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Pré-Publication, Document De Travail Année : 2023

Asymptotic Behavior of Degenerate Linear Kinetic Equations with Non-Isothermal Boundary Conditions

Résumé

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.
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Dates et versions

hal-04279269 , version 1 (10-11-2023)

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Armand Bernou. Asymptotic Behavior of Degenerate Linear Kinetic Equations with Non-Isothermal Boundary Conditions. 2023. ⟨hal-04279269⟩

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UNIV-AMU CIVIS3I
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