Global semiclassical limit from Hartree to Vlasov equation for concentrated initial data

Lafleche, Laurent

doi:10.1016/j.anihpc.2021.01.004

Lafleche, Laurent ^1,²

¹ a CEREMADE, UMR CNRS 7534, Université Paris-Dauphine, PSL Research University, Place du Maréchal de Lattre de Tassigny, 75775 Paris cedex 16, France
² b CMLS, École polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau cedex, France

Annales de l'I.H.P. Analyse non linéaire, novembre – décembre 2021, Tome 38 (2021) no. 6, pp. 1739-1762.

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Résumé

We prove a quantitative and global in time semiclassical limit from the Hartree to the Vlasov equation in the case of a singular interaction potential in dimension $d \geq 3$ , including the case of a Coulomb singularity in dimension $d = 3$ . This result holds for initial data concentrated enough in the sense that some space moments are initially sufficiently small. As an intermediate result, we also obtain quantitative bounds on the space and velocity moments of even order and the asymptotic behavior of the spatial density due to dispersion effects, uniform in the Planck constant ħ.

DOI : 10.1016/j.anihpc.2021.01.004

Classification : 82C10, 35Q41, 35Q55, 82C05, 35Q83
Mots-clés : Hartree equation, Nonlinear Schrödinger equation, Vlasov equation, Coulomb interaction, Gravitational interaction, Semiclassical limit

Affiliations des auteurs :

Lafleche, Laurent ^{1, 2}

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     title = {Global semiclassical limit from {Hartree} to {Vlasov} equation for concentrated initial data},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1739--1762},
     publisher = {Elsevier},
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Lafleche, Laurent. Global semiclassical limit from Hartree to Vlasov equation for concentrated initial data. Annales de l'I.H.P. Analyse non linéaire, novembre – décembre 2021, Tome 38 (2021) no. 6, pp. 1739-1762. doi : 10.1016/j.anihpc.2021.01.004. https://www.numdam.org/articles/10.1016/j.anihpc.2021.01.004/

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