A Geometric approximation to the euler equations: The Vlasov-Monge- Ampère system

Y. Brenier; G. Loeper

doi:10.1007/s00039-004-0488-1

A Geometric approximation to the euler equations: The Vlasov-Monge- Ampère system

Y. Brenier, G. Loeper

Research output: Contribution to journal › Article › Research › peer-review

20 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 / ∂ _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.

Original language	English
Pages (from-to)	1182-1218
Number of pages	37
Journal	Geometric and Functional Analysis
Volume	14
Issue number	6
DOIs	https://doi.org/10.1007/s00039-004-0488-1
Publication status	Published - Dec 2004
Externally published	Yes

Access to Document

10.1007/s00039-004-0488-1

Cite this

@article{72ac939ccbd44d53bb4002a6c9afcf17,

title = "A Geometric approximation to the euler equations: The Vlasov-Monge- Amp{\`e}re system",

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.",

author = "Y. Brenier and G. Loeper",

year = "2004",

month = dec,

doi = "10.1007/s00039-004-0488-1",

language = "English",

volume = "14",

pages = "1182--1218",

journal = "Geometric and Functional Analysis",

issn = "1016-443X",

publisher = "Birkhauser",

number = "6",

}

TY - JOUR

T1 - A Geometric approximation to the euler equations

T2 - The Vlasov-Monge- Ampère system

AU - Brenier, Y.

AU - Loeper, G.

PY - 2004/12

Y1 - 2004/12

N2 - 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.

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.

UR - http://www.scopus.com/inward/record.url?scp=16244390838&partnerID=8YFLogxK

U2 - 10.1007/s00039-004-0488-1

DO - 10.1007/s00039-004-0488-1

M3 - Article

AN - SCOPUS:16244390838

SN - 1016-443X

VL - 14

SP - 1182

EP - 1218

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

IS - 6

ER -

A Geometric approximation to the euler equations: The Vlasov-Monge- Ampère system

Abstract

Access to Document

Other files and links

Cite this