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### Open Access Journals

DCDS

The present volume is related to the conference "Geometric Mechanics'', to
be held from November 19 to November 23, 2007, at the CIRM (Centre International de Rencontres Mathématiques)
in Marseille, France.

For more information please click the “Full Text” above.

For more information please click the “Full Text” above.

keywords:

JGM

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.

DCDS

This paper investigates different Poisson structures that have been proposed to give a Hamiltonian formulation to evolution equations issued from fluid mechanics. Our aim is to explore the main brackets which have been proposed and to discuss the difficulties which arise when one tries to give a rigorous meaning to these brackets. Our main interest is in the definition of a

*valid and usable*bracket to study*rotational fluid flows with a free boundary*. We discuss some results which have emerged in the literature to solve some of the difficulties that arise. It appears to the author that the main problems are still open.
CPAA

We show that the modified Constantin-Lax-Majda equation modeling vortex and quasi-geostrophic dynamics [27] can be recast as the geodesic flow on the subgroup $\mathrm{Diff}_{1}^{\infty}(\mathbb{S})$ of orientation-preserving diffeomorphisms $\varphi \in \mathrm{Diff}^{\infty}(\mathbb{S})$ such that $\varphi(1) = 1$ equipped with the right-invariant metric induced by the homogeneous Sobolev norm $\dot H^{1/2}$. On the extended group of diffeomorphisms of Sobolev class $H^{k}$ with $k\ge 2$, this induces a

*weak Riemannian*structure. We establish that the geodesic spray is smooth and we obtain local existence and uniqueness of the geodesics.
JGM

The geodesic equations of a class of

*right invariant*metrics on the semi-direct product $Diff(\mathbb{S}^1)$Ⓢ$Diff(\mathbb{S}^1)$ are studied. The equations are explicitly described, they have the form of a system of coupled equations of Camassa-Holm type and possess singular (peakon) solutions. Their integrability is further investigated, however no compatible bi-Hamiltonian structures on the corresponding dual Lie algebra $(Vect(\mathbb{S}^1)$Ⓢ$Vect(\mathbb{S}^1))^{*}$ are found.## Year of publication

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