ARNAUDON, Marc; CRUZEIRO, Ana Bela; FANG, Shizan

doi:10.2422/2036-2145.201602_006

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ARNAUDON, Marc
Institut de Mathématiques de Bordeaux [IMB]

CRUZEIRO, Ana Bela

FANG, Shizan
Institut de Mathématiques de Bourgogne [Dijon] [IMB]

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Annali della Scuola Normale Superiore di Pisa, Classe di Scienze. 2018, vol. 18, n° 3, p. 1033-1060

Scuola Normale Superiore

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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.< Réduire

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Generalized stochastic Lagrangian paths for the Navier-Stokes equation

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