Nash embedding, shape operator and Navier-Stokes equation on a Riemannian manifold
Differential Geometry
2019-10-14 v2 Analysis of PDEs
Probability
Abstract
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
Cite
@article{arxiv.1907.13519,
title = {Nash embedding, shape operator and Navier-Stokes equation on a Riemannian manifold},
author = {Shizan Fang},
journal= {arXiv preprint arXiv:1907.13519},
year = {2019}
}