Global Semiclassical Limit from Hartree to Vlasov Equation for Concentrated Initial Data

Laurent Lafleche (1, 2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 CMLS - Centre de Mathématiques Laurent Schwartz

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 ≥ 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 semiclassical bounds on the space and velocity moments of even order and the asymptotic behaviour of the spatial density due to dispersion effects.

Mots clés: Hartree equation; Nonlinear Schrödinger equation; Vlasov equation; Coulomb interaction; gravitational interaction; semiclassical limit

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]; Physique Quantique [quant-ph]

Laurent Lafleche. Global Semiclassical Limit from Hartree to Vlasov Equation for Concentrated Initial Data. 2019. ⟨hal-02046481v1⟩