Constantin and Iyer's representation formula for the Navier--Stokes equations on manifolds

TL;DR

This paper extends Constantin and Iyer's probabilistic representation formula for the Navier--Stokes equations from flat spaces to compact Riemannian manifolds, using the de Rham--Hodge Laplacian and Lie derivatives.

Contribution

It introduces a probabilistic representation formula for Navier--Stokes equations on manifolds, utilizing the de Rham--Hodge Laplacian and Lie derivative decomposition.

Findings

Formulated a probabilistic representation on manifolds

Used de Rham--Hodge Laplacian for vector fields

Applied Lie derivatives decomposition approach

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.