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 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 . This improves the rate of convergence in 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 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 . Ceci améliore le taux de convergence dans 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 corps vers l’équation de Hartree-Fock pour des états mixtes.
Accepted:
Published online:
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
Jacky J. Chong 1; Laurent Lafleche 2; Chiara Saffirio 3
@article{JEP_2023__10__703_0, author = {Jacky J. Chong and Laurent Lafleche and Chiara Saffirio}, 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}, pages = {703--726}, publisher = {\'Ecole polytechnique}, volume = {10}, year = {2023}, doi = {10.5802/jep.230}, language = {en}, url = {https://jep.centre-mersenne.org/articles/10.5802/jep.230/} }
TY - JOUR AU - Jacky J. Chong AU - Laurent Lafleche AU - Chiara Saffirio TI - On the $L^2$ rate of convergence in the limit from the Hartree to the Vlasov–Poisson equation JO - Journal de l’École polytechnique — Mathématiques PY - 2023 SP - 703 EP - 726 VL - 10 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.230/ DO - 10.5802/jep.230 LA - en ID - JEP_2023__10__703_0 ER -
%0 Journal Article %A Jacky J. Chong %A Laurent Lafleche %A Chiara Saffirio %T On the $L^2$ rate of convergence in the limit from the Hartree to the Vlasov–Poisson equation %J Journal de l’École polytechnique — Mathématiques %D 2023 %P 703-726 %V 10 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.230/ %R 10.5802/jep.230 %G en %F JEP_2023__10__703_0
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, Volume 10 (2023), pp. 703-726. doi : 10.5802/jep.230. https://jep.centre-mersenne.org/articles/10.5802/jep.230/
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