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

TL;DR

This paper proves the global existence of Lagrangian solutions for the ionic Vlasov--Poisson system, introducing new techniques for well-posedness and stability, and establishing the equivalence of different solution concepts.

Contribution

It establishes the global existence of Lagrangian solutions under mild conditions, with novel decomposition methods ensuring uniqueness and stability, and links between solution types.

Findings

Proved well-posedness of the Poisson--Boltzmann equation for $L^p$ densities.

Constructed global-in-time Lagrangian solutions with bounded energy.

Showed renormalized solutions coincide with Lagrangian and weak solutions under certain conditions.

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.