Uniqueness of the Non-Equilibrium Steady State for a 1d BGK Model in Kinetic Theory
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Uniqueness of the Non-Equilibrium Steady State for a 1d BGK Model in Kinetic Theory E. Carlen1 · R. Esposito2 · J. Lebowitz3 · R. Marra4 · C. Mouhot5
Received: 8 June 2019 / Accepted: 5 September 2019 © Springer Nature B.V. 2019
Abstract 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 Lp 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. Keywords Kinetic equation · Uniqueness · Non-equilibrium steady state
B E. Carlen
[email protected] R. Esposito [email protected] J. Lebowitz [email protected] R. Marra [email protected] C. Mouhot [email protected]
1
Department of Mathematics, Rutgers University, 110 Felinghuysen Rd., Piscataway, NJ 08541, USA
2
International Research Center, Università di l’Aquila, L’Aquila, (AQ) 67100, Italy
3
Department of Mathematics & Department of Physics, Rutgers University, 110 Felinghuysen Rd., Piscataway, NJ 08541, USA
4
Dipartimento di Fisica and Unità INFN, Università di Roma Tor Vergata, 00133 Roma, Italy
5
DPMMS, Centre for Mathematical Sciences, University of Cambridge, Wilberforce road, Cambridge CB3 0WA, UK
E. Carlen et al.
1 Introduction This paper is a contribution to the theory of non-equilibrium steady states (NESS), of open systems in the particular context of kinetic theory. The understanding of NESS, their properties, uniqueness or lack thereof and stability or lack thereof, represents a challenge in mathematical physics due to the fact that the dynamics are nonlinear, non-Markovian and the absence of an entropy principle. Our main result is a uniqueness and stability theorem for the NESS in a simple nonlinear model.
1.1 The Model We briefly describe the sort of underlying particle model that would lead to the type of kinetic equation that we study here. It consists of a gas of particles on the one-dimensional torus T, that interact only through binary energy conserving collisions, however we also suppose that there are two types of scatterers distributed on the torus according to some Poisson distribution, as in a Lorentz model, except that each scatterer has a temperature, T1 or T2 depending on its type, and a certain radius of interaction, so that when a gas particle travels across the interaction interval, a Poisson clock runs and if it goes off, the particle assumes a new velocity chosen at random according to the Maxwellian distribution for the temper
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