Vol. 3, No. 1, 2021

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Existence of a nonequilibrium steady state for the nonlinear BGK equation on an interval

Josephine Evans and Angeliki Menegaki

Vol. 3 (2021), No. 1, 223–252
Abstract

We show existence of a nonequilibrium steady state for the one-dimensional, nonlinear BGK model on an interval with diffusive boundary conditions. These boundary conditions represent the coupling of the system with two heat reservoirs at different temperatures. The result holds when the boundary temperatures at the two ends are away from the equilibrium case, as our analysis is not perturbative around the equilibrium. We employ a fixed point argument to reduce the study of the model with nonlinear collisional interactions to the linear BGK model.

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Keywords
nonequilibrium steady state, nonlinear BGK model, diffusive boundary conditions, existence results, heat transfer
Mathematical Subject Classification
Primary: 35Q20, 35Q49, 35Q82
Milestones
Received: 10 August 2020
Revised: 5 October 2020
Accepted: 19 January 2021
Published: 28 May 2021
Authors
Josephine Evans
Warwick Mathematics Institute
University of Warwick
Coventry
United Kingdom
Angeliki Menegaki
Department of Pure Mathematics and Mathematical Statistics
Centre for Mathematical Sciences
University of Cambridge
United Kingdom