Details
| Original language | English |
|---|---|
| Pages (from-to) | 335-372 |
| Number of pages | 38 |
| Journal | Journal of Geometric Mechanics |
| Volume | 6 |
| Issue number | 3 |
| Publication status | Published - Sept 2014 |
Abstract
In this paper, we study the geodesic flow of a right-invariant metric induced by a general Fourier multiplier on the diffeomorphism group of the circle and on some of its homogeneous spaces. This study covers in particular right-invariant metrics induced by Sobolev norms of fractional order. We show that, under a certain condition on the symbol of the inertia operator (which is satisfied for the fractional Sobolev norm Hsfor s ≥ 1/2), the corresponding initial value problem is well-posed in the smooth category and that the Riemannian exponential map is a smooth local diffeomorphism. Paradigmatic examples of our general setting cover, besides all traditional Euler equations induced by a local inertia operator, the Constantin-Lax-Majda equation, and the Euler-Weil-Petersson equation.
Keywords
- Diffeomorphism group of the circle, Euler equation, Sobolev metrics of fractional order
ASJC Scopus subject areas
- Engineering(all)
- Mechanics of Materials
- Mathematics(all)
- Geometry and Topology
- Mathematics(all)
- Control and Optimization
- Mathematics(all)
- Applied Mathematics
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In: Journal of Geometric Mechanics, Vol. 6, No. 3, 09.2014, p. 335-372.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle
AU - Escher, Joachim
AU - Kolev, Boris
PY - 2014/9
Y1 - 2014/9
N2 - In this paper, we study the geodesic flow of a right-invariant metric induced by a general Fourier multiplier on the diffeomorphism group of the circle and on some of its homogeneous spaces. This study covers in particular right-invariant metrics induced by Sobolev norms of fractional order. We show that, under a certain condition on the symbol of the inertia operator (which is satisfied for the fractional Sobolev norm Hsfor s ≥ 1/2), the corresponding initial value problem is well-posed in the smooth category and that the Riemannian exponential map is a smooth local diffeomorphism. Paradigmatic examples of our general setting cover, besides all traditional Euler equations induced by a local inertia operator, the Constantin-Lax-Majda equation, and the Euler-Weil-Petersson equation.
AB - In this paper, we study the geodesic flow of a right-invariant metric induced by a general Fourier multiplier on the diffeomorphism group of the circle and on some of its homogeneous spaces. This study covers in particular right-invariant metrics induced by Sobolev norms of fractional order. We show that, under a certain condition on the symbol of the inertia operator (which is satisfied for the fractional Sobolev norm Hsfor s ≥ 1/2), the corresponding initial value problem is well-posed in the smooth category and that the Riemannian exponential map is a smooth local diffeomorphism. Paradigmatic examples of our general setting cover, besides all traditional Euler equations induced by a local inertia operator, the Constantin-Lax-Majda equation, and the Euler-Weil-Petersson equation.
KW - Diffeomorphism group of the circle
KW - Euler equation
KW - Sobolev metrics of fractional order
UR - http://www.scopus.com/inward/record.url?scp=84907552304&partnerID=8YFLogxK
U2 - 10.3934/jgm.2014.6.335
DO - 10.3934/jgm.2014.6.335
M3 - Article
AN - SCOPUS:84907552304
VL - 6
SP - 335
EP - 372
JO - Journal of Geometric Mechanics
JF - Journal of Geometric Mechanics
SN - 1941-4889
IS - 3
ER -