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

2019/07/29 by Shizan Fang, Fang, Shizan · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Probability (math.PR) #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1907.13519

openalex publication_date 2019/07/29 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28

Abstract

What is the suitable Laplace operator on vector fields for the Navier-Stokes\nequation on a Riemannian manifold? In this note, by considering Nash embedding,\nwe will try to elucidate different aspects of different Laplace operators such\nas de Rham-Hodge Laplacian as well as Ebin-Marsden's Laplacian. A probabilistic\nrepresentation formula for Navier-Stokes equations on a general compact\nRiemannian manifold is obtained when de Rham-Hodge Laplacian is involved. MSC\n2010: 35Q30, 58J65\n

Citations

• The incompressible Navier-Stokes equations on non-compact manifolds

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• On the Navier-Stokes equations and the Hamilton-Jacobi-Bellman equation on the group of volume preserving diffeomorphisms

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