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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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

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
J. M. Rodríguez; R. Taboada-Vázquez Numerical behaviour of a new LES model with nonlinear viscosity, Journal of Computational and Applied Mathematics, Volume 377 (2020), p. 12 (Id/No 112868) | DOI:10.1016/j.cam.2020.112868 | Zbl:1434.76029
Aibin Zang Existence of weak solutions for non-stationary flows of fluids with shear thinning dependent viscosities under slip boundary conditions in half space, Science China. Mathematics, Volume 61 (2018) no. 4, pp. 727-744 | DOI:10.1007/s11425-016-0686-1 | Zbl:1388.35003
J. M. Rodríguez; R. Taboada-Vázquez A new LES model derived from generalized Navier-Stokes equations with nonlinear viscosity, Computers Mathematics with Applications, Volume 73 (2017) no. 2, pp. 294-303 | DOI:10.1016/j.camwa.2016.11.024 | Zbl:1370.76032
Tomás Chacón Rebollo; Roger Lewandowski Evolutionary NS-TKE Model, Mathematical and Numerical Foundations of Turbulence Models and Applications (2014), p. 247 | DOI:10.1007/978-1-4939-0455-6_8
Martin Fuchs Stationary flows of shear thickening fluids in 2D, Journal of Mathematical Fluid Mechanics, Volume 14 (2012) no. 1, pp. 43-54 | DOI:10.1007/s00021-010-0044-8 | Zbl:1294.35097
Miroslav Bulíček; Piotr Gwiazda; Josef Málek; Agnieszka Świerczewska-Gwiazda On Unsteady Flows of Implicitly Constituted Incompressible Fluids, SIAM Journal on Mathematical Analysis, Volume 44 (2012) no. 4, p. 2756 | DOI:10.1137/110830289
H. Beirão da Veiga; F. Crispo; C. R. Grisanti Reducing slip boundary value problems from the half to the whole space. Applications to inviscid limits and to non-Newtonian fluids, Journal of Mathematical Analysis and Applications, Volume 377 (2011) no. 1, pp. 216-227 | DOI:10.1016/j.jmaa.2010.10.045 | Zbl:1223.35254

Cité par 8 documents. Sources : Crossref, zbMATH

^☆ 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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