Particles interacting with a vibrating medium: existence of solutions and convergence to the Vlasov--Poisson system

TL;DR

This paper studies a kinetic model of particles interacting with a vibrational environment, proving the existence of solutions and deriving the Vlasov--Poisson system in a specific regime.

Contribution

It establishes the existence of weak solutions for a broad class of initial data and external forces, and derives the Vlasov--Poisson system from coupled Vlasov-Wave equations.

Findings

Existence of weak solutions for the kinetic equation.

Derivation of the Vlasov--Poisson system in a specific regime.

Identification of conditions linking the coupled system to classical models.

Abstract

We are interested in a kinetic equation intended to describe the interaction of particles with their environment. The environment is modeled by a collection of local vibrational degrees of freedom. We establish the existence of weak solutions for a wide class of initial data and external forces. We also identify a relevant regime which allows us to derive, quite surprisingly, the attractive Vlasov--Poisson system from the coupled Vlasov-Wave equations.