Global existence of classical solutions for a Vlasov-Schrödinger-Poisson system

Naoufel Ben Abdallah (1) , Florian Méhats (1, 2) , Géraldine Quinio (1)

1 MIP - Mathématiques pour l'Industrie et la Physique
2 IRMAR - Institut de Recherche Mathématique de Rennes

Global existence and uniqueness of a classical solution of the two dimensional Vlasov equation coupled to the Schrödinger-Poisson system is proven. The two dimensional driving forces appearing in the Vlasov equation are deduced from the electrostatic potential in the Born-Oppenheimer approximation and take into account the quantum behaviour in the third direction. The electrostatic potential is computed by solving a three dimensional Poisson equation. The existence and uniqueness of the solution is proven by a fixed point argument on the Vlasov equation. It relies on the use of an a priori energy estimate, and on the resolution of the Schrödinger-Poisson system by convex minimization.

Domaines: Equations aux dérivées partielles [math.AP]

Naoufel Ben Abdallah, Florian Méhats, Géraldine Quinio. Global existence of classical solutions for a Vlasov-Schrödinger-Poisson system. Indiana University Mathematics Journal, 2006, 55 (4), pp.1423-1448. ⟨10.1512/iumj.2006.55.2659⟩. ⟨hal-00378541⟩