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Global existence of Lagrangian solutions to the ionic Vlasov--Poisson system

Young-Pil Choi, Dowan Koo, Sihyun Song

TL;DR

The paper establishes global existence of Lagrangian (hence renormalized) solutions to the ionic Vlasov–Poisson system with massless electrons on the torus, under mild integrability assumptions on the initial data. The authors introduce a novel Poisson–Boltzmann decomposition that yields well-posedness, strong stability, and a strict lower bound for the electron density, enabling control of the coupled system. They construct global-in-time Lagrangian solutions by regularization and level-set decomposition, prove temporal continuity of all relevant quantities, and derive an energy inequality showing the total energy remains bounded by its initial value. Under additional integrability, these Lagrangian solutions coincide with weak solutions, connecting Eulerian and particle-trajectory formulations and highlighting the transport structure of the model. The results advance the mathematical understanding of kinetic models with self-consistent fields and low-regularity data, providing tools for the rigorous analysis of ion/electron dynamics at the ion timescale.

Abstract

In this paper, we establish the global existence of Lagrangian solutions to the ionic Vlasov--Poisson system under mild integrability assumptions on the initial data. Our approach involves proving the well-posedness of the Poisson--Boltzmann equation for densities in $L^p$ with $p>1$, introducing a novel decomposition technique that ensures uniqueness, stability, and improved bounds for the thermalized electron density. Using this result, we construct global-in-time Lagrangian solutions while demonstrating that the energy functional remains uniformly bounded by its initial value. Additionally, we show that renormalized solutions coincide with Lagrangian solutions, highlighting the transport structure of the system, and prove that renormalized solutions coincide with weak solutions under additional integrability assumptions.

Global existence of Lagrangian solutions to the ionic Vlasov--Poisson system

TL;DR

The paper establishes global existence of Lagrangian (hence renormalized) solutions to the ionic Vlasov–Poisson system with massless electrons on the torus, under mild integrability assumptions on the initial data. The authors introduce a novel Poisson–Boltzmann decomposition that yields well-posedness, strong stability, and a strict lower bound for the electron density, enabling control of the coupled system. They construct global-in-time Lagrangian solutions by regularization and level-set decomposition, prove temporal continuity of all relevant quantities, and derive an energy inequality showing the total energy remains bounded by its initial value. Under additional integrability, these Lagrangian solutions coincide with weak solutions, connecting Eulerian and particle-trajectory formulations and highlighting the transport structure of the model. The results advance the mathematical understanding of kinetic models with self-consistent fields and low-regularity data, providing tools for the rigorous analysis of ion/electron dynamics at the ion timescale.

Abstract

In this paper, we establish the global existence of Lagrangian solutions to the ionic Vlasov--Poisson system under mild integrability assumptions on the initial data. Our approach involves proving the well-posedness of the Poisson--Boltzmann equation for densities in with , introducing a novel decomposition technique that ensures uniqueness, stability, and improved bounds for the thermalized electron density. Using this result, we construct global-in-time Lagrangian solutions while demonstrating that the energy functional remains uniformly bounded by its initial value. Additionally, we show that renormalized solutions coincide with Lagrangian solutions, highlighting the transport structure of the system, and prove that renormalized solutions coincide with weak solutions under additional integrability assumptions.
Paper Structure (29 sections, 28 theorems, 281 equations)

This paper contains 29 sections, 28 theorems, 281 equations.

Table of Contents

  1. Introduction
  2. Overview of related works
  3. Main results
  4. Structure of the paper
  5. Poisson--Boltzmann Equation
  6. Solvability via Calculus of Variations
  7. Weak stability
  8. Comparison principle and uniform lower bound of $e^{\Phi[\rho]}$
  9. Strong stability
  10. Energy functionals
  11. Preliminaries: renormalized and Lagrangian solutions
  12. Renormalized solutions
  13. Proof of Proposition \ref{['cor: weak']}
  14. Maximal regular flows and Lagrangian solutions
  15. Regular flows
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $d \geq 2$ and $\rho \in L_+^p(\mathbb T^d)$ with $p > 1$. The Poisson--Boltzmann equation admits a unique solution $\Phi[\rho] \in W^{1,2}(\mathbb T^d)$, satisfying the following properties:

Theorems & Definitions (67)

  • Theorem 1.1: Well-posedness and properties of the Poisson--Boltzmann equation
  • Theorem 1.2: Global existence of Lagrangian and renormalized solutions
  • Remark 1.3
  • Proposition 1.4: Global existence of weak solutions
  • Remark 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • ...and 57 more