A Geometric approximation to the euler equations

The Vlasov-Monge- Ampère system

Y. Brenier, G. Loeper

Research output: Contribution to journalArticleResearchpeer-review

14 Citations (Scopus)

Abstract

This paper studies the Vlasov-Monge-Ampère system (VMA), a fully non-linear version of the Vlasov-Poisson system (VP) where the (real) Monge-Ampère equation det ∂2Ψ / ∂ xixj = ρ substitutes for the usual Poisson equation. This system can be derived as a geometric approximation of the Euler equations of incompressible fluid mechanics in the spirit of Arnold and Ebin. Global existence of weak solutions and local existence of smooth solutions are obtained. Links between the VMA system, the VP system and the Euler equations are established through rigorous asymptotic analysis.

Original languageEnglish
Pages (from-to)1182-1218
Number of pages37
JournalGeometric and Functional Analysis
Volume14
Issue number6
DOIs
Publication statusPublished - Dec 2004
Externally publishedYes

Cite this

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abstract = "This paper studies the Vlasov-Monge-Amp{\`e}re system (VMA), a fully non-linear version of the Vlasov-Poisson system (VP) where the (real) Monge-Amp{\`e}re equation det ∂2Ψ / ∂ xi∂xj = ρ substitutes for the usual Poisson equation. This system can be derived as a geometric approximation of the Euler equations of incompressible fluid mechanics in the spirit of Arnold and Ebin. Global existence of weak solutions and local existence of smooth solutions are obtained. Links between the VMA system, the VP system and the Euler equations are established through rigorous asymptotic analysis.",
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A Geometric approximation to the euler equations : The Vlasov-Monge- Ampère system. / Brenier, Y.; Loeper, G.

In: Geometric and Functional Analysis, Vol. 14, No. 6, 12.2004, p. 1182-1218.

Research output: Contribution to journalArticleResearchpeer-review

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AU - Loeper, G.

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AB - This paper studies the Vlasov-Monge-Ampère system (VMA), a fully non-linear version of the Vlasov-Poisson system (VP) where the (real) Monge-Ampère equation det ∂2Ψ / ∂ xi∂xj = ρ substitutes for the usual Poisson equation. This system can be derived as a geometric approximation of the Euler equations of incompressible fluid mechanics in the spirit of Arnold and Ebin. Global existence of weak solutions and local existence of smooth solutions are obtained. Links between the VMA system, the VP system and the Euler equations are established through rigorous asymptotic analysis.

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