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

TL;DR

This paper investigates right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle, establishing well-posedness and smoothness of the exponential map under certain conditions, with applications to various Euler equations.

Contribution

It generalizes the analysis of right-invariant metrics to fractional Sobolev norms and proves well-posedness and smoothness of the geodesic flow for these metrics.

Findings

Well-posedness of the initial value problem for fractional Sobolev metrics.

Smoothness of the Riemannian exponential map under certain conditions.

Applicability to classical Euler equations and specific models like Constantin-Lax-Majda.

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.