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

TL;DR

This paper analyzes the long-term behavior of degenerate linear kinetic equations with non-uniform boundary conditions, establishing existence, uniqueness, and convergence rates of steady states under broad assumptions.

Contribution

It proves for the first time the existence of steady states and convergence rates for degenerate linear Boltzmann equations with general boundary conditions without temperature assumptions.

Findings

Existence of steady states established.

Exponential convergence under certain conditions.

Polynomial convergence when degeneracy conditions are not met.

Abstract

We study the degenerate linear Boltzmann equation inside a bounded domain with a generalized diffuse reflection at the boundary and variable temperature, including the Maxwell boundary conditions with the wall Maxwellian or heavy-tailed reflection kernel and the Cercignani-Lampis boundary condition. Our abstract collisional setting applies to the linear BGK model, the relaxation towards a space-dependent steady state, and collision kernels with fat tails. 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.