{
  "id": "2501.07393",
  "version": "v2",
  "published": "2025-01-13T15:08:40.000Z",
  "updated": "2025-01-24T07:11:09.000Z",
  "title": "On the axially symmetric solutions to the spatially homogeneous Landau equation",
  "authors": [
    "Jin Woo Jang",
    "Junha Kim"
  ],
  "comment": "22 pages. This version includes additional references on the grazing collision limit and the semi-classical limit",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we consider the spatially homogeneous Landau equation, which is a variation of the Boltzmann equation in the grazing collision limit. For the Landau equation for hard potentials in the style of Desvillettes-Villani (Comm. Partial Differential Equations, 2000), we provide the proof of the existence of axisymmetric measure-valued solution for any axisymmetric $\\mathcal{P}_p(\\mathbb{R}^3)$ initial profile for any $p\\ge 2$. Moreover, we prove that if the initial data is not a single Dirac mass, then the solution instantaneously becomes analytic for any time $t>0$ in the hard potential case. In the soft potential and the Maxwellian molecule cases, we show that there are no solutions whose support is contained in a fixed line even for any given line-concentrated data.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2025-01-24T07:11:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q20"
    ],
    "keywords": [
      "spatially homogeneous landau equation",
      "axially symmetric solutions",
      "partial differential equations",
      "single dirac mass",
      "hard potential case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}