An estimate for the vorticity of the Navier–Stokes equation

Zhongmin Qian

doi:10.1016/j.crma.2008.11.007

Probability Theory

An estimate for the vorticity of the Navier–Stokes equation
[Une estimation de la vorticité de l'équation de Navier–Stokes]

Présenté par : Marc Yor

Zhongmin Qian ¹

¹ Mathematical Institute, University of Oxford, 24–29, St Giles', Oxford OX1 3LB, UK

Comptes Rendus. Mathématique, Volume 347 (2009) no. 1-2, pp. 89-92.

Résumé
Abstract

Supposons que $\vec{u} (\cdot, t)$ soit une solution de l'équation de Navier–Stokes sur le torus $T^{3}$ de la dimension 3, et soit $\vec{ω} (\cdot, t) = \nabla \times \vec{u} (\cdot, t)$ la vorticité, nous démontrons dans cette Note que l'application

‖ \vec{ω} (\cdot, t) ‖_{1} + \frac{\sqrt{2}}{4 ν} ‖ \vec{u} (\cdot, t) ‖_{2}^{2}

est une fonction décroissante en t.

Let $\vec{u} (\cdot, t)$ be a strong solution of the Navier–Stokes equation on 3-dimensional torus $T^{3}$ , and $\vec{ω} (\cdot, t) = \nabla \times \vec{u} (\cdot, t)$ be the vorticity. In this Note we show that

‖ \vec{ω} (\cdot, t) ‖_{1} + \frac{\sqrt{2}}{4 ν} ‖ \vec{u} (\cdot, t) ‖_{2}^{2}

is decreasing in t as long as the solution

\vec{u} (\cdot, t)

exists, where

ν > 0

is the viscosity constant and

‖ \cdot ‖_{q}

denotes the

L^{q}

-norm.

Reçu le : 2008-04-01
Accepté le : 2008-11-04
Publié le : 2008-11-26

DOI : 10.1016/j.crma.2008.11.007

Affiliations des auteurs :

Zhongmin Qian ¹

¹ Mathematical Institute, University of Oxford, 24–29, St Giles', Oxford OX1 3LB, UK

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     title = {An estimate for the vorticity of the {Navier{\textendash}Stokes} equation},
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Zhongmin Qian. An estimate for the vorticity of the Navier–Stokes equation. Comptes Rendus. Mathématique, Volume 347 (2009) no. 1-2, pp. 89-92. doi : 10.1016/j.crma.2008.11.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.11.007/

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