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 d 3 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 d 4 are transported by a global flow (in particular, they pre- serve all the natural Casimir invariants), and we obtain the global existence of weak solutions in any dimension under minimal assumptions on the initial data.

On the Lagrangian structure of transport equations: The Vlasov–Poisson system

Ambrosio, Luigi
;
Colombo, Maria;
2017

Abstract

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 d 3 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 d 4 are transported by a global flow (in particular, they pre- serve all the natural Casimir invariants), and we obtain the global existence of weak solutions in any dimension under minimal assumptions on the initial data.
Settore MAT/05 - Analisi Matematica
File in questo prodotto:
File Dimensione Formato  
Vlasov-Poisson-final.pdf

accesso aperto

Tipologia: Submitted version (pre-print)
Licenza: Accesso gratuito (sola lettura)
Dimensione 1.06 MB
Formato Adobe PDF
1.06 MB Adobe PDF Visualizza/Apri
Duke_Math_J_2017.pdf

Accesso chiuso

Tipologia: Published version
Licenza: Non pubblico
Dimensione 425.29 kB
Formato Adobe PDF
425.29 kB Adobe PDF   Visualizza/Apri   Richiedi una copia

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11384/68936
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 7
  • ???jsp.display-item.citation.isi??? 8
social impact