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

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

References in corpus (1)

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

Cited by in corpus (5)

• Asymptotically almost periodic solutions to parabolic equations on the real hyperbolic manifold
• On asymptotically almost periodic solutions to the Navier-Stokes equations on hyperbolic manifolds
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• Generalized gravitational fields and well-posedness of the Boussinesq systems on non-compact Riemannian Manifolds
• Well-posedness and exponential stability for Boussinesq systems on real hyperbolic Manifolds and application