From the Hartree equation to the Vlasov-Poisson system: strong convergence for a class of mixed states

We consider the evolution of $N$ fermions interacting through a Coulomb or gravitational potential in the mean-field limit as governed by the nonlinear Hartree equation with Coulomb or gravitational interaction. In the limit of large $N$, we study the convergence in trace norm towards the classical Vlasov-Poisson equation for a special class of mixed quasi-free states.Comment: 21 pages. Typos corrected, references updated and detailed proof of Lemma 2.4 adde

From microscopic dynamics to kinetic equations

Starting from a system of N particles at a microscopic scale, we describe different scaling limits which lead to kinetic equations in a macroscopic regime: the low-density limit, the weak-coupling limit, the grazing collision limit and the mean-field limit. A particular relevance is given to the rigorous derivation of the Boltzmann equation (starting from a system of N particles interacting via a short range potential) and to a consistency result concerning the Landau equation. A Kac's model for the Landau equation is presented as well. The last part of the work is dedicated to the Vlasov-Poisson system, in particular we discuss the Cauchy problem related to this equation in presence of a point charge

On the validity of the Boltzmann equation for short range potentials

We consider a classical system of point particles interacting by means of a short range potential. We prove that, in the low--density (Boltzmann--Grad) limit, the system behaves, for short times, as predicted by the associated Boltzmann equation. This is a revisitation and an extension of the thesis of King (unpublished), appeared after the well known result of Lanford for hard spheres, and of a recent paper by Gallagher et al (arXiv: 1208.5753v1). Our analysis applies to any stable and smooth potential. In the case of repulsive potentials (with no attractive parts), we estimate explicitly the rate of convergence

Polynomial propagation of moments and global existence for a Vlasov-Poisson system with a point charge

In this paper, we extend to the case of initial data constituted of a Dirac mass plus a bounded density (with finite moments) the theory of Lions and Perthame [6] for the Vlasov-Poisson equation. Our techniques also provide polynomially growing in time estimates for moments of the bounded density.Comment: 27 pages; new version: few typos have been corrected, the introduction has been modifie