Existence of a Non-Equilibrium Steady State for the Non-linear BGK equation on an interval

TL;DR

This paper proves the existence of a non-equilibrium steady state in a one-dimensional non-linear BGK model with boundary heat reservoirs, using a fixed point approach without relying on perturbation methods.

Contribution

It establishes the existence of non-equilibrium steady states for the non-linear BGK equation on an interval with diffusive boundary conditions, extending previous results beyond perturbative regimes.

Findings

Existence of non-equilibrium steady states proven

Non-perturbative analysis employed

Fixed point method reduces non-linear to linear BGK

Abstract

We show existence of a non-equilibrium steady state for the one-dimensional, non-linear BGK model on an interval with diffusive boundary conditions. These boundary conditions represent the coupling of the system with two heat reservoirs at different temperatures. Our analysis is not perturbative around the equilibrium. We employ a fixed point argument to reduce the study of the model with non-linear collisional interactions to the linear BGK.