Title: Global existence of classical solutions for a Vlasov-Schrodinger-Poisson system

Authors: Naoufel Ben Abdallah, Florian Mehats and Geraldine Quinio

Issue: Volume 55 (2006), Issue 4, 1423-1448

Abstract: Global existence and uniqueness of a classical solution of the two dimensional Vlasov equation coupled to the Schr\"odinger-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 Schr\"odinger-Poisson system by convex minimization.