Existence and uniqueness of global solutions for the modified anisotropic 3D Navier−Stokes equations
Bessaih, Hakima1 ; Trabelsi, Saber2 ; Zorgati, Hamdi3
1 University of Wyoming, Department of Mathematics, Dept. 3036, 1000 East University Avenue, Laramie WY 82071, US.
2 Division of Mathematical and Computer Science and Engineering, King Abdullah University of Science and Technology, 23955-6900 Thuwal, Saudi Arabia.
3 Département de Mathématiques, Campus Universitaire, Université Tunis El Manar 2092, Tunisia.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 6, pp. 1817-1823.

We study a modified three-dimensional incompressible anisotropic Navier−Stokes equations. The modification consists in the addition of a power term to the nonlinear convective one. This modification appears naturally in porous media when a fluid obeys the Darcy−Forchheimer law instead of the classical Darcy law. We prove global in time existence and uniqueness of solutions without assuming the smallness condition on the initial data. This improves the result obtained for the classical 3D incompressible anisotropic Navier−Stokes equations.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2016008
Classification : 35Q30, 35Q35, 76D05, 76D03, 76S05
Mots-clés : Navier−Stokes equations, Brinkman−Forchheimer-extended Darcy model, anisotropic viscosity
Bessaih, Hakima 1 ; Trabelsi, Saber 2 ; Zorgati, Hamdi 3

1 University of Wyoming, Department of Mathematics, Dept. 3036, 1000 East University Avenue, Laramie WY 82071, US.
2 Division of Mathematical and Computer Science and Engineering, King Abdullah University of Science and Technology, 23955-6900 Thuwal, Saudi Arabia.
3 Département de Mathématiques, Campus Universitaire, Université Tunis El Manar 2092, Tunisia. 
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     title = {Existence and uniqueness of global solutions for the modified anisotropic {3D} {Navier\ensuremath{-}Stokes} equations},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1817--1823},
     publisher = {EDP-Sciences},
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Bessaih, Hakima; Trabelsi, Saber; Zorgati, Hamdi. Existence and uniqueness of global solutions for the modified anisotropic 3D Navier−Stokes equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 6, pp. 1817-1823. doi : 10.1051/m2an/2016008. http://www.numdam.org/articles/10.1051/m2an/2016008/

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations. Vol. 343 of Grundl. Math. Wiss. [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg (2011). | MR | Zbl

R. Bennacer, A. Tobbal and H. Beji, Convection naturelle Thermosolutale dans une Cavité Poreuse Anisotrope: Formulation de Darcy-Brinkman.Rev. Energ. Ren. 5 (2002) 1–21.

X. Cai and Q. Jiu, Weak and strong solutions for the incompressible Navier−Stokes equations with damping. J. Math. Ana. Appl. 343 (2008) 799–809. | DOI | MR | Zbl

J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Fluids with anisotropic viscosity. ESAIM: M2AN 34 (2000) 315–335. | DOI | Numdam | MR | Zbl

J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Mathematical Geophysics. An Introduction to Rotating Fluids and the Navier−Stokes Equations. Vol. 32 Oxford Lect. Ser. Math. Appl. (2006). | MR | Zbl

E. Grenier and N. Masmoudi, Ekman layers of rotating fluid, the case of well prepared initial data. Commun. Partial Differ. Eq. 22 (1997) 953–975. | DOI | MR | Zbl

V. Kalantarov and S. Zelik, Smooth attractors for the Brinkman−Forchheimer equations with fast growing nonlinearities. Commun. Pure Appl. Anal. 11 (2012) 2037–2054. | DOI | MR | Zbl

D. Iftimie, A uniqueness result for the Navier−Stokes equations with vanishing vertical viscosity. SIAM J. Math. Anal. 33 1483–1493. | DOI | MR | Zbl

O.A. Ladyžhenskaya, The Mathematical Theory Of Viscous Incompressible Flow. Second English edition, revised and enlarged. Vol. 2 of Mathematics and its Applications. Gordon and Breach Science Publishers, New York (1969). | MR | Zbl

P.A. Markowich, E.S. Titi and S. Trabelsi, Continuous data assimilation for the three-dimensional Brinkman−Forchheimer-extended Darcy model. Nonlinearity 29 (2016) 1292. | DOI | MR | Zbl

M. Paicu, Équation anisotrope de Navier−Stokes dans des espaces critiques. Rev. Mat. Iberoamer. 21 (2005) 179–235. | DOI | MR | Zbl

J. Pedlosky, Geophysical Fluids Dynamics. Springer Verlag, New York (1987). | Zbl

R. Temam, Infinite Dimensional Dynamical Systems In Mechanics and Physics. Springer-Verlag, New York (1997). | MR | Zbl

J. Simon, Compact sets in the space L p (0,T;B). Ann. Mat. Pura Appl. 146 (1987) 65–96. | DOI | MR | Zbl

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