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.

Abstract (VO)
Résumé

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.

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.

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

@article{CRMATH_2009__347_1-2_89_0,
     author = {Zhongmin Qian},
     title = {An estimate for the vorticity of the {Navier{\textendash}Stokes} equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {89--92},
     publisher = {Elsevier},
     volume = {347},
     number = {1-2},
     year = {2009},
     doi = {10.1016/j.crma.2008.11.007},
     language = {en},
}

TY  - JOUR
AU  - Zhongmin Qian
TI  - An estimate for the vorticity of the Navier–Stokes equation
JO  - Comptes Rendus. Mathématique
PY  - 2009
SP  - 89
EP  - 92
VL  - 347
IS  - 1-2
PB  - Elsevier
DO  - 10.1016/j.crma.2008.11.007
LA  - en
ID  - CRMATH_2009__347_1-2_89_0
ER  -

%0 Journal Article
%A Zhongmin Qian
%T An estimate for the vorticity of the Navier–Stokes equation
%J Comptes Rendus. Mathématique
%D 2009
%P 89-92
%V 347
%N 1-2
%I Elsevier
%R 10.1016/j.crma.2008.11.007
%G en
%F CRMATH_2009__347_1-2_89_0

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/

Bibliographie
Cité par

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

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

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

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

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

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

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

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

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

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

Zhaoxia Liu A decreasing property of the 3D magneto-hydrodynamic flows on a torus, Acta Mathematica Scientia. Series B. (English Edition), Volume 45 (2025) no. 4, pp. 1384-1390 | DOI:10.1007/s10473-025-0408-z | Zbl:8039885
Guillaume Lévy Retracted: A uniqueness lemma with applications to regularization and incompressible fluid mechanics, Science China. Mathematics, Volume 64 (2021) no. 4, pp. 711-724 | DOI:10.1007/s11425-018-9477-3 | Zbl:1464.35185
Siran Li On vortex alignment and the boundedness of the $L^{q}$ -norm of vorticity in incompressible viscous fluids, Acta Mathematica Scientia. Series B. (English Edition), Volume 40 (2020) no. 6, pp. 1700-1708 | DOI:10.1007/s10473-020-0606-7 | Zbl:1499.35477
Li Ma Eigenvalue estimates and $L^{1}$ energy on closed manifolds, Acta Mathematica Sinica. English Series, Volume 30 (2014) no. 10, pp. 1729-1734 | DOI:10.1007/s10114-014-1726-6 | Zbl:1304.58017
Terence Tao Localisation and compactness properties of the Navier–Stokes global regularity problem, Analysis PDE, Volume 6 (2013) no. 1, p. 25 | DOI:10.2140/apde.2013.6.25

Cité par 5 documents. Sources : Crossref, zbMATH

Commentaires - Politique