Stability of the Front under a Vlasov-Fokker-Planck Dynamics

TL;DR

This paper proves the asymptotic stability of a phase boundary front in a two-species particle system modeled by coupled Vlasov-Fokker-Planck equations, under small symmetric disturbances.

Contribution

It establishes the stability of a key phase boundary solution in a kinetic model with long-range interactions and thermal reservoir, a novel result for such systems.

Findings

Proved asymptotic stability of the front solution.

Analyzed the behavior under small symmetric perturbations.

Provided mathematical framework for phase boundary stability.

Abstract

We consider a kinetic model for a system of two species of particles interacting through a longrange repulsive potential and a reservoir at given temperature. The model is described by a set of two coupled Vlasov-Fokker-Plank equations. The important front solution, which represents the phase boundary, is a one-dimensional stationary solution on the real line with given asymptotic values at infinity. We prove the asymptotic stability of the front for small symmetric perturbations.