{
  "abstract": "Averaging lemmas were introduced as a tool of the mathematical analysis of kinetic equations, i.e. PDEs for functions in phase space ( x , v ) containing a transport (“advection”) term. By integrating over v in velocity space ℝ v d (velocity averaging), one gains regularity for the density in position space ℝ x d . The concept was invented independently by V. I. Agoshkov and by F. Golse, B. Perthame, R. Sentis and P.-L. Lions, and successfully applied to the analysis of Vlasov or Boltzmann equations in classical kinetic theory. In quantum kinetic theory, the Schrödinger equation for the complex-valued wave function in the physical space is converted into the Wigner equation for the real-valued Wigner function in phase space (which can take negative values). The Wigner ( quantum Vlasov ) equation contains the transport term of classical kinetic equations plus a pseudo-differential operator containing the potential. We give answers to the long-standing question of whether and to which extent averaging lemmas apply to the quantum case of the Wigner equation. The hard part are the semi-classical averaging lemmas, where one considers the asymptotics of vanishing Planck constant towards the nonnegative Wigner measure. In that context pure vs. mixed states play a crucial role, as well as the connection between the Schrödinger equation and Quantum Hydrodynamics (QHD). We present the results for the classical and quantum cases and sketch the semi-classical case which is worked out in full detail in a follow-up article.",
  "abstract_shown": true,
  "affiliations": [
    "CMLS, École polytechnique, F-91128 Palaiseau, France",
    "Research platform MMM c/o Fak. Math. Univ. Wien, Oskar Morgenstern Platz 1, A-1090 Wien, Austria",
    "Wolfgang Pauli Inst. c/o Fak. Math. Univ. Wien, Oskar Morgenstern Platz 1, A-1090 Wien, Austria"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 3,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "François Golse"
    },
    {
      "affs": [
        1
      ],
      "name": "Norbert J. Mauser"
    },
    {
      "affs": [
        0,
        1,
        2
      ],
      "name": "Jakob Möller",
      "orcid": "0009-0003-4272-439X"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/brjdmmm9affz3.bib",
  "cited_by": 0,
  "doi": "10.5802/crmath.853",
  "fulltext_url": null,
  "id": "brjdmmm9affz3",
  "issns": [
    "1631-073X",
    "1778-3569"
  ],
  "landing_url": "https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.853/",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2604|Gas Dynamics and Kinetic Theory"
    ]
  ],
  "openalex": "W4416964993",
  "pdf_url": "https://comptes-rendus.academie-sciences.fr/mathematique/item/10.5802/crmath.853.pdf",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": "MathDoc/Centre Mersenne",
  "ris": "https://research.draconitic.com/paper/brjdmmm9affz3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:27Z",
      "native_id": "10.5802/crmath.853",
      "source": "crossref",
      "url": "https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.853/"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2604|Gas Dynamics and Kinetic Theory",
      "subfield": "general"
    }
  ],
  "title": "Velocity averaging lemmas: classical, quantum and semi-classical",
  "title_text": "Velocity averaging lemmas: classical, quantum and semi-classical",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:18:27Z",
  "url": "https://research.draconitic.com/paper/brjdmmm9affz3",
  "venue": "Comptes Rendus. Mathématique",
  "versions": []
}