Abstract
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.
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More From: Annales de l'Institut Henri Poincaré C, Analyse non linéaire
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