Constantin and Iyer’s representation formula for the Navier–Stokes equations on manifolds

Abstract : The purpose of this paper is to establish a probabilistic representation formula for the Navier--Stokes equations on compact Riemannian manifolds. Such a formula has been provided by Constantin and Iyer in the flat case of $\mathbb R^n$ or of $\mathbb T^n$. On a Riemannian manifold, however, there are several different choices of Laplacian operators acting on vector fields. In this paper, we shall use the de Rham--Hodge Laplacian operator which seems more relevant to the probabilistic setting, and adopt Elworthy--Le Jan--Li's idea to decompose it as a sum of the square of Lie derivatives.
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Submitted on : Monday, February 5, 2018 - 4:54:22 PM
Last modification on : Friday, October 19, 2018 - 10:34:37 AM

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Shizan Fang, Dejun Luo. Constantin and Iyer’s representation formula for the Navier–Stokes equations on manifolds. Potential Analysis, Springer Verlag, 2018, 48 (2), pp.181-206. ⟨10.1007/s11118-017-9631-0⟩. ⟨hal-01701332⟩

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