Skip to content
DraconiticResearch
Search Sources API
184,937 studies31 sourcesUpdated 46 min ago
Peer-reviewedCC BY 4.0Comptes Rendus. Mathématique

Velocity averaging lemmas: classical, quantum and semi-classical

  • François Golse1
  • Norbert J. Mauser2
  • Jakob Möller1,2,3
  1. 1CMLS, École polytechnique, F-91128 Palaiseau, France
  2. 2Research platform MMM c/o Fak. Math. Univ. Wien, Oskar Morgenstern Platz 1, A-1090 Wien, Austria
  3. 3Wolfgang Pauli Inst. c/o Fak. Math. Univ. Wien, Oskar Morgenstern Platz 1, A-1090 Wien, Austria
Open at sourcePDFDOICite

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.

Related

  1. Velocity Averaging for the Wigner Equation with Electromagnetic Fields
    François Golse, Jakob Möller
    arXivPreprintarXiv licenceFull textAnalysis & PDEsMathematical physicsarXiv
  2. Extremal projectors in the semi-classical case
    Sophie Chemla
    Annales de l'Institut FourierPeer-reviewedLicence not statedFull textGeometry & topologyHAL
  3. Global classical solutions for the general planar Broadwell model with arbitrary velocity orientation: a fixed-point framework
    Koudzo Togbévi Selom Sobah, Amah Séna d'Almeida
    arXivPreprintCC BY-NC-ND 4.0Full textAnalysis & PDEsarXiv
  4. Quasi-exponentials and semi-classical limits for the chordal BPZ system
    Mo Chen, Hao Wu
    arXivPreprintarXiv licenceFull textMathematical physicsProbabilityarXiv
  5. Semi-classical Szegő kernel asymptotics on CR manifolds of high codimension
    Chin-Yu Hsiao, Chin Jui Yang
    arXivPreprintarXiv licenceFull textAnalysis & PDEsGeometry & topologyarXiv

Cite

BibTeXRIS
BibTeX
@article{golse2026velocity,
  title = {{Velocity averaging lemmas: classical, quantum and semi-classical}},
  author = {François Golse and Norbert J. Mauser and Jakob Möller},
  journal = {Comptes Rendus. Mathématique},
  year = {2026},
  month = oct,
  doi = {10.5802/crmath.853},
  url = {https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.853/},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Velocity averaging lemmas: classical, quantum and semi-classical
AU  - François Golse
AU  - Norbert J. Mauser
AU  - Jakob Möller
PY  - 2026
DA  - 2026/10/20
JO  - Comptes Rendus. Mathématique
DO  - 10.5802/crmath.853
UR  - https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.853/
AB  - 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.
SN  - 1631-073X
SN  - 1778-3569
ER  -