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.
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Journal of Geometric Mechanics, American Institute of Mathematical Sciences (AIMS), 2014, 6 (3), pp.335-372. 〈10.3934/jgm.2014.6.335〉
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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〉

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