The Incompressible Navier–Stokes Equations on Non-compact Manifolds

Pierfelice, Vittoria

doi:10.1007/s12220-016-9691-1

Cited by 35 publications

(47 citation statements)

References 52 publications

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“…However, in [6] it is argued that ∆ H should be used, at least in some situations. Also in [1] ∆ H is used while in [8] Bochner Laplacian is used. We do not know how the choice of Bochner Laplacian or the Hodge Laplacian should be interpreted from the point of view of continuum mechanics.…”

Section: Model and The Diffusion Operatormentioning

confidence: 99%

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Navier–Stokes equations on Riemannian manifolds

Samavaki

Tuomela

2020

Journal of Geometry and Physics

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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.

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Section: Model and The Diffusion Operatormentioning

confidence: 99%

“…Navier Stokes equations have been widely studied both form theoretical and applied points of view [12]. In recent years there has been a growing interest of this system on Riemannian manifolds [1,2,6,8,9,11] and the many references therein. There seems to be two different reasons for this interest.…”

Section: Introductionmentioning

confidence: 99%

Navier–Stokes equations on Riemannian manifolds

Samavaki

Tuomela

2020

Journal of Geometry and Physics

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“…When Ric is negative, the above relation yields the existence of Leray's weak solution (see for instance [26]). For the general case, the existence of Leray's weak solution to (1.9) was proved in [27] (Theorem 4.6, p.498 and p.504).…”

Section: Introductionmentioning

confidence: 99%

Generalized stochastic Lagrangian paths for the Navier-Stokes equation

Arnaudon¹,

Cruzeiro²,

Fang³

2018

Annali Scuola Normale Superiore - Classe Di Scienze

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In the note added in proof of the seminal paper [Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. 92 (1970), 102-163], Ebin and Marsden introduced the so-called correct Laplacian for the Navier-Stokes equation on a compact Riemannian manifold. In spirit of Brenier's generalized flows for the Euler equation, we introduce a class of semimartingales on a compact Riemannian manifold. We prove that these semimartingales are critical points to the corresponding kinetic energy if and only if its drift term solves weakly the Navier-Stokes equation defined with Ebin-Marsden's Laplacian. We also show that for the case of torus, classical solutions of the Navier-Stokes equation realize the minimum of the kinetic energy in a suitable class.

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“…The following formula holds (see [25] or [3]), A = −∆A − Ric(A) − ∇div(A), A ∈ X (M ) (1.4) where ∆A = Trace(∇ 2 A). The Ebin-Marsden's Laplacianˆ has been used recently in [23] to solve Navier-Stokes equation on a Riemannian manifold with negative Ricci curvature : it is quite convenient with sign minus in formula (1.4). In [7], the authors gave severals arguments from physics in order to convive the relevance ofˆ .…”

Section: Introductionmentioning

confidence: 99%

Nash Embedding, Shape Operator and Navier-Stokes Equation on a Riemannian Manifold

Fang

2020

Acta Math. Appl. Sin. Engl. Ser.

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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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The Incompressible Navier–Stokes Equations on Non-compact Manifolds

Cited by 35 publications

References 52 publications

Navier–Stokes equations on Riemannian manifolds

Navier–Stokes equations on Riemannian manifolds

Generalized stochastic Lagrangian paths for the Navier-Stokes equation

Nash Embedding, Shape Operator and Navier-Stokes Equation on a Riemannian Manifold

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