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.

40 pages. Corrected typos and improved redaction

References in corpus (3)

• The Lagrangian Radon Transform and the Weil representation
• Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group. II
• Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms

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