Details
| Original language | English |
|---|---|
| Pages (from-to) | 1407-1419 |
| Number of pages | 13 |
| Journal | Communications on Pure and Applied Analysis |
| Volume | 11 |
| Issue number | 4 |
| Publication status | Published - Jul 2012 |
Abstract
We show that the modified Constantin-Lax-Majda equation modeling vortex and quasi-geostrophic dynamics [27] can be recast as the geodesic flow on the subgroup Diff∞1(S) of orientation-preserving diffeomorphisms φ ∈ Diff∞(S) such that Φ(1) = 1 equipped with the right-invariant metric induced by the homogeneous Sobolev norm Ḣ1/2. On the extended group of diffeomorphisms of Sobolev class Hk with k ≥ 2, this induces a weak Riemannian structure. We establish that the geodesic spray is smooth and we obtain local existence and uniqueness of the geodesics.
Keywords
- CLM equation, Euler equation on diffeomorphisms group of the circle
ASJC Scopus subject areas
- Mathematics(all)
- Analysis
- Mathematics(all)
- Applied Mathematics
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In: Communications on Pure and Applied Analysis, Vol. 11, No. 4, 07.2012, p. 1407-1419.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - The geometry of a vorticity model equation
AU - Escher, Joachim
AU - Kolev, Boris
AU - Wunsch, Marcus
PY - 2012/7
Y1 - 2012/7
N2 - We show that the modified Constantin-Lax-Majda equation modeling vortex and quasi-geostrophic dynamics [27] can be recast as the geodesic flow on the subgroup Diff∞1(S) of orientation-preserving diffeomorphisms φ ∈ Diff∞(S) such that Φ(1) = 1 equipped with the right-invariant metric induced by the homogeneous Sobolev norm Ḣ1/2. On the extended group of diffeomorphisms of Sobolev class Hk with k ≥ 2, this induces a weak Riemannian structure. We establish that the geodesic spray is smooth and we obtain local existence and uniqueness of the geodesics.
AB - We show that the modified Constantin-Lax-Majda equation modeling vortex and quasi-geostrophic dynamics [27] can be recast as the geodesic flow on the subgroup Diff∞1(S) of orientation-preserving diffeomorphisms φ ∈ Diff∞(S) such that Φ(1) = 1 equipped with the right-invariant metric induced by the homogeneous Sobolev norm Ḣ1/2. On the extended group of diffeomorphisms of Sobolev class Hk with k ≥ 2, this induces a weak Riemannian structure. We establish that the geodesic spray is smooth and we obtain local existence and uniqueness of the geodesics.
KW - CLM equation
KW - Euler equation on diffeomorphisms group of the circle
UR - http://www.scopus.com/inward/record.url?scp=84861936438&partnerID=8YFLogxK
U2 - 10.3934/cpaa.2012.11.1407
DO - 10.3934/cpaa.2012.11.1407
M3 - Article
AN - SCOPUS:84861936438
VL - 11
SP - 1407
EP - 1419
JO - Communications on Pure and Applied Analysis
JF - Communications on Pure and Applied Analysis
SN - 1534-0392
IS - 4
ER -