Nash embedding, shape operator and Navier-Stokes equation on a Riemannian manifold

Shizan Fang (1)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]

What is the suitable Laplace operator on vector fields for the Navier-Stokes equation on a Riemannian manifold? In this note, by considering Nash embedding, we will try to elucidate different aspects of different Laplace operators such as de Rham-Hodge Laplacian as well as Ebin-Marsden's Laplacian. A probabilistic representation formula for Navier-Stokes equations on a general compact Riemannian manifold is obtained when de Rham-Hodge Laplacian is involved. MSC 2010: 35Q30, 58J65

Mots clés: Nash embedding; shape operator; vector valued Laplacians; Navier-Stokes equa- tions; stochastic representation

Domaines: Probabilités [math.PR]; Géométrie différentielle [math.DG]

Shizan Fang. Nash embedding, shape operator and Navier-Stokes equation on a Riemannian manifold. 2019. ⟨hal-02196468v2⟩