An estimate for the vorticity of the Navier–Stokes equation

Qian, Zhongmin

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

Qian, Zhongmin ¹

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

Comptes Rendus. Mathématique, Tome 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 :

Qian, Zhongmin 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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Qian, Zhongmin. An estimate for the vorticity of the Navier–Stokes equation. Comptes Rendus. Mathématique, Tome 347 (2009) no. 1-2, pp. 89-92. doi : 10.1016/j.crma.2008.11.007. http://archive.numdam.org/articles/10.1016/j.crma.2008.11.007/

Bibliographie
Cité par

[1] Constantin, P. Navier–Stokes equations and area of interfaces, Commun. Math. Phys., Volume 129 (1990), pp. 241-266

[2] Fujita, H.; Kato, T. On the Navier–Stokes initial value problem. I, Arch. Rational Mech. Anal., Volume 16 (1964), pp. 269-315

[3] Heywood, J.G. Viscous incompressible fluids: mathematical theory, Encyclopaedia of Mathematical Physics, 2006, pp. 369-379

[4] Hopf, E. Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Mach. Nachr., Volume 4 (1950–1951), pp. 213-231

[5] Kreiss, H.O.; Lorenz, J. Initial-Boundary Value Problems and the Navier–Stokes Equations, Classics in Applied Mathematics, vol. 47, SIAM, 2004

[6] Ladyzenskaya, O.A. The Mathematical Theory of Viscous Incompressible Flow, Gordan and Breach, New York, 1969 (Translation from the Russian)

[7] Leray, J. Sur le mouvement d'un liquide visquex emplissent l'espace, Acta Math., Volume 63 (1934), pp. 193-248

[8] Majda, A.J.; Bertozzi, A.L. Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2002

[9] Serrin, J. The initial value problem for the Navier–Stokes equations (Langer, R.T., ed.), Nonlinear Problems, Proceedings of a Symposium, Madison, WI, University of Wisconsin, Madison, WI, 1963, pp. 69-98

[10] Temam, R. Navier–Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, American Mathematical Society, 2001

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