Asymptotic Behavior of Degenerate Linear Kinetic Equations with Non-Isothermal Boundary Conditions
Feb 1, 2025·
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1 min read
Armand Bernou
Abstract
We study the degenerate linear Boltzmann equation inside a bounded domain with the Maxwell and the Cercignani-Lampis boundary conditions, two generalizations of the diffuse reflection, with variable temperature. This includes a model of relaxation towards a space-dependent steady state. For both boundary conditions, we prove for the first time the existence of a steady state and a rate of convergence towards it without assumptions on the temperature variations. Our results for the Cercignani-Lampis boundary condition make also no hypotheses on the accommodation coefficients. The proven rate is exponential when a control condition on the degeneracy of the collision operator is satisfied, and only polynomial when this assumption is not met, in line with our previous results regarding the free-transport equation. We also provide a precise description of the different convergence rates, including lower bounds, when the steady state is bounded. Our method yields constructive constants.
Type
Publication
Journal of Differential Equations, in press
This article is part of a larger project, in which I study rigorously kinetic equations in a domain with boundary conditions and non-variable temperature, with a particular emphasis on the Cercignani-Lampis boundary condition. This latter model for the reflection of gas particles at the boundary has gained popularity in the physics community following recent experiments (see references in the paper). After the study of the free-transport setting, this paper extends to collisional models, and in particular to the degenerate linear Boltzmann equations, and provide proofs of existence of non-equilibrium steady states along with a rate of convergence towards them.
