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Global stability of vacuum for the relativistic Vlasov-Maxwell-Boltzmann system

Chuqi Cao, Xingyu Li

TL;DR

This work proves global-in-time existence and nonlinear stability of vacuum for the three-dimensional relativistic Vlasov-Maxwell-Boltzmann system with arbitrary $c\ge 1$, for small initial data and $\gamma\in(-2,0]$, with bounds uniform in $c$. The authors introduce a $c$-dependent vector-field method, Glassey–Strauss decomposition, and a novel chain rule for the relativistic Boltzmann operator, enabling high-order commutations without compact data. A key innovation is the null decomposition of the electromagnetic field and carefully crafted weight functions that yield optimal decay rates for the distribution function and EM fields, despite strong coupling and nonlocal collision terms. The results bridge relativistic and nonrelativistic regimes, provide a robust framework for vacuum stability in RVMB, and establish tools likely applicable to related kinetic-field systems. Overall, the paper delivers a rigorous, $c$-uniform analysis of vacuum stability in a fundamentally nonlinear, nonlocal kinetic model, with potential implications for Newtonian limits and scattering behavior.

Abstract

We consider the three-dimensional relativistic Vlasov-Maxwell-Boltzmann system, where the speed of light $c$ is an arbitrary constant no less than 1, and we establish global existence and nonlinear stability of the vacuum for small initial data, with bounds that are uniform in $c$. The analysis is based on the vector field method combined with the Glassey-Strauss decomposition of the electromagnetic field, and does not require any compact support assumption on the initial data. A key ingredient of the proof is the derivation of a chain rule for the relativistic Boltzmann collision operator that is compatible with the commutation properties of the vector fields. These tools allow us to control the coupled kinetic and electromagnetic equations and to obtain global stability near vacuum.

Global stability of vacuum for the relativistic Vlasov-Maxwell-Boltzmann system

TL;DR

This work proves global-in-time existence and nonlinear stability of vacuum for the three-dimensional relativistic Vlasov-Maxwell-Boltzmann system with arbitrary , for small initial data and , with bounds uniform in . The authors introduce a -dependent vector-field method, Glassey–Strauss decomposition, and a novel chain rule for the relativistic Boltzmann operator, enabling high-order commutations without compact data. A key innovation is the null decomposition of the electromagnetic field and carefully crafted weight functions that yield optimal decay rates for the distribution function and EM fields, despite strong coupling and nonlocal collision terms. The results bridge relativistic and nonrelativistic regimes, provide a robust framework for vacuum stability in RVMB, and establish tools likely applicable to related kinetic-field systems. Overall, the paper delivers a rigorous, -uniform analysis of vacuum stability in a fundamentally nonlinear, nonlocal kinetic model, with potential implications for Newtonian limits and scattering behavior.

Abstract

We consider the three-dimensional relativistic Vlasov-Maxwell-Boltzmann system, where the speed of light is an arbitrary constant no less than 1, and we establish global existence and nonlinear stability of the vacuum for small initial data, with bounds that are uniform in . The analysis is based on the vector field method combined with the Glassey-Strauss decomposition of the electromagnetic field, and does not require any compact support assumption on the initial data. A key ingredient of the proof is the derivation of a chain rule for the relativistic Boltzmann collision operator that is compatible with the commutation properties of the vector fields. These tools allow us to control the coupled kinetic and electromagnetic equations and to obtain global stability near vacuum.
Paper Structure (33 sections, 59 theorems, 324 equations)

This paper contains 33 sections, 59 theorems, 324 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Related Works
  4. The relativistic Vlasov-Maxwell system
  5. Boltzmann and Landau Equations
  6. Stability Near Vacuum
  7. Key ideas of proof
  8. Null structure of electricmagnetic field
  9. Vector field method and $c$-dependent vector field
  10. 1: The $c$-Dependent Formulation.
  11. 2: Advantages for Decay and Commutation.
  12. Weight Functions and the Distribution Function Estimate
  13. The relativistic Boltzmann collision operator
  14. 1: Chain rule for the relativistic Boltzmann operator.
  15. 2: Estimate of the weight function in relativistic regime.
  16. ...and 18 more sections

Key Result

Theorem 1.1

Let $\gamma\in (-2,0]$. For any $c \ge 1$, $M>0$, there exists a constant $\epsilon_0 >0$ independent of $c$, such that for any $\epsilon_1\in (0,\epsilon_0)$, if the initial data $(f_0, E_0, B_0)$ satisfy then the relativistic Vlasov-Maxwell-Boltzmann system RVMB1-RVMB2 admits a global solution. Moreover, we have the following estimates in $t$ for distribution function where $\hat{Z}$ is define

Theorems & Definitions (124)

  • Theorem 1.1: Global stability of the relativistic Vlasov-Maxwell-Boltzmann system
  • Theorem 1.2: Chain rule for the relativistic Boltzmann operator
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Definition 2.1
  • Remark 2.2
  • ...and 114 more