The Hartree and Vlasov equations at positive density. Issue 12 (9th September 2020)

• Record Type: Journal Article
• Authors: Lewin, Mathieu; Sabin, Julien
• Is Part Of: Communications in partial differential equations. Volume 45:Issue 12(2020)
• Page Start: 1702
• Page End: 1754
• Publication Date: 2020-09-09
• Subjects: Hartree equation -- positive density -- semiclassical analysis; Differential equations, Partial -- Periodicals; 515.353
• DOI: 10.1080/03605302.2020.1803355
• Languages: English
• ISSNs: 0360-5302

Abstract

Abstract: 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.