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

Presented by: 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.

Abstract
Résumé

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.

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.

Accepted: 2007-07-10
Published online: 2007-08-24

DOI: 10.1016/j.crma.2007.07.015

Author's affiliations:

Hugo Beirão da Veiga ¹

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

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     author = {Hugo Beir\~ao da Veiga},
     title = {Concerning the {Ladyzhenskaya{\textendash}Smagorinsky} turbulence model of the {Navier{\textendash}Stokes} equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {249--252},
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     year = {2007},
     doi = {10.1016/j.crma.2007.07.015},
     language = {en},
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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/

References
Cited by

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[2] H. Beirão da Veiga, Navier–Stokes equations with shear thickening viscosity. Regularity up to the boundary, J. Math. Fluid Mech., in press

[3] H. Beirão da Veiga, Navier–Stokes equations with shear thinning viscosity. Regularity up to the boundary, J. Math. Fluid Mech., in press

[4] H. Beirão da Veiga, On the Ladyzhenskaya–Smagorinsky turbulence model of the Navier–Stokes equations in smooth domains. The regularity problem, J. Eur. Math. Soc., submitted for publication

[5] H. Beirão da Veiga, On the global regularity of shear thinning flows in smooth domains, in press

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