Asymptotic Behavior of Degenerate Linear Kinetic Equations with Non-Isothermal Boundary Conditions
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2023
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866912164186423296 |
|---|---|
| author | Bernou, Armand |
| author_facet | Bernou, Armand |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_01694 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Asymptotic Behavior of Degenerate Linear Kinetic Equations with Non-Isothermal Boundary Conditions Bernou, Armand Analysis of PDEs Mathematical Physics 35B40, 35Q20 (82C40, 82D05) 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. |
| title | Asymptotic Behavior of Degenerate Linear Kinetic Equations with Non-Isothermal Boundary Conditions |
| topic | Analysis of PDEs Mathematical Physics 35B40, 35Q20 (82C40, 82D05) |
| url | https://arxiv.org/abs/2308.01694 |