# Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle

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 $H^{s}$ for $s \ge 1/2$), 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 :
Type de document :
Article dans une revue
Journal of Geometric Mechanics, American Institute of Mathematical Sciences (AIMS), 2014, 6 (3), pp.335-372. 〈10.3934/jgm.2014.6.335〉
Domaine :

Littérature citée [41 références]

https://hal.archives-ouvertes.fr/hal-00673137
Contributeur : Boris Kolev <>
Soumis le : mardi 26 août 2014 - 20:44:07
Dernière modification le : mercredi 14 février 2018 - 14:40:06
Document(s) archivé(s) le : jeudi 27 novembre 2014 - 16:37:11

### Fichiers

EK2012.pdf
Fichiers produits par l'(les) auteur(s)

### Citation

Joachim Escher, Boris Kolev. Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle. Journal of Geometric Mechanics, American Institute of Mathematical Sciences (AIMS), 2014, 6 (3), pp.335-372. 〈10.3934/jgm.2014.6.335〉. 〈hal-00673137v4〉

### Métriques

Consultations de la notice

## 286

Téléchargements de fichiers