On the Lagrangian structure of transport equations: the Vlasov-Poisson system

TL;DR

This paper investigates the Lagrangian structure of the Vlasov-Poisson system, demonstrating that weak solutions are Lagrangian and establishing global existence under minimal initial data assumptions.

Contribution

It develops tools for the Lagrangian analysis of transport equations and applies them to prove weak solutions of Vlasov-Poisson are Lagrangian and globally existing.

Findings

Weak solutions are Lagrangian

Global existence under minimal assumptions

Tools for Lagrangian analysis of transport equations

Abstract

The Vlasov-Poisson system is a classical model in physics used to describe the evolution of particles under their self-consistent electric or gravitational field. The existence of classical solutions is limited to dimensions $d\leq 3$ under strong assumptions on the initial data, while weak solutions are known to exist under milder conditions. However, in the setting of weak solutions it is unclear whether the Eulerian description provided by the equation physically corresponds to a Lagrangian evolution of the particles. In this paper we develop several general tools concerning the Lagrangian structure of transport equations with non-smooth vector fields and we apply these results: (1) to show that weak solutions of Vlasov-Poisson are Lagrangian; (2) to obtain global existence of weak solutions under minimal assumptions on the initial data.