Concerning the Ladyzhenskaya–Smagorinsky turbulence model of the Navier–Stokes equations

Hugo Beirão da Veiga

doi:10.1016/j.crma.2007.07.015

Partial Differential Equations

Concerning the Ladyzhenskaya–Smagorinsky turbulence model of the Navier–Stokes equations
[Sur le modèle de turbulence de Ladyzhenskaya–Smagorinsky des équations de Navier–Stokes]

Présenté par : Haïm Brezis

Hugo Beirão da Veiga ¹

¹ Dipartimento di Matematica Applicata “U.Dini”, Via Buonarrotti, 1/C, 56127 Pisa, Italy

Comptes Rendus. Mathématique, Volume 345 (2007) no. 5, pp. 249-252.

Résumé
Abstract

Dans des articles récents (voir ci-après) nous avons démontré des résultats de régularité dans des espaces $L^{q}$ pour les dérivées secondes de la vitesse et les dérivées premières de la presssion, pour des systèmes de Stokes et Navier–Stokes avec des viscosités qui dépendent de la partie symétrique du gradient des vitesses. Nous considérons seulement des résultats de régularité valables jusqu'au bord. Dans ces articles nous avons considéré des frontières plates. Tous récemment nous avons généraliser ces résultats aux cas des frontières arbitraires. Le but de cette note est de décrire ces résultats, avec des commentaires adéquats.

In some recent papers (see below) we prove regularity results in $L^{q} (Ω)$ spaces for the second order derivatives of the velocity and the first order derivatives of the pressure for solutions to Stokes and Navier–Stokes systems of equations with shear thickening viscosity. We take into account only regularity results that hold up to the boundary. In the above references we consider flat boundaries. More recently, we have extended the above results to the case of curvilinear boundaries. The aim of this note is to describe these last results, together with suitable comments.

Accepté le : 2007-07-10
Publié le : 2007-08-24

DOI : 10.1016/j.crma.2007.07.015

Affiliations des auteurs :

Hugo Beirão da Veiga ¹

¹ Dipartimento di Matematica Applicata “U.Dini”, Via Buonarrotti, 1/C, 56127 Pisa, Italy

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     pages = {249--252},
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Hugo Beirão da Veiga. Concerning the Ladyzhenskaya–Smagorinsky turbulence model of the Navier–Stokes equations. Comptes Rendus. Mathématique, Volume 345 (2007) no. 5, pp. 249-252. doi : 10.1016/j.crma.2007.07.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.07.015/

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Cité par

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Songmao He; Xin-Guang Yang Asymptotic behavior of 3D Ladyzhenskaya-type fluid flow model with delay, Discrete and Continuous Dynamical Systems. Series S, Volume 18 (2025) no. 3, pp. 832-853 | DOI:10.3934/dcdss.2024135 | Zbl:7989693
João Carlos F. Barreira; Juan Límaco Local Null Controllability of the Smagorinsky-Boussinesq Model, Analysis and PDE in Latin America, Volume 10 (2024), p. 49 | DOI:10.1007/978-3-031-73274-4_7
Hyeong-Ohk Bae; Jörg Wolf Existence of strong solutions to the equations of unsteady motion of shear thickening incompressible fluids, Nonlinear Analysis. Real World Applications, Volume 23 (2015), pp. 160-182 | DOI:10.1016/j.nonrwa.2014.12.002 | Zbl:1311.35204
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Francesca Crispo Global regularity of a class of $p$ -fluid flows in cylinders, Journal of Mathematical Analysis and Applications, Volume 341 (2008) no. 1, pp. 559-574 | DOI:10.1016/j.jmaa.2007.10.034 | Zbl:1135.35061

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