Comptes Rendus
no. 15-16
Partial Differential Equations
Reproductive solution of a second-grade fluid system
[Solution reproductive d'un système de fluide de grade deux]

Présenté par : Philippe G. Ciarlet

Luis Friz1 ; Francisco Guillén-González2 ; M.A. Rojas-Medar1
1 Depto. Ciencias Básicas, Facultad de Ciencias, Universidad del Bío-Bío, Avenida Andrés Bello s/n, Casilla 447, Chillán, Chile
2 Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apto. 1160, 41080 Sevilla, Spain
Comptes Rendus. Mathématique, Volume 348 (2010) no. 15-16, pp. 879-883.

On traite l'existence et l'unicité de la solution reproductive d'un système de fluide de grade deux avec des termes sources suffisamments petits, en utilisant la méthode d'approximation de Galerkin et des arguments de compacité.

We treat the existence and uniqueness of reproductive solution (weak time-periodic solution) of a second-grade fluid system for small enough source terms, by using the Galerkin approximation method and compactness arguments.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.07.021
Luis Friz 1 ; Francisco Guillén-González 2 ; M.A. Rojas-Medar 1

1 Depto. Ciencias Básicas, Facultad de Ciencias, Universidad del Bío-Bío, Avenida Andrés Bello s/n, Casilla 447, Chillán, Chile
2 Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apto. 1160, 41080 Sevilla, Spain 
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     title = {Reproductive solution of a second-grade fluid system},
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Luis Friz; Francisco Guillén-González; M.A. Rojas-Medar. Reproductive solution of a second-grade fluid system. Comptes Rendus. Mathématique, Volume 348 (2010) no. 15-16, pp. 879-883. doi : 10.1016/j.crma.2010.07.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.07.021/

[1] D. Cioranescu; V. Girault Weak and classical solutions of a family of second grade fluids, Internat. J. Non-Linear Mech., Volume 32 (1997), pp. 317-335

[2] D. Cioranescu; O. El Hacène Existence and uniqueness for fluids of second grade, Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar, vol. 109, Pitman, 1984, pp. 178-197

[3] J.E. Dunn; R.L. Fosdick Thermodynamics, stability and boundedness of fluids of complexity two and fluids of second grade, Arch. Ration. Mech. Anal., Volume 56 (1974), pp. 191-252

[4] R.L. Fosdick; K.R. Rajapogal Anomalous features in the model of second grade fluids, Arch. Ration. Mech. Anal., Volume 70 (1978), pp. 145-152

[5] G.P. Galdi; A. Sequeira Further existence results for classical solutions of the equations of second-grade fluid, Arch. Ration. Mech. Anal., Volume 128 (1994), pp. 297-312

[6] V. Girault; P.A. Raviart Finite Element Methods for Navier–Stokes Equations. Theory and Algorithms, SCM, vol. 5, Springer-Verlag, Berlin, 1986

[7] R. Temam Navier–Stokes Equations, North-Holland, Amsterdam, 1977

Cité par Sources :

☆ The first author is supported by FONDECYT-Chile Grant 1090510. The second and third authors were supported by project DGI-MEC Grant MTM2006-07932, Spain and FONDECYT-Chile Grant 1080628. Moreover, the second author also was supported by Junta de Andalucía project P06-FQM-02373, Spain.

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