Details
| Original language | English |
|---|---|
| Article number | 1250016 |
| Journal | Communications in Contemporary Mathematics |
| Volume | 14 |
| Issue number | 3 |
| Publication status | Published - Jun 2012 |
Abstract
We prove that several evolution equations arising as mathematical models for fluid motion cannot be realized as metric Euler equations on the Lie group DIFF ∞( 1) of all smooth and orientation-preserving diffeomorphisms on the circle. These include the quasi-geostrophic model equation, cf. [A. Córdoba, D. Córdoba and M. A. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math. 162 (2005) 13771389], the axisymmetric Euler flow in d (see [H. Okamoto and J. Zhu, Some similarity solutions of the NavierStokes equations and related topics, Taiwanese J. Math. 4 (2000) 65103]), and De Gregorio's vorticity model equation as introduced in [S. De Gregorio, On a one-dimensional model for the three-dimensional vorticity equation, J. Stat. Phys. 59 (1990) 12511263].
Keywords
- diffeomorphism group of the circle, geodesic flow, Non-metric Euler equation
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
- Mathematics(all)
- Applied Mathematics
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In: Communications in Contemporary Mathematics, Vol. 14, No. 3, 1250016, 06.2012.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Restrictions on the geometry of the periodic vorticity equation
AU - Escher, Joachim
AU - Wunsch, Marcus
N1 - Funding information: We are grateful to Bogdan V. Matioc for carefully checking laborious computations. The first author expresses his gratitude to Hisashi Okamoto and the Research Institute for Mathematical Sciences at Kyoto University, whose hospitality he highly appreciated during his visit in March 2010. The second author acknowledges financial support by the Postdoctoral Fellowship P09024 of the Japan Society for the Promotion of Science.
PY - 2012/6
Y1 - 2012/6
N2 - We prove that several evolution equations arising as mathematical models for fluid motion cannot be realized as metric Euler equations on the Lie group DIFF ∞( 1) of all smooth and orientation-preserving diffeomorphisms on the circle. These include the quasi-geostrophic model equation, cf. [A. Córdoba, D. Córdoba and M. A. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math. 162 (2005) 13771389], the axisymmetric Euler flow in d (see [H. Okamoto and J. Zhu, Some similarity solutions of the NavierStokes equations and related topics, Taiwanese J. Math. 4 (2000) 65103]), and De Gregorio's vorticity model equation as introduced in [S. De Gregorio, On a one-dimensional model for the three-dimensional vorticity equation, J. Stat. Phys. 59 (1990) 12511263].
AB - We prove that several evolution equations arising as mathematical models for fluid motion cannot be realized as metric Euler equations on the Lie group DIFF ∞( 1) of all smooth and orientation-preserving diffeomorphisms on the circle. These include the quasi-geostrophic model equation, cf. [A. Córdoba, D. Córdoba and M. A. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math. 162 (2005) 13771389], the axisymmetric Euler flow in d (see [H. Okamoto and J. Zhu, Some similarity solutions of the NavierStokes equations and related topics, Taiwanese J. Math. 4 (2000) 65103]), and De Gregorio's vorticity model equation as introduced in [S. De Gregorio, On a one-dimensional model for the three-dimensional vorticity equation, J. Stat. Phys. 59 (1990) 12511263].
KW - diffeomorphism group of the circle
KW - geodesic flow
KW - Non-metric Euler equation
UR - http://www.scopus.com/inward/record.url?scp=84861942681&partnerID=8YFLogxK
U2 - 10.1142/S0219199712500162
DO - 10.1142/S0219199712500162
M3 - Article
AN - SCOPUS:84861942681
VL - 14
JO - Communications in Contemporary Mathematics
JF - Communications in Contemporary Mathematics
SN - 0219-1997
IS - 3
M1 - 1250016
ER -