Hydrodynamic limit from kinetic models with massless electrons to the ionic Euler--Poisson system

TL;DR

This paper rigorously derives the ionic Euler--Poisson system from kinetic models with massless electrons, addressing mathematical challenges posed by nonlinear elliptic equations and establishing error estimates and existence of solutions.

Contribution

It provides the first quantitative hydrodynamic limit from kinetic models with massless electrons to the ionic Euler--Poisson system, including error bounds and global existence results.

Findings

Hydrodynamic limit established with error estimates

Global-in-time existence of weak entropy solutions proven

Mathematical difficulties from nonlinear elliptic equations overcome

Abstract

We study the derivation of ion dynamics, namely, the ionic Euler--Poisson system, from kinetic descriptions. The kinetic framework consists of the ionic Vlasov--Poisson equation coupled with either a nonlinear Fokker--Planck operator or a local alignment term. In both kinetic and fluid models, the massless electrons are assumed to be in thermodynamic equilibrium, leading to an electric potential governed by the Poisson--Boltzmann equation. The exponential nonlinearity in this semilinear elliptic problem creates significant mathematical difficulties, which we overcome by exploiting the physical structure of the system, in particular, the role of the electron velocity field hidden in the limiting equation. Our first main result establishes the hydrodynamic limit from the kinetic model to the ionic Euler--Poisson system, providing quantitative error estimates via the modulated energy method. As a second contribution, we prove the global-in-time existence of weak entropy solutions to the kinetic equations, ensuring consistency with the hydrodynamic limit framework.