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.

26 pages. We add Section 4 discussing the Killing vector fields on Riemannian symmetric spaces which satisfy the conditions in Section 3

References in corpus (1)

• Geodesics in the space of measure-preserving maps and plans

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