Existence of Eulerian Solutions to the Semigeostrophic Equations in Physical Space: The 2-Dimensional Periodic Case

Luigi Ambrosio; Maria Colombo; Guido de Philippis; Alessio Figalli

doi:10.1080/03605302.2012.669443

Existence of Eulerian Solutions to the Semigeostrophic Equations in Physical Space: The 2-Dimensional Periodic Case

Luigi Ambrosio, Maria Colombo, Guido de Philippis, Alessio Figalli

Mathematics

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

Abstract

In this article we use new regularity and stability estimates for Alexandrov solutions to Monge-Ampère equations, recently established by De Philippis and Figalli [14], to provide global in time existence of distributional solutions to the semigeostrophic equations on the 2-dimensional torus, under very mild assumptions on the initial data. A link with Lagrangian solutions is also discussed.

Original language	English (US)
Pages (from-to)	2209-2227
Number of pages	19
Journal	Communications in Partial Differential Equations
Volume	37
Issue number	12
DOIs	https://doi.org/10.1080/03605302.2012.669443
State	Published - Jan 2012

Keywords

Lagrangian flow
Optimal transporation
Semigeostrophic equations

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1080/03605302.2012.669443

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title = "Existence of Eulerian Solutions to the Semigeostrophic Equations in Physical Space: The 2-Dimensional Periodic Case",

abstract = "In this article we use new regularity and stability estimates for Alexandrov solutions to Monge-Amp{\`e}re equations, recently established by De Philippis and Figalli [14], to provide global in time existence of distributional solutions to the semigeostrophic equations on the 2-dimensional torus, under very mild assumptions on the initial data. A link with Lagrangian solutions is also discussed.",

keywords = "Lagrangian flow, Optimal transporation, Semigeostrophic equations",

author = "Luigi Ambrosio and Maria Colombo and {de Philippis}, Guido and Alessio Figalli",

note = "Funding Information: L.A., G.D.P., and A.F. acknowledge the support of the ERC ADG GeMeThNES. A.F. was also supported by the NSF Grant DMS-0969962.",

year = "2012",

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AU - Colombo, Maria

AU - de Philippis, Guido

AU - Figalli, Alessio

N1 - Funding Information: L.A., G.D.P., and A.F. acknowledge the support of the ERC ADG GeMeThNES. A.F. was also supported by the NSF Grant DMS-0969962.

PY - 2012/1

Y1 - 2012/1

N2 - In this article we use new regularity and stability estimates for Alexandrov solutions to Monge-Ampère equations, recently established by De Philippis and Figalli [14], to provide global in time existence of distributional solutions to the semigeostrophic equations on the 2-dimensional torus, under very mild assumptions on the initial data. A link with Lagrangian solutions is also discussed.

AB - In this article we use new regularity and stability estimates for Alexandrov solutions to Monge-Ampère equations, recently established by De Philippis and Figalli [14], to provide global in time existence of distributional solutions to the semigeostrophic equations on the 2-dimensional torus, under very mild assumptions on the initial data. A link with Lagrangian solutions is also discussed.

KW - Lagrangian flow

KW - Optimal transporation

KW - Semigeostrophic equations

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JF - Communications in Partial Differential Equations

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ER -

Existence of Eulerian Solutions to the Semigeostrophic Equations in Physical Space: The 2-Dimensional Periodic Case

Abstract

Keywords

ASJC Scopus subject areas

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