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Local well-posedness of the Vlasov-Poisson-Landau system and related models

Patrick Flynn

TL;DR

The paper proves local well-posedness for the Vlasov-Poisson-Landau system and its massless-electron variant on the 3D torus for large initial data by propagating a weighted anisotropic $L^2$-based Sobolev norm in velocity and space. The key technique combines a diffusion-transport decomposition of the Landau operator with energy and commutator estimates in the $\mathfrak E$ framework, and uses a contraction-based iteration to construct solutions. It also analyzes the associated Poincaré-Poisson subsystem to handle the massless-electron limit, establishing existence, bounds, and regularity for the electrostatic potential and the electron-parameter $\beta$, and proving compatibility with the main VPL results. The results extend local well-posedness to large data in a setting without perturbative smallness assumptions and provide a structured pathway toward potential global well-posedness, contingent on further advances in related homogeneous Landau theory. The work also clarifies the role of velocity derivatives in the nonlinear analysis and sets the stage for the massless limit and related kinetic models to be treated rigorously within this anisotropic Sobolev framework.

Abstract

We prove local well-posedness for the Vlasov-Poisson-Landau system and the variant with massless electrons in a 3D periodic spatial domain for large initial data. This is accomplished by propagating weighted anisotropic L2-based Sobolev norms. In the case of the massless electron system, we also carry out an analysis of the Poincare-Poisson system. This is a companion paper to the author's previous work with Yan Guo.

Local well-posedness of the Vlasov-Poisson-Landau system and related models

TL;DR

The paper proves local well-posedness for the Vlasov-Poisson-Landau system and its massless-electron variant on the 3D torus for large initial data by propagating a weighted anisotropic -based Sobolev norm in velocity and space. The key technique combines a diffusion-transport decomposition of the Landau operator with energy and commutator estimates in the framework, and uses a contraction-based iteration to construct solutions. It also analyzes the associated Poincaré-Poisson subsystem to handle the massless-electron limit, establishing existence, bounds, and regularity for the electrostatic potential and the electron-parameter , and proving compatibility with the main VPL results. The results extend local well-posedness to large data in a setting without perturbative smallness assumptions and provide a structured pathway toward potential global well-posedness, contingent on further advances in related homogeneous Landau theory. The work also clarifies the role of velocity derivatives in the nonlinear analysis and sets the stage for the massless limit and related kinetic models to be treated rigorously within this anisotropic Sobolev framework.

Abstract

We prove local well-posedness for the Vlasov-Poisson-Landau system and the variant with massless electrons in a 3D periodic spatial domain for large initial data. This is accomplished by propagating weighted anisotropic L2-based Sobolev norms. In the case of the massless electron system, we also carry out an analysis of the Poincare-Poisson system. This is a companion paper to the author's previous work with Yan Guo.
Paper Structure (11 sections, 6 theorems, 169 equations)

This paper contains 11 sections, 6 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. Main Result
  3. Method of the proof
  4. A priori estimates
  5. Lower and upper bounds on the diffusion matrix
  6. Energy estimates
  7. Contraction estimates
  8. Construction of Solutions
  9. Solutions to the linear system
  10. Construction of solutions to the VPL system
  11. Extension to the VPL-Ion system and Landau equation

Key Result

Theorem 1.1

Let $F_{+,in}(x,v), F_{-,in}(x,w)$ be nonnegative functions in $L^1(\mathbf T^3\times \mathbf R^3)$ with unit $L^1$ norm.

Theorems & Definitions (14)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 4 more