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.

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 . This improves the 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 . 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.

Received:
Accepted:
Published online:
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

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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@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/

[1] L. Amour, M. Khodja & J. Nourrigat - “The semiclassical limit of the time dependent Hartree-Fock equation: the Weyl symbol of the solution”, Anal. PDE 6 (2013) no. 7, p. 1649-1674 | DOI | MR | Zbl

[2] A. Athanassoulis, T. Paul, F. Pezzotti & M. Pulvirenti - “Strong semiclassical approximation of Wigner functions for the Hartree dynamics”, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 22 (2011) no. 4, p. 525-552 | DOI | MR | Zbl

[3] N. Benedikter, M. Porta, C. Saffirio & B. Schlein - “From the Hartree dynamics to the Vlasov equation”, Arch. Rational Mech. Anal. 221 (2016) no. 1, p. 273-334 | DOI | MR | Zbl

[4] N. Benedikter, M. Porta & B. Schlein - “Mean-field regime for fermionic systems”, in Effective evolution equations from quantum dynamics, Springer, Cham, 2016, p. 57-78 | DOI

[5] J. Bergh & J. Löfström - Interpolation spaces. An introduction, Grundlehren Math. Wiss., vol. 223, Springer, Berlin, Heidelberg, 1976 | DOI

[6] F. Castella - “L 2 - solutions to the Schrödinger-Poisson system: existence, uniqueness, time behaviour, and smoothing effects”, Math. Models Methods Appl. Sci. 7 (1997) no. 8, p. 1051-1083 | DOI | Zbl

[7] L. Chen, J. Lee & M. Liew - “Combined mean-field and semiclassical limits of large fermionic systems”, J. Statist. Phys. 182 (2021) no. 2, p. 24 | DOI | MR | Zbl

[8] J. J. Chong, L. Lafleche & C. Saffirio - “From many-body quantum dynamics to the Hartree-Fock and Vlasov equations with singular potentials”, 2021 | arXiv

[9] J. J. Chong, L. Lafleche & C. Saffirio - “Global-in-time semiclassical regularity for the Hartree-Fock equation”, J. Math. Phys. 63 (2022) no. 8, article ID 081904, 9 pages | DOI | MR | Zbl

[10] A. Figalli, M. Ligabò & T. Paul - “Semiclassical limit for mixed states with singular and rough potentials”, Indiana Univ. Math. J. 61 (2012) no. 1, p. 193-222 | DOI | MR | Zbl

[11] F. Golse & T. Paul - “The Schrödinger equation in the mean-field and semiclassical regime”, Arch. Rational Mech. Anal. 223 (2017) no. 1, p. 57-94 | DOI | Zbl

[12] F. Golse & T. Paul - “Mean-field and classical limit for the N-body quantum dynamics with Coulomb interaction”, Comm. Pure Appl. Math. (2021), p. 1-35 | DOI

[13] S. Graffi, A. Martinez & M. Pulvirenti - “Mean-field approximation of quantum systems and classical limit”, Math. Models Methods Appl. Sci. 13 (2003) no. 1, p. 59-73 | DOI | MR

[14] R. A. Hunt - “On L(p,q) spaces”, Enseign. Math. 12 (1966), p. 249-276 | DOI | MR | Zbl

[15] L. Lafleche - “Propagation of moments and semiclassical limit from Hartree to Vlasov equation”, J. Statist. Phys. 177 (2019) no. 1, p. 20-60 | DOI | MR | Zbl

[16] L. Lafleche - “Global semiclassical limit from Hartree to Vlasov equation for concentrated initial data”, Ann. Inst. H. Poincaré C Anal. Non Linéaire 38 (2021) no. 6, p. 1739-1762 | DOI | MR | Zbl

[17] L. Lafleche & C. Saffirio - “Strong semiclassical limit from Hartree and Hartree-Fock to Vlasov-Poisson equation”, Anal. PDE (2021), to appear

[18] N. Lerner & Y. Morimoto - “On the Fefferman-Phong inequality and a Wiener-type algebra of pseudodifferential operators”, Publ. RIMS, Kyoto Univ. 43 (2007) no. 2, p. 329-371 | DOI | MR | Zbl

[19] M. Lewin & J. Sabin - “The Hartree and Vlasov equations at positive density”, Comm. Partial Differential Equations 45 (2020) no. 12, p. 1702-1754 | DOI | MR | Zbl

[20] P.-L. Lions & T. Paul - “Sur les mesures de Wigner”, Rev. Mat. Iberoamericana 9 (1993) no. 3, p. 553-618 | DOI | MR | Zbl

[21] P.-L. Lions & B. Perthame - “Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system”, Invent. Math. 105 (1991) no. 2, p. 415-430 | DOI | MR

[22] P. A. Markowich & N. J. Mauser - “The classical limit of a self-consistent quantum Vlasov equation”, Math. Models Methods Appl. Sci. 3 (1993) no. 1, p. 109-124 | DOI | MR | Zbl

[23] H. Narnhofer & G. L. Sewell - “Vlasov hydrodynamics of a quantum mechanical model”, Comm. Math. Phys. 79 (1981) no. 1, p. 9-24 | DOI | MR

[24] K. Pfaffelmoser - “Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data”, J. Differential Equations 95 (1992) no. 2, p. 281-303 | DOI | MR | Zbl

[25] R. T. Powers & E. Størmer - “Free states of the canonical anticommutation relations”, Comm. Math. Phys. 16 (1970) no. 1, p. 1-33 | DOI | MR | Zbl

[26] C. Saffirio - “Semiclassical limit to the Vlasov equation with inverse power law potentials”, Comm. Math. Phys. 373 (2019) no. 2, p. 571-619 | DOI | MR | Zbl

[27] C. Saffirio - “From the Hartree equation to the Vlasov-Poisson system: strong convergence for a class of mixed states”, SIAM J. Math. Anal. 52 (2020) no. 6, p. 5533-5553 | DOI | MR | Zbl

[28] H. Spohn - “On the Vlasov hierarchy”, Math. Methods Appl. Sci. 3 (1981) no. 1, p. 445-455 | DOI | MR | Zbl

Cited by Sources: