The Hartree and Vlasov equations at positive density

TL;DR

This paper studies the relationship between the Hartree and Vlasov equations at positive density, proving convergence in a semi-classical limit and establishing global well-posedness in energy space.

Contribution

It demonstrates the convergence of Hartree solutions to Vlasov solutions in a semi-classical limit and proves global well-posedness for the Vlasov equation in energy space.

Findings

Hartree solutions converge to Vlasov solutions in the semi-classical limit

Global well-posedness of the Vlasov equation in energy space

Analysis around translation-invariant stationary states

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.