The Hartree and Vlasov equations at positive density

Mathieu Lewin (1, 2) , Julien Sabin (3, 4)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 PSL - Université Paris Sciences et Lettres
3 LMO - Laboratoire de Mathématiques d'Orsay
4 Université Paris-Saclay

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.

Domaines: Physique mathématique [math-ph]

Mathieu Lewin, Julien Sabin. The Hartree and Vlasov equations at positive density. Communications in Partial Differential Equations, 2020, 45 (12), pp.1702--1754. ⟨10.1080/03605302.2020.1803355⟩. ⟨hal-02964229⟩