Regularity theorems, up to the boundary, for shear thickening flows

Hugo Beirão da Veiga; Petr Kaplický; Michael Růžička

doi:10.1016/j.crma.2010.04.010

Partial Differential Equations

Regularity theorems, up to the boundary, for shear thickening flows
[Théorèmes de régularité, jusqu'à la frontière des solutions de problèmes aux limites pour des fluides visqueux dilatants]

Présenté par : Haïm Brezis

Hugo Beirão da Veiga ¹ ; Petr Kaplický ² ; Michael Růžička ³

¹ Dipartimento di Matematica Applicata “U. Dini”, Via Buonarrotti, 1/C, 56127 Pisa, Italy
² Charles University in Prague, Sokolovská 83, Praha 8, 18675, Czech Republic
³ Mathematical Institut, University Freiburg, Eckerstr. 1, 79104 Freiburg, Germany

Comptes Rendus. Mathématique, Volume 348 (2010) no. 9-10, pp. 541-544.

Résumé
Abstract

Dans cette Note on étudie la régularité jusqu'à la frontière des solutions faibles de systèmes décrivant le mouvement de fluides newtoniens généralisés visqueux dilatants, dans le cas de conditions aux limites de Dirichlet homogènes. Le tenseur des contraintes supplémentaires $S (D)$ , voir (2), est donné par une loi de puissance avec un exposant $p ⩾ 2$ . Des résultats détaillés présentés ici sont donnés dans un article à paraître [4] (H. Beirão da Veiga et al., in press). Dans cette Note on se limite à l'énoncé des résultats démontrés dans [4] (H. Beirão da Veiga et al., in press) suivis de commentaires.

This Note concerns the regularity up to the boundary of weak solutions to systems describing the flow of generalized Newtonian shear thickening fluids under the homogeneous Dirichlet boundary condition. The extra stress tensor $S (D)$ , see (2) below, is given by a power law with shear exponent $p ⩾ 2$ . Complete proofs of the results presented here are given in the forthcoming paper [4] (H. Beirão da Veiga et al., in press). The aim of this Note is to describe the results proved in H. Beirão da Veiga et al. (in press) [4], together with suitable comments.

Reçu le : 2010-03-17
Publié le : 2010-04-27

DOI : 10.1016/j.crma.2010.04.010

Affiliations des auteurs :

Hugo Beirão da Veiga ¹ ; Petr Kaplický ² ; Michael Růžička ³

¹ Dipartimento di Matematica Applicata “U. Dini”, Via Buonarrotti, 1/C, 56127 Pisa, Italy
² Charles University in Prague, Sokolovská 83, Praha 8, 18675, Czech Republic
³ Mathematical Institut, University Freiburg, Eckerstr. 1, 79104 Freiburg, Germany

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     title = {Regularity theorems, up to the boundary, for shear thickening flows},
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Hugo Beirão da Veiga; Petr Kaplický; Michael Růžička. Regularity theorems, up to the boundary, for shear thickening flows. Comptes Rendus. Mathématique, Volume 348 (2010) no. 9-10, pp. 541-544. doi : 10.1016/j.crma.2010.04.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.04.010/

Bibliographie
Cité par

[1] H. Beirão da Veiga On the regularity of flows with Ladyzhenskaya shear-dependent viscosity and slip or nonslip boundary conditions, Comm. Pure Appl. Math., Volume 58 (2005) no. 4, pp. 552-577

[2] 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., Volume 11 (2009), pp. 127-167

[3] H. Beirão da Veiga Turbulence models, p-fluid flows, and $W^{2, l}$ -regularity of solutions, Comm. Pure Appl. Anal., Volume 8 (2009), pp. 769-783

[4] H. Beirão da Veiga, P. Kaplický, M. Růžička, Boundary regularity of shear thickening flows, J. Math. Fluid Mech., in press, | DOI

[5] L.C. Berselli, L. Diening, M. Růžička, Existence of strong solutions for incompressible fluids with shear dependent viscosities, J. Math. Fluid Mech., in press, | DOI

[6] M. Bulíček, F. Ettwein, P. Kaplický, D. Pražák, On uniqueness and time regularity of flows of power-law like non-Newtonian fluids, Math. Methods Appl. Sci. (2010), | DOI

[7] F. Crispo; C. Grisanti On the $C^{1, γ} (\bar{Ω}) \cap W^{2, 2} (Ω)$ regularity for a class of electro-rheological fluids, J. Math. Anal. Appl., Volume 356 (2009), pp. 119-132

[8] L. Diening; M. Růžička Strong solutions for generalized Newtonian fluids, J. Math. Fluid Mech., Volume 7 (2005), pp. 413-450

[9] P. Kaplický Regularity of flows of a non-Newtonian fluid subject to Dirichlet boundary conditions, Z. Anal. Anwendungen, Volume 24 (2005) no. 3, pp. 467-486

[10] P. Kaplický; J. Málek; J. Stará $C^{1, α}$ -regularity of weak solutions to a class of nonlinear fluids in two dimensions – stationary Dirichlet problem, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), Volume 259 (1999), pp. 89-121

[11] O.A. Ladyzhenskaya The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969

[12] J.L. Lions Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969

[13] J. Málek; J. Nečas; M. Rokyta; M. Růžička Weak and Measure-Valued Solutions to Evolutionary PDEs, Applied Mathematics and Mathematical Computations, vol. 13, Chapman & Hall, London, 1996

[14] J. Málek; J. Nečas; M. Růžička On weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains: the case $p ⩾ 2$ , Adv. Differential Equations, Volume 6 (2001) no. 3, pp. 257-302

Cité par Sources :

^☆ Michael Růžička has been supported by DFG Forschergruppe “Nonlinear Partial Differential Equations: Theoretical and Numerical Analysis”. Hugo Beirão da Veiga and Petr Kaplický thank the University of Freiburg for the kind hospitality during part of the preparation of the Note. Research of Petr Kaplický was also supported by the grant GACR 201/09/0917 and partially also by the research project MSM 0021620839.

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