{
  "id": "2308.01694",
  "version": "v1",
  "published": "2023-08-03T11:24:21.000Z",
  "updated": "2023-08-03T11:24:21.000Z",
  "title": "Asymptotic Behavior of Degenerate Linear Kinetic Equations with Non-Isothermal Boundary Conditions",
  "authors": [
    "Armand Bernou"
  ],
  "comment": "35 pages, 1 table. Comments are welcome",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-03T11:24:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B40",
      "35Q20",
      "82C40",
      "82D05"
    ],
    "keywords": [
      "degenerate linear kinetic equations",
      "non-isothermal boundary conditions",
      "asymptotic behavior",
      "steady state",
      "cercignani-lampis boundary condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}