Deformation and symmetry in the inviscid SQG and the 3D euler equations

Dongho Chae; Peter Constantin; Jiahong Wu

doi:10.1007/s00332-012-9124-7

Deformation and symmetry in the inviscid SQG and the 3D euler equations

Dongho Chae, Peter Constantin, Jiahong Wu

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

9 Scopus citations

Abstract

The global regularity problem concerning the inviscid SQG and the 3D Euler equations remains an outstanding open question. This paper presents several geometric observations on solutions of these equations. One observation stems from a relation between what we call Eulerian and Lagrangian deformations and reflects the alignment of the stretching directions of these deformations and the tangent direction of the level curves for the SQG equation. Various spatial symmetries in solutions to the 3D Euler equations are exploited. In addition, two observations on the curvature of the level curves of the SQG equation are also included.

Original language	English (US)
Pages (from-to)	665-688
Number of pages	24
Journal	Journal of Nonlinear Science
Volume	22
Issue number	5
DOIs	https://doi.org/10.1007/s00332-012-9124-7
State	Published - Oct 2012

All Science Journal Classification (ASJC) codes

Modeling and Simulation
General Engineering
Applied Mathematics

Keywords

3D Euler equation
Geometric property
Surface quasi-geostrophic equation

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10.1007/s00332-012-9124-7

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title = "Deformation and symmetry in the inviscid SQG and the 3D euler equations",

abstract = "The global regularity problem concerning the inviscid SQG and the 3D Euler equations remains an outstanding open question. This paper presents several geometric observations on solutions of these equations. One observation stems from a relation between what we call Eulerian and Lagrangian deformations and reflects the alignment of the stretching directions of these deformations and the tangent direction of the level curves for the SQG equation. Various spatial symmetries in solutions to the 3D Euler equations are exploited. In addition, two observations on the curvature of the level curves of the SQG equation are also included.",

keywords = "3D Euler equation, Geometric property, Surface quasi-geostrophic equation",

author = "Dongho Chae and Peter Constantin and Jiahong Wu",

note = "Funding Information: Acknowledgements The authors thank the referees for their useful comments and suggestions. Chae{\textquoteright}s research was partially supported by NRF grant No. 2006-0093854. Constantin{\textquoteright}s research was partially supported by NSF grant DMS 0804380. Wu{\textquoteright}s research was partially supported by NSF grant DMS 0907913 and the AT&T Foundation at Oklahoma State University.",

year = "2012",

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doi = "10.1007/s00332-012-9124-7",

language = "English (US)",

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journal = "Journal of Nonlinear Science",

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AU - Chae, Dongho

AU - Constantin, Peter

AU - Wu, Jiahong

N1 - Funding Information: Acknowledgements The authors thank the referees for their useful comments and suggestions. Chae’s research was partially supported by NRF grant No. 2006-0093854. Constantin’s research was partially supported by NSF grant DMS 0804380. Wu’s research was partially supported by NSF grant DMS 0907913 and the AT&T Foundation at Oklahoma State University.

PY - 2012/10

Y1 - 2012/10

N2 - The global regularity problem concerning the inviscid SQG and the 3D Euler equations remains an outstanding open question. This paper presents several geometric observations on solutions of these equations. One observation stems from a relation between what we call Eulerian and Lagrangian deformations and reflects the alignment of the stretching directions of these deformations and the tangent direction of the level curves for the SQG equation. Various spatial symmetries in solutions to the 3D Euler equations are exploited. In addition, two observations on the curvature of the level curves of the SQG equation are also included.

AB - The global regularity problem concerning the inviscid SQG and the 3D Euler equations remains an outstanding open question. This paper presents several geometric observations on solutions of these equations. One observation stems from a relation between what we call Eulerian and Lagrangian deformations and reflects the alignment of the stretching directions of these deformations and the tangent direction of the level curves for the SQG equation. Various spatial symmetries in solutions to the 3D Euler equations are exploited. In addition, two observations on the curvature of the level curves of the SQG equation are also included.

KW - 3D Euler equation

KW - Geometric property

KW - Surface quasi-geostrophic equation

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Deformation and symmetry in the inviscid SQG and the 3D euler equations

Abstract

All Science Journal Classification (ASJC) codes

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