On the $L^2$ rate of convergence in the limit from the Hartree to the Vlasov–Poisson equation

Jacky J. Chong; Laurent Lafleche; Chiara Saffirio

doi:10.5802/jep.230

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]

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

Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 703-726.

Résumé
Abstract

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.

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.

Reçu le : 2022-04-08
Accepté le : 2023-03-20
Publié le : 2023-04-07

DOI : 10.5802/jep.230

Classification : 81Q20, 35Q55, 35Q83, 82C10, 82C05
Keywords: Semiclassical limit, Hartree equation, Vlasov equation, Coulomb potential, gravitational potential.
Mot clés : Limite semi-classique, équation de Hartree, équation de Vlasov, potentiel de Coulomb, potentiel gravitationnel

Affiliations des auteurs :

Jacky J. Chong ¹ ; Laurent Lafleche ² ; Chiara Saffirio 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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Jacky J. Chong; Laurent Lafleche; Chiara Saffirio. 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. https://jep.centre-mersenne.org/articles/10.5802/jep.230/

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