Uniqueness of the Non-Equilibrium Steady State for a 1d BGK model in kinetic theory

2019/04/10 by Carlen, Eric, Esposito, Raffaele, Leibowitz, Joel +2 · 1 citation
#35Q20 #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1904.05142

Abstract

We continue our investigation of kinetic models of a one-dimensional gas in contact with homogeneous thermal reservoirs at different temperatures. Nonlinear collisional interactions between particles are modeled by a so-called BGK dynamics which conserves local energy and particle density. Weighting the nonlinear BGK term with a parameter α∈ [0,1], and the linearinteraction with the reservoirs by (1-α), we prove that for all α close enough to zero, the explicit spatially uniform non-equilibrium stable state (NESS) is unique, and there are no spatially non-uniform NESS with a spatial density ρ belonging to L^p for any p>1. We also show that for all α∈ [0,1], the spatially uniform NESS is dynamically stable, with small perturbation converging to zero exponentially fast.

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• Existence and stability of non-equilibrium steady states of a weakly non-linear kinetic Fokker-Planck equation in a domain

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