Geometric Analysis of the Linear Boltzmann Equation I. Trend to Equilibrium

Abstract

Abstract. This work is devoted to the analysis of the linear Boltzmann equation in a bounded domain, in the presence of a force deriving from a potential. The collision operator is allowed to be degenerate in the following two senses: (1) the associated collision kernel may vanish in a large subset of the phase space; (2) we do not assume that it is bounded below by a Maxwellian at infinity in velocity.We study how the association of transport and collision phenomena can lead to convergence to equilibrium, using concepts and ideas from control theory. We prove two main classes of results. On the one hand, we show that convergence towards an equilibrium is equivalent to an almost everywhere geometric control condition. The equilibria (which are not necessarily Maxwellians with our general assumptions on the collision kernel) are described in terms of the equivalence classes of an appropriate equivalence relation. On the other hand, we characterize the exponential convergence to equilibrium in terms of the Lebeau constant, which involves some averages of the collision frequency along the flow of the transport. We handle several cases of phase spaces, including those associated to specular reflection in a bounded domain, or to a compact Riemannian manifold.

Trend to equilibrium and spectral localization properties for the linear Boltzmann equation

“…The aim of this note is to present the results from [11,12], which deal with the linear Boltzmann equation, set in a bounded domain and in the presence of an external force. A specificity of these works is that the collision operator is allowed to be degenerate in the following two senses: (1) the associated collision kernel may vanish in a large subset of the phase space; (2) we do not assume that it is bounded below by a Maxwellian at infinity in velocity.…”

“…We study: -the large time behavior of solutions of the linear Boltzmann equation, by giving criteria (inspired from control theory) which ensure converge towards an equilibrium and when possible, convergence at an exponential rate [11] ; -some properties of localization for the spectrum of the associated operator [12].…”

“…Résumé (Relaxation vers l'équilibre et propriétés de localisation spectrale pour l'équation de Boltzmann linéaire) L'objectif de cette note est de présenter les résultats de [11,12], qui concernent l'équation de Boltzmann linéaire, posée dans un domaine borné et en présence d'une force extérieure. Une spécificité de ces travaux réside dans la prise en compte d'opé-rateurs de collision dégénérés aux deux sens suivants : (1) le noyau de collision associé peut s'annuler sur un grand sous-ensemble de l'espace des phases ; (2) le noyau de collision n'est pas supposé être minoré par une Maxwellienne à l'infini en vitesse.…”

“…Nous étudions : -le comportement en temps grand des solutions l'équation de Boltzmann linéaire, en donnant des critères (inspirés par la théorie du contrôle) pour assurer la convergence vers un équilibre et quand cela est possible, convergence à un taux exponentiel [11] ; -les propriétés de localisation du spectre de l'opérateur associé [12].…”

“…We actually handle somek which are for instance polynomially growing in the v and v variables, see [11], but we stick to this simple assumption for the sake of clarity of exposition.…”