Navier–Stokes equations on Riemannian manifolds

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...

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Bibliographic Details
Published in:Journal of geometry and physics Vol. 148; p. 103543
Main Authors: Samavaki, Maryam, Tuomela, Jukka
Format: Journal Article
Language:English
Published: Elsevier B.V 01.02.2020
Subjects:
Curvature tensor
Killing vector fields
Navier–Stokes equations
Riemannian manifolds
Killing vector fields
Curvature tensor
Navier–Stokes equations
Riemannian manifolds
57N16
53C21
76D05
ISSN:0393-0440, 1879-1662
Online Access:Get full text
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Summary: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.
ISSN:0393-0440
1879-1662
DOI:10.1016/j.geomphys.2019.103543