Existence and Stability of Non-Equilibrium Steady States of a Weakly Non-Linear Kinetic Fokker-Planck Equation in a Domain

We study a weakly non-linear Fokker-Planck equation with BGK heat thermostats in a spatially bounded domain with conservative Maxwell boundary conditions, presenting a space-dependent accommodation coefficient and a space-dependent temperature on the spatial boundary. The model is based from a problem introduced in [E. A. Carlen, R. Esposito, J. L. Lebowitz, R. Marra, and C. Mouhot. Approach to the steady state in kinetic models with thermal reservoirs at different temperatures. J . Stat. Phys., 172(2):522–543, 2018] where the authors studied the properties of the non-equilibrium steady states for non-linear kinetic Fokker-Planck equations with BGK thermostats in the torus. We generalize those results for bounded domains using the recent results presented in [K. Carrapatoso, P. Gabriel, R. Medina, and S. Mischler. Constructive Krein-Rutman result for kinetic Fokker-Planck equations in a domain , 2024] for the study of general kinetic Fokker-Planck equations with Maxwell boundary conditions. More precisely, in a weakly non-linear regime, we obtain the existence of a non-equilibrium steady state and its stability in the perturbative regime.