{
  "id": "2208.09755",
  "version": "v1",
  "published": "2022-08-20T22:20:15.000Z",
  "updated": "2022-08-20T22:20:15.000Z",
  "title": "Global Dynamics and Photon Loss in the Kompaneets Equation",
  "authors": [
    "Joshua Ballew",
    "Gautam Iyer",
    "C. David Levermore",
    "Hailiang Liu",
    "Robert L. Pego"
  ],
  "comment": "36 pages, 3 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "The Kompaneets equation governs dynamics of the photon energy spectrum in certain high temperature (or low density) plasmas. We prove several results concerning the long-time convergence of solutions to Bose--Einstein equilibria and the failure of photon conservation. In particular, we show the total photon number can decrease with time via an outflux of photons at the zero-energy boundary. The ensuing accumulation of photons at zero energy is analogous to Bose--Einstein condensation. We provide two conditions that guarantee that photon loss occurs, and show that once loss is initiated then it persists forever. We prove that as $t\\to \\infty$, solutions necessarily converge to equilibrium and we characterize the limit in terms of the total photon loss. Additionally, we provide a few results concerning the behavior of the solution near the zero-energy boundary, an Oleinik inequality, a comparison principle, and show that the solution operator is a contraction in $L^1$. None of these results impose a boundary condition at the zero-energy boundary.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-20T22:20:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B40",
      "35K65",
      "35Q85"
    ],
    "keywords": [
      "global dynamics",
      "zero-energy boundary",
      "kompaneets equation governs dynamics",
      "total photon loss",
      "total photon number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}