A note on Constantin and Iyer's representation formula for the Navier–Stokes equations

Shizan Fang (1) , Dejun Luo (2)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]
2 Institute of Applied Mathematics, Chinese Academy of Sciences

The purpose of this note is to establish a probabilistic representation formula for Navier–Stokes equations on a compact Riemannian manifold. To this end, we first give a geometric interpretation of Constantin and Iyer's representation formula for the Navier–Stokes equation, then extend it to a compact Riemannian manifold. We shall use Elworthy–Le Jan–Li's idea to decompose de Rham–Hodge Laplacian operator on a manifold as a sum of the square of vector fields. MSC 2010: 35Q30, 58J65 Keywords: Navier–Stokes equations, stochastic representation, de Rham–Hodge Lapla-cian, stochastic flow, pull-back vector field

Domaines: Probabilités [math.PR]

Shizan Fang, Dejun Luo. A note on Constantin and Iyer's representation formula for the Navier–Stokes equations. 2015. ⟨hal-01192831⟩