Abstract. We show that certain right-invariant metrics endow the infinite-dimensional Lie group of all smooth orientation-preserving diffeomorphisms of the circle with a Riemannian structure. The study of the Riemannian exponential map allows us to prove infinite-dimensional counterparts of results from classical Riemannian geometry: the Riemannian exponential map is a smooth local diffeomorphism and the length-minimizing property of the geodesics holds. Mathematics Subject Classification (2000). 35Q35, 58B25.

According to the principle of least action, the spatially periodic motions of one-dimensional mechanical systems with no external forces are described in the Lagrangian formalism by geodesics on a manifold-configuration space, the group D of smooth orientation-preserving diffeomorphisms of the circle. The periodic inviscid Burgers equation is the geodesic equation on D with the L 2 right-invariant metric. However, the exponential map for this right-invariant metric is not a C 1 local diffeomorphism and the geometric structure is therefore deficient. On the other hand, the geodesic equation on D for the H 1 right-invariant metric is also a re-expression of a model in mathematical physics. We show that in this case the exponential map is a C 1 local diffeomorphism and that if two diffeomorphisms are sufficiently close on D, they can be joined by a unique length-minimizing geodesic-a state of the system is transformed to another nearby state by going through a uniquely determined flow that minimizes the energy. We also analyze for both metrics the breakdown of the geodesic flow.

40 pages. Corrected typos and improved redaction.International audienceIn 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

Abstract. In this paper we present a geometric interpretation of the periodic Degasperis-Procesi equation as the geodesic flow of a right invariant symmetric linear connection on the diffeomorphism group of the circle. We also show that for any evolution in the family of b-equations there is neither gain nor loss of the spatial regularity of solutions. This in turn allows us to view the Degasperis-Procesi and the Camassa-Holm equation as an ODE on the Fréchet space of all smooth functions on the circle.

We study the geodesic exponential maps corresponding to Sobolev type right-invariant (weak) Riemannian metrics µ (k) (k ≥ 0) on the Virasoro group Vir and show that for k ≥ 2, but not for k = 0, 1, each of them defines a smooth Fréchet chart of the unital element e ∈ Vir. In particular, the geodesic exponential map corresponding to the KdV equation (k = 0) is not a local diffeomorphism near the origin.

No abstract

Abstract. This article consists of a detailed geometric study of the one-dimensional vorticity model equationwhich is a particular case of the generalized Constantin-Lax-Majda equation. Wunsch showed that this equation is the Euler-Arnold equation on Diff(S 1 ) when the latter is endowed with the rightinvariant homogeneousḢ 1/2 -metric. In this article we prove that the exponential map of this Riemannian metric is not Fredholm and that the sectional curvature is locally unbounded. Furthermore, we prove a Beale-Kato-Majda-type blow-up criterion, which we then use to demonstrate a link to our non-Fredholmness result. Finally, we extend a blow-up result of Castro-Córdoba to the periodic case and to a much wider class of initial conditions, using a new generalization of an inequality for Hilbert transforms due to Córdoba-Córdoba.

Of concern is the study of fractional order Sobolev-type metrics on the group of H ∞ -diffeomorphism of R d and on its Sobolev completions D q (R d ). It is shown that the H s -Sobolev metric induces a strong and smooth Riemannian metric on the Banach manifolds D s (R d ) for s > 1 + d 2 . As a consequence a global well-posedness result of the corresponding geodesic equations, both on the Banach manifold D s (R d ) and on the smooth regular Fréchet-Lie group of all H ∞ -diffeomorphisms is obtained. In addition a local existence result for the geodesic equation for metrics of order 1 2 ≤ s < 1 + d/2 is derived.