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

Shizan Fang

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arXiv · 1907.13519

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

Shizan Fang

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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

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Shizan Fang. 2019-10-11. Nash embedding, shape operator and Navier-Stokes equation on a Riemannian manifold. https://arxiv.org/abs/1907.13519

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