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    • Plusieurs versions

    Geodesics On The Symplectomorphism Group

    Ebin, David
    Geometric and Functional Analysis, 2012, Vol.22(1), pp.202-212 [Revue évaluée par les pairs]

    • Plusieurs versions

    Comparison theorems in Riemannian geometry

    Cheeger, Jeff
    • Plusieurs versions

    Global Solutions of the Equations of Elastodynamics of Incompressible Neo-Hookean Materials

    Ebin, David G.
    Proceedings of the National Academy of Sciences of the United States of America, 01 May 1993, Vol.90(9), pp.3802-3805 [Revue évaluée par les pairs]

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    Motion of slightly compressible fluids in a bounded domain. II

    Disconzi, Marcelo M., Ebin, David G.
    Commun. Contemp. Math. Vol. 19, Issue 04, 1650054 (2017) [57 pages] [Revue évaluée par les pairs]

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    On the Limit of Large Surface Tension for a Fluid Motion with Free Boundary

    Disconzi, Marcelo M, Ebin, David G
    Communications in Partial Differential Equations, 03 April 2014, Vol.39(4), pp.740-779 [Revue évaluée par les pairs]

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    Riemannian geometry of the contactomorphism group

    Ebin, David G., Preston, Stephen C.
    Cornell University
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    Titre: Riemannian geometry of the contactomorphism group
    Auteur: Ebin, David G.; Preston, Stephen C.
    Sujet: Mathematics - Analysis Of Pdes
    Description: We define a right-invariant Riemannian metric on the group of contactomorphisms and study its Euler-Arnold equation. If the metric is associated to the contact form, the Euler-Arnold equation reduces to $m_t + u(m) + (n+2) mE(f) = 0$, in terms of the Reeb field $E$, a stream function $f$, the contact vector field $u$ defined by $f$, and the momentum $m = f - \Delta f$. Here the equation is considered on a compact manifold $M$ of dimension $2n+1$. When $n=0$ this reduces to the Camassa-Holm equation, and we emphasize the analogy with the higher-order equation. We use the usual momentum conservation law for Euler-Arnold equations to rewrite the geodesic equation as a smooth first-order equation on the contactomorphism group of Sobolev class $H^s$, and thus obtain local existence in time of solutions which depend smoothly on initial data. In addition we prove a global existence criterion analogous to the Beale-Kato-Majda criterion in fluid mechanics, and show how this criterion is automatically satisfied on the totally geodesic subgroup of quantomorphisms. Finally we briefly discuss singular solutions and conservation laws of the Euler-Arnold equation.
    Identifiant: 1409.2197 (ARXIV ID)

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    Riemannian Geometry of the Contactomorphism Group

    Ebin, David, Preston, Stephen
    Arnold Mathematical Journal, 2015, Vol.1(1), pp.5-36

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    The free boundary Euler equations with large surface tension

    Disconzi, Marcelo M, Ebin, David G
    Journal of Differential Equations, 15 July 2016, Vol.261(2), pp.821-889 [Revue évaluée par les pairs]

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    Riemannian geometry on the quantomorphism group

    Ebin, David G., Preston, Stephen C.
    Cornell University
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    Titre: Riemannian geometry on the quantomorphism group
    Auteur: Ebin, David G.; Preston, Stephen C.
    Sujet: Mathematics - Differential Geometry
    Description: We are interested in the geometry of the group $\mathcal{D}_q(M)$ of diffeomorphisms preserving a contact form $\theta$ on a manifold $M$. We define a Riemannian metric on $\mathcal{D}_q(M)$, compute the corresponding geodesic equation, and show that solutions exist for all time and depend smoothly on initial conditions. In certain special cases (such as on the 3-sphere), the geodesic equation is a simplified version of the quasigeostrophic equation, so we obtain a new geodesic interpretation of this geophysical system. We also show that the genuine quasigeostrophic equation on $S^2$ can be obtained as an Euler-Arnold equation on a one-dimensional central extension of $T_{\id}\mathcal{D}_q(M)$, and that our global existence result extends to this case. If $E$ is the Reeb field of $\theta$ and $\mu$ is the volume form, assumed compatible in the sense that $\text{div} E=0$, we show that $\mathcal{D}_q(M)$ is a smooth submanifold of $\mathcal{D}_{E,\mu}(M)$, the space of diffeomorphisms preserving the vector field $E$ and the volume form $\mu$, in the sense of $H^s$ Sobolev completions. The latter manifold is related to symmetric motion of ideal fluids. We further prove that the corresponding geodesic equations and projections are $C^{\infty}$ objects in the Sobolev topology. Comment: 36 pages
    Identifiant: 1302.5075 (ARXIV ID)

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    Thomae’s formula for Z n curves

    Ebin, David, Farkas, Hershel
    Journal d'Analyse Mathématique, 2010, Vol.111(1), pp.289-320 [Revue évaluée par les pairs]