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1 December 2017 On the Lagrangian structure of transport equations: The Vlasov–Poisson system
Luigi Ambrosio, Maria Colombo, Alessio Figalli
Duke Math. J. 166(18): 3505-3568 (1 December 2017). DOI: 10.1215/00127094-2017-0032


The Vlasov–Poisson system is an important nonlinear transport equation, used to describe the evolution of particles under their self-consistent electric or gravitational field. The existence of classical solutions is limited to dimensions d3 under strong assumptions on the initial data, whereas 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 article we develop several general tools concerning the Lagrangian structure of transport equations with nonsmooth vector fields, and we apply these results to show that weak/renormalized solutions of Vlasov–Poisson are Lagrangian and actually that the concepts of renormalized and Lagrangian solutions are equivalent. As a corollary, we prove that finite-energy solutions in dimension d4 are transported by a global flow (in particular, they preserve all the natural Casimir invariants), and we obtain the global existence of weak solutions in any dimension under minimal assumptions on the initial data.


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Luigi Ambrosio. Maria Colombo. Alessio Figalli. "On the Lagrangian structure of transport equations: The Vlasov–Poisson system." Duke Math. J. 166 (18) 3505 - 3568, 1 December 2017.


Received: 29 April 2016; Revised: 7 May 2017; Published: 1 December 2017
First available in Project Euclid: 8 September 2017

zbMATH: 06837466
MathSciNet: MR3732882
Digital Object Identifier: 10.1215/00127094-2017-0032

Primary: 35F25
Secondary: 34A12 , 35Q83 , 37C10

Keywords: Lagrangian flows , renormalized solutions , transport equations , Vlasov–Poisson

Rights: Copyright © 2017 Duke University Press


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Vol.166 • No. 18 • 1 December 2017
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