Uniqueness of the Non-Equilibrium Steady State for a 1d BGK Model in Kinetic Theory

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 linear interaction with the reservoirs by (1 − α) , we prove that for some α close enough to zero, the explicit spatially uniform non-equilibrium steady 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.