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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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Marc Arnaudon, Ana Bela Cruzeiro, Shizan Fang. Generalized stochastic Lagrangian paths for the Navier-Stokes equation. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Scuola Normale Superiore 2018, 18 (3), pp.1033-1060. ⟨10.2422/2036-2145.201602_006⟩. ⟨hal-01197123v2⟩

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