Solvability of the Navier-Stokes equations on manifolds with boundary

Priebe, Volker

doi:10.1007/bf02567605

Cited by 21 publications

(12 citation statements)

References 7 publications

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“…The Navier-Stokes equations on a Riemannian manifold were introduced in [18,53,84] and have been studied over the years (see e.g. [9,15,39,42,52,61,67,68]). When a manifold is a stationary surface in R 3 , numerical methods for the surface Navier-Stokes and Stokes equations have been developed in the recent works [64,65,[73][74][75].…”

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confidence: 99%

Navier-Stokes equations in a curved thin domain, Part III: thin-film limit

Miura¹

2020

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We consider the Navier-Stokes equations with Navier's slip boundary conditions in a three-dimensional curved thin domain around a given closed surface. Under suitable assumptions we show that the average in the thin direction of a strong solution to the bulk Navier-Stokes equations converges weakly in appropriate function spaces on the limit surface as the thickness of the thin domain tends to zero. Moreover, we characterize the limit as a weak solution to limit equations, which are the damped and weighted Navier-Stokes equations on the limit surface. We also prove the strong convergence of the average of a strong solution to the bulk equations towards a weak solution to the limit equations by showing estimates for the difference between them. In some special case our limit equations agree with the Navier-Stokes equations on a Riemannian manifold in which the viscous term contains the Ricci curvature. This is the first result on a rigorous derivation of the surface Navier-Stokes equations on a general closed surface by the thin-film limit.

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confidence: 99%

Navier-Stokes equations in a curved thin domain, Part III: thin-film limit

Miura¹

2020

Preprint

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Maekawa¹,

Miura²

2021

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We study the enhanced dissipation for the two-jet Kolmogorov type flow which is a stationary solution to the Navier-Stokes equations on the two-dimensional unit sphere given by the zonal spherical harmonic function of degree two. Based on the pseudospectral bound method developed by Ibrahim, Maekawa, and Masmoudi [15] and a modified version of the Gearhart-Prüss type theorem shown by Wei [48], we derive an estimate for the resolvent of the linearized operator along the imaginary axis and show that a solution to the linearized equation rapidly decays at the rate O(e − √ ν t ) when the viscosity coefficient ν is sufficiently small as in the case of the plane Kolmogorov flow.

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“…In that case, we also recover (1) in the Euclidean case because the curvature vanishes. Writing the Navier-Stokes as (2), we are following [Ily91,Ily94,CRT99,FF05,Rod08,Rod07]; for other writings we refer to [Pri94,CF96].…”

Section: Introductionmentioning

confidence: 99%