Generalized stochastic Lagrangian paths for the Navier-Stokes equation

Abstract : In the note added in proof of the seminal paper [Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. 92 (1970), 102-163], Ebin and Marsden introduced the so-called correct Laplacian for the Navier-Stokes equation on a compact Riemannian manifold. In the spirit of Brenier's generalized flows for the Euler equation, we introduce a class of semimartingales on a compact Riemannian manifold. We prove that these semimartingales are critical points to the corresponding kinetic energy if and only if its drift term solves weakly the Navier-Stokes equation defined with Ebin-Marsden's Laplacian. We also show that for the torus case, classical solutions of the Navier-Stokes equation realize the minimum of the kinetic energy in a suitable class.
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Pré-publication, Document de travail
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Contributeur : Shizan Fang <>
Soumis le : vendredi 19 février 2016 - 17:14:59
Dernière modification le : mercredi 24 février 2016 - 08:43:12
Document(s) archivé(s) le : vendredi 20 mai 2016 - 10:35:05


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  • HAL Id : hal-01197123, version 2
  • ARXIV : 1509.03491



Marc Arnaudon, Ana Bela Cruzeiro, Shizan Fang. Generalized stochastic Lagrangian paths for the Navier-Stokes equation. 2016. 〈hal-01197123v2〉



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