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Global existence for the relativistic Vlasov-Poisson system in a two-dimensional bounded domain

Yanmin Mu, Dehua Wang

TL;DR

The paper proves global existence and uniqueness of classical solutions to the two-dimensional relativistic Vlasov-Poisson system in convex bounded domains under specular reflection, with either Neumann or homogeneous Dirichlet boundary conditions for the electric potential. The authors develop a boundary-adapted geometric framework using arc-length coordinates and Frenet-Serret frames, introduce the auxiliary variables $\alpha$ and $\beta$ to control distance to the singular set and boundary collisions, and construct a convergent iterative scheme based on a linearized Vlasov equation coupled to a Poisson solve. A velocity lemma guarantees finite boundary interactions and supports uniform estimates that propagate in time; a Green’s-function-based Poisson solver provides the necessary regularity for $\varphi$ and $E=\nabla\varphi$. The results establish global existence for general initial data in 2D convex domains, clarifying boundary effects in kinetic PDEs and extending known Cauchy-problem results to bounded domains with boundaries of varying boundary conditions. The techniques offer a robust framework for handling boundary geometry in relativistic kinetic models with self-consistent fields, with potential applicability to broader bounded-domain Vlasov-type systems.

Abstract

In this paper, we prove the global existence of solutions to the relativistic Vlasov-Poisson system for general initial data in convex bounded domains of two space dimensions, assuming the specular reflection boundary conditions for the distribution density. The boundary conditions for the electric potential are considered in two cases: Neumann boundary conditions and homogeneous Dirichlet boundary conditions. The core ideas involve constructing suitable velocity lemmas and applying geometric techniques. In the two-dimensional case, it is crucial to select the arc length as the parameter of the curve and to further combine this with the Frenet-Serret formulas, enabling us to effectively describe the distribution density equation near the boundary and thus establishing a vital connection in the geometric representation.

Global existence for the relativistic Vlasov-Poisson system in a two-dimensional bounded domain

TL;DR

The paper proves global existence and uniqueness of classical solutions to the two-dimensional relativistic Vlasov-Poisson system in convex bounded domains under specular reflection, with either Neumann or homogeneous Dirichlet boundary conditions for the electric potential. The authors develop a boundary-adapted geometric framework using arc-length coordinates and Frenet-Serret frames, introduce the auxiliary variables and to control distance to the singular set and boundary collisions, and construct a convergent iterative scheme based on a linearized Vlasov equation coupled to a Poisson solve. A velocity lemma guarantees finite boundary interactions and supports uniform estimates that propagate in time; a Green’s-function-based Poisson solver provides the necessary regularity for and . The results establish global existence for general initial data in 2D convex domains, clarifying boundary effects in kinetic PDEs and extending known Cauchy-problem results to bounded domains with boundaries of varying boundary conditions. The techniques offer a robust framework for handling boundary geometry in relativistic kinetic models with self-consistent fields, with potential applicability to broader bounded-domain Vlasov-type systems.

Abstract

In this paper, we prove the global existence of solutions to the relativistic Vlasov-Poisson system for general initial data in convex bounded domains of two space dimensions, assuming the specular reflection boundary conditions for the distribution density. The boundary conditions for the electric potential are considered in two cases: Neumann boundary conditions and homogeneous Dirichlet boundary conditions. The core ideas involve constructing suitable velocity lemmas and applying geometric techniques. In the two-dimensional case, it is crucial to select the arc length as the parameter of the curve and to further combine this with the Frenet-Serret formulas, enabling us to effectively describe the distribution density equation near the boundary and thus establishing a vital connection in the geometric representation.

Paper Structure

This paper contains 18 sections, 19 theorems, 213 equations.

Table of Contents

  1. Introduction
  2. A more convenient coordinate system near $\partial\Omega\times \mathbb{R}^{2}$
  3. Compatibility conditions and assumptions for the initial and boundary data
  4. Main results
  5. Difficulties and strategy of the proofs
  6. Organization of the paper
  7. Iterative Procedure
  8. Linear Problem
  9. Velocity lemma
  10. Well-posedness of the linear problem
  11. On the Convergence of the Sequence {$f^{n}$}
  12. The solution of the Poisson equation with Neumann boundary conditions
  13. The iterative sequence {$f^{n}$} is globally defined in time
  14. Convergence of the iterative sequences {$f^{n}$}
  15. Prolongability of uniform estimates for the functions $f^{n}$
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $f_{0} \in C^{1,\mu}_{0}(\bar{\Omega}\times\mathbb{R}^{2}), f_{0}\geq 0$ for some $0 <\mu< 1$, satisfying v2.8. Suppose that $h \in C^{2,\mu} (\partial\Omega)$ satisfies v1.12 and $h > 0$. Then there exists a unique solution $f \in C^{1;1,\lambda}_{t;(x,v)} ((0,\infty)\times \bar{\Omega}\times \

Theorems & Definitions (44)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 3.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Lemma 3.1
  • proof
  • ...and 34 more