Properties of non-equilibrium steady states for the non-linear BGK equation on the torus

We study the non-linear BGK model in one dimension coupled to a spatially varying thermostat. We show existence, local uniqueness, and linear stability of a steady state when the linear coupling term is large compared to the non-linear self-interaction term. This model possesses a non-explicit spatially dependent non-equilibrium steady state. For the existence and the local uniqueness we utilise a fixed point argument, reducing the study of the non-linear model to the linear BGK model, while for the stability we adapt the L^{2} hypocoercivity theory, yielding existence of a spectral gap for the linearised operator around this non-equilibrium steady state.