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

Joachim Escher; Boris Kolev

doi:10.3934/jgm.2014.6.335

Article Contents

2014, Volume 6, Issue 3: 335-372. Doi: 10.3934/jgm.2014.6.335

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

Joachim Escher^1, and
Boris Kolev^2,

1.
Institute for Applied Mathematics, University of Hanover, D-30167 Hanover
2.
Institut de Mathématiques de Marseille, Aix Marseille Université, CNRS, Centrale Marseille, 13453 Marseille

Received: November 2013

Revised: March 2014

Published: September 2014

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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.

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Mathematics Subject Classification: Primary: 58D05; Secondary: 35Q53.

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