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Long-Time Dynamics of the 3D Vlasov-Maxwell System with Boundaries

Jin Woo Jang, Chanwoo Kim

TL;DR

The paper proves global-in-time classical solutions for the 3D Vlasov–Maxwell system in a half-space with gravity and perfectly conducting boundaries, addressing long-standing challenges from wave–particle interactions. The authors introduce a new magnetic-field representation in Coulomb gauge that cancels nonlinear $S$-type terms, enabling a nearly linear treatment of magnetic effects and allowing precise control of particle travel times. They construct non-vacuum steady states via a Lagrangian approach with boundary data and gravity, establishing uniqueness and sharp weighted $L^ty$ bounds. Building on this, they prove dynamical asymptotic stability: small perturbations produce a global solution that decays like $(1+t)^{-1}$ in both particle distributions and electromagnetic fields, with detailed decay channels for homogeneous, boundary, and nonlinear contributions. The results highlight gravity’s stabilizing role in stellar-like plasmas and provide a rigorous framework for long-time VM dynamics with boundaries and magnetic fields, potentially informing solar wind modeling and astrophysical plasmas.

Abstract

We construct global-in-time classical solutions to the nonlinear Vlasov-Maxwell system in a three-dimensional half-space beyond the vacuum scattering regime. Our approach combines the construction of stationary solutions to the associated boundary-value problem with a proof of their asymptotic dynamical stability in $L^\infty$ under small perturbations, providing a new framework for understanding long-time wave-particle interactions in the presence of boundaries and interacting magnetic fields. To the best of our knowledge, this work presents the first construction of asymptotically stable non-vacuum steady states under general perturbations in the full three-dimensional nonlinear Vlasov-Maxwell system.

Long-Time Dynamics of the 3D Vlasov-Maxwell System with Boundaries

TL;DR

The paper proves global-in-time classical solutions for the 3D Vlasov–Maxwell system in a half-space with gravity and perfectly conducting boundaries, addressing long-standing challenges from wave–particle interactions. The authors introduce a new magnetic-field representation in Coulomb gauge that cancels nonlinear -type terms, enabling a nearly linear treatment of magnetic effects and allowing precise control of particle travel times. They construct non-vacuum steady states via a Lagrangian approach with boundary data and gravity, establishing uniqueness and sharp weighted bounds. Building on this, they prove dynamical asymptotic stability: small perturbations produce a global solution that decays like in both particle distributions and electromagnetic fields, with detailed decay channels for homogeneous, boundary, and nonlinear contributions. The results highlight gravity’s stabilizing role in stellar-like plasmas and provide a rigorous framework for long-time VM dynamics with boundaries and magnetic fields, potentially informing solar wind modeling and astrophysical plasmas.

Abstract

We construct global-in-time classical solutions to the nonlinear Vlasov-Maxwell system in a three-dimensional half-space beyond the vacuum scattering regime. Our approach combines the construction of stationary solutions to the associated boundary-value problem with a proof of their asymptotic dynamical stability in under small perturbations, providing a new framework for understanding long-time wave-particle interactions in the presence of boundaries and interacting magnetic fields. To the best of our knowledge, this work presents the first construction of asymptotically stable non-vacuum steady states under general perturbations in the full three-dimensional nonlinear Vlasov-Maxwell system.

Paper Structure

This paper contains 92 sections, 43 theorems, 760 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Main Theorems, Difficulties and Our Strategies
  3. Heuristic Explanation of Main Results, Difficulties, and Ideas
  4. Main Theorem 1: Unique Solvability of the Steady Problem
  5. Main Theorem 2: Dynamical Asymptotic Stability
  6. Characteristics for Wave and Transport Dynamics
  7. New Magnetic Field Representation via the Potentials
  8. Retarded Solutions
  9. Potential Representation in the Half Space $\mathbb{R}^3_+$
  10. Homogeneous solution
  11. Particular solution
  12. Relativistic Trajectory
  13. Boundary Exit Time
  14. Weight Comparison
  15. Weight Comparison in the Stationary Case
  16. ...and 77 more sections

Key Result

Theorem 2.1

Fix $g>0$ with $\min\{m_-,m_+\}g\ge 8$ and choose $\beta>1$ such that $\min\{m_-,m_+\}g \beta^3\gg 1.$ Suppose that the inflow boundary data $G_\pm$ is a $C^1$ exponentially localized: Then we construct a unique classical solution to the stationary Vlasov--Maxwell system 2speciesVM-steady with the incoming boundary condition 2species-perturbabsorbing.st and the perfect conductor boundary conditio

Figures (1)

  • Figure :

Theorems & Definitions (104)

  • Conjecture 1.1: Linear Instability Threshold
  • Theorem 2.1: Unique Solvability of the Steady Problem
  • Remark 2.2
  • Theorem 2.3: Asymptotic Stability
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Proposition 3.2: Representation of Magnetic field
  • proof
  • Lemma 3.3: Lemma 1.6 of MR1142472
  • ...and 94 more