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We study the geometry of the space of densities Dens(M ), which is the quotient space Diff(M )/Diff µ (M ) of the diffeomorphism group of a compact manifold M by the subgroup of volume-preserving diffemorphisms, endowed with a right-invariant homogeneous SobolevḢ 1 -metric. We construct an explicit isometry from this space to (a subset of) an infinite-dimensional sphere and show that the associated Euler-Arnold equation is a completely integrable system in any space dimension. We also prove that its smooth solutions break down in finite time.Furthermore, we show that theḢ 1 -metric induces the Fisher-Rao (information) metric on the space of probability distributions, and thus its Riemannian distance is the spherical version of Hellinger distance. We compare it to the Wasserstein distance in optimal transport which is induced by an L 2 -metric on Diff(M ). TheḢ 1 geometry we introduce in this paper can be seen as an infinite-dimensional version of the geometric theory of statistical manifolds.

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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.

We study the motion of an inextensible string (a whip) fixed at one point in the absence of gravity, satisfying the equations $$ \eta_{tt} = \partial_s(\sigma \eta_s), \qquad \sigma_{ss}-\lvert \eta_{ss}\rvert^2 = -\lvert \eta_{st}\rvert^2, \qquad \lvert \eta_s\rvert^2 \equiv 1 $$ with boundary conditions $\eta(t,1)=0$ and $\sigma(t,0)=0$. We prove local existence and uniqueness in the space defined by the weighted Sobolev energy $$ \sum_{\ell=0}^m \int_0^1 s^{\ell} \lvert \partial_s^{\ell}\eta_t\rvert^2 \, ds + \int_0^1 s^{\ell+1} \lvert \partial_s^{\ell+1}\eta\rvert^2 \, ds, $$ when $m\ge 3$. In addition we show persistence of smooth solutions as long as the energy for $m=3$ remains bounded. We do this via the method of lines, approximating with a discrete system of coupled pendula (a chain) for which the same estimates hold.Comment: 47 pages, 8 figure