global-in-time result of this type for the non-cutoff Vlasov-Poisson-Boltzmann system. Our approach is inspired by the recent work of Chaturvedi-Luk-Nguyen ({\it J. Amer. Math. Soc.} {\bf 36} (2023), no. 4, 1103--1189.)

Paper Structure

This paper contains 21 sections, 17 theorems, 203 equations.

Table of Contents

  1. Introduction and main results
  2. The problem
  3. The notations
  4. The weights and the corresponding functionals
  5. The main result
  6. Literatures
  7. Difficulties and strategies
  8. Outline of the paper
  9. Linearized Boltzmann equation
  10. Density estimates
  11. Linear density estimates
  12. Nonlinear density estimates
  13. Nonlinear energy estimates and global existence
  14. Enhanced dissipation
  15. Weighted energy estimates
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $0<q\ll 1$, $\iota_0 \geq N_{max}+9$, and $N_{max}\geq 9$. Denote under the condition case. Assume the initial datum $f_0$ satisfies the following conservation laws: There exist positive constants $\mathcal{M}\in (0,\mathcal{M}_0]$, and $\nu\in(0,\nu_0]$ such that if: then the Cauchy problem f and f-initial admits a unique global smooth solution $f(t,x,v)$. Moreover, The solution $f(t,x,v)$

Theorems & Definitions (35)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Proposition 2.1
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.1
  • ...and 25 more