Calculus of Variations and Geometric Measure Theory
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L. Ambrosio - M. Colombo - A. Figalli

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

created by colombom on 11 Dec 2014
modified by ambrosio on 16 Jul 2015


Submitted Paper

Inserted: 11 dec 2014
Last Updated: 16 jul 2015

Year: 2014


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.

Tags: GeMeThNES
Keywords: Renormalized solutions, Transport equations, Vlasov-Poisson system, Lagrangian flows


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