Navier–Stokes equations on Riemannian manifolds

Abstract

We study properties of the solutions to Navier-Stokes system on compact Riemannian manifolds. The motivation for such a formulation comes from atmospheric models as well as some thin film flows on curved surfaces. There are different choices of the diffusion operator which have been used in previous studies, and we make a few comments why the choice adopted below seems to us the correct one. This choice leads to the conclusion that Killing vector fields are essential in analyzing the qualitative properties of the flow. We give several results illustrating this and analyze also the linearized version of Navier-Stokes system which is interesting in numerical applications. Finally we consider the 2 dimensional case which has specific characteristics, and treat also the Coriolis effect which is essential in atmospheric flows.

On zonal steady solutions to the 2D Euler equations on the rotating unit sphere

“…A complete introduction to the theory of solutions to Navier-Stokes and Euler equations on the two dimensional unit sphere can be found in [31] and the references therein. See also [30] and the references therein for a discussion regarding the Navier-Stokes and Euler equations on compact Riemannian manifolds. Just like shear flows in the Euclidean setting, zonal flows are basic for understanding the long time dynamics of the equations.…”

$H^\infty$-calculus for the surface Stokes operator and applications

“…where ∆ Σ is the Bochner (the connection) Laplacian and Ric Σ u is the Ricci (1, 1)tensor, given in local coordinates by (Ric Σ ) ℓ k = g iℓ R ik , so that Ric Σ u = R ℓ k u k τ ℓ . It is well-known that (∆ Σ u|u) Σ = −|∇u| 2 L2(Σ) for u ∈ H 2 2 (Σ, TΣ), see for instance [30,Lemma 3.5]. Let u ∈ H 2 2,σ (Σ, TΣ).…”

Rate of the enhanced dissipation for the two-jet Kolmogorov type flow on the unit sphere

“…The Navier-Stokes equations on spheres and more general manifolds with this kind of viscous term have been studied by many authors (see e.g. [43,36,33,31,9,21,5,7,39,40,35,22,37,38]). There are also several works on the Navier-Stokes equations on manifolds in which the viscous term is taken to be ν∆ H u by analogy of the flat domain case (see e.g.…”