On the L 2 rate of convergence in the limit from the Hartree to the Vlasov–Poisson equation
[Sur le taux de convergence dans L 2 dans la limite de l’équation de Hartree à l’équation de Vlasov-Poisson]
Chong, Jacky J.1 ; Lafleche, Laurent2 ; Saffirio, Chiara3
1 Department of Mathematics, The University of Texas at Austin Austin, TX 78712, USA & Department of Mathematics, School of Mathematical Sciences, Peking University Beijing, China
2 Department of Mathematics, The University of Texas at Austin Austin, TX 78712, USA & Institut Camille Jordan, CNRS, Université Claude Bernard Lyon 1 43 boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France
3 Department of Mathematics and Computer Science, University of Basel 4051 Basel, Switzerland
Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 703-726

Using a new stability estimate for the difference of the square roots of two solutions of the Vlasov–Poisson equation, we obtain the convergence in the L 2 norm of the Wigner transform of a solution of the Hartree equation with Coulomb potential to a solution of the Vlasov–Poisson equation, with a rate of convergence proportional to ℏ. This improves the ℏ 3/4-ε rate of convergence in L 2 obtained in [L. Lafleche, C. Saffirio: Analysis & PDE, to appear]. Another reason of interest of this paper is the new method, reminiscent of the ones used to prove the mean-field limit from the many-body Schrödinger equation towards the Hartree–Fock equation for mixed states.

Grâce à une nouvelle estimée de stabilité pour la différence entre les racines carrées de deux solutions de l’équation de Vlasov-Poisson, nous obtenons la convergence en norme L 2 de la transformée de Wigner d’une solution de l’équation de Hartree avec potentiel de Coulomb vers une solution de l’équation de Vlasov-Poisson, avec un taux de convergence proportionnel à la constante de Planck ℏ. Ceci améliore le taux de convergence h 3/4-ε dans L 2 obtenu dans [L. Lafleche, C. Saffirio : Analysis & PDE, à paraître]. Un autre intérêt de cet article est la nouvelle méthode, réminiscente de celles utilisées pour prouver la limite de champ moyen de l’équation de Schrödinger à N corps vers l’équation de Hartree-Fock pour des états mixtes.

Reçu le :
Accepté le :
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DOI : 10.5802/jep.230
Classification : 81Q20, 35Q55, 35Q83, 82C10, 82C05
Keywords: Semiclassical limit, Hartree equation, Vlasov equation, Coulomb potential, gravitational potential.
Mots-clés : Limite semi-classique, équation de Hartree, équation de Vlasov, potentiel de Coulomb, potentiel gravitationnel

Chong, Jacky J. 1 ; Lafleche, Laurent 2 ; Saffirio, Chiara 3

1 Department of Mathematics, The University of Texas at Austin Austin, TX 78712, USA & Department of Mathematics, School of Mathematical Sciences, Peking University Beijing, China
2 Department of Mathematics, The University of Texas at Austin Austin, TX 78712, USA & Institut Camille Jordan, CNRS, Université Claude Bernard Lyon 1 43 boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France
3 Department of Mathematics and Computer Science, University of Basel 4051 Basel, Switzerland
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {On the $L^2$ rate of convergence in the limit from the {Hartree} to the {Vlasov{\textendash}Poisson} equation},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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Chong, Jacky J.; Lafleche, Laurent; Saffirio, Chiara. On the $L^2$ rate of convergence in the limit from the Hartree to the Vlasov–Poisson equation. Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 703-726. doi: 10.5802/jep.230

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