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The Hartree and Vlasov equations at positive density

Lewin, Mathieu; Sabin, Julien (2020), The Hartree and Vlasov equations at positive density, Communications in Partial Differential Equations, 45, 12, p. 1702-1754. 10.1080/03605302.2020.1803355

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Type
Article accepté pour publication ou publié
Date
2020
Nom de la revue
Communications in Partial Differential Equations
Volume
45
Numéro
12
Éditeur
Taylor & Francis
Pages
1702-1754
Identifiant publication
Métadonnées
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Auteur(s)
Lewin, Mathieu cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Sabin, Julien
Laboratoire de Mathématiques d'Orsay [LMO]
Résumé (EN)
We consider the nonlinear Hartree and Vlasov equations around a translation-invariant (homogeneous) stationary state in infinite volume, for a short range interaction potential. For both models, we consider time-dependent solutions which have a finite relative energy with respect to the reference translation-invariant state. We prove the convergence of the Hartree solutions to the Vlasov ones in a semi-classical limit and obtain as a by-product global well-posedness of the Vlasov equation in the (relative) energy space.
Mots-clés
Hartree equation; positive density; semiclassical analysis