Asymptotic Behavior of Degenerate Linear Kinetic Equations with Non-Isothermal Boundary Conditions
arXiv:2308.01694
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.
37 pages, 1 table. Comments are welcome. Extended with a generalized Maxwell boundary condition encompassing (in addition to the classical wall Maxwellian) heavy-tailed reflection kernels at the boundary, and with a heavy-tailed collisional kernel inside the domain