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 incompressible anisotropic Navier−Stokes equations.
Accepté le :
DOI : 10.1051/m2an/2016008
Mots clés : Navier−Stokes equations, Brinkman−Forchheimer-extended Darcy model, anisotropic viscosity
@article{M2AN_2016__50_6_1817_0, author = {Bessaih, Hakima and Trabelsi, Saber and Zorgati, Hamdi}, 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}, volume = {50}, number = {6}, year = {2016}, doi = {10.1051/m2an/2016008}, zbl = {1356.35155}, mrnumber = {3580123}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2016008/} }
TY - JOUR AU - Bessaih, Hakima AU - Trabelsi, Saber AU - Zorgati, Hamdi TI - Existence and uniqueness of global solutions for the modified anisotropic 3D Navier−Stokes equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 1817 EP - 1823 VL - 50 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2016008/ DO - 10.1051/m2an/2016008 LA - en ID - M2AN_2016__50_6_1817_0 ER -
%0 Journal Article %A Bessaih, Hakima %A Trabelsi, Saber %A Zorgati, Hamdi %T Existence and uniqueness of global solutions for the modified anisotropic 3D Navier−Stokes equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 1817-1823 %V 50 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2016008/ %R 10.1051/m2an/2016008 %G en %F M2AN_2016__50_6_1817_0
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
Convection naturelle Thermosolutale dans une Cavité Poreuse Anisotrope: Formulation de Darcy-Brinkman.Rev. Energ. Ren. 5 (2002) 1–21.
, and ,Weak and strong solutions for the incompressible Navier−Stokes equations with damping. J. Math. Ana. Appl. 343 (2008) 799–809. | DOI | MR | Zbl
and ,Fluids with anisotropic viscosity. ESAIM: M2AN 34 (2000) 315–335. | DOI | Numdam | MR | Zbl
, , and ,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
Ekman layers of rotating fluid, the case of well prepared initial data. Commun. Partial Differ. Eq. 22 (1997) 953–975. | DOI | MR | Zbl
and ,Smooth attractors for the Brinkman−Forchheimer equations with fast growing nonlinearities. Commun. Pure Appl. Anal. 11 (2012) 2037–2054. | DOI | MR | Zbl
and ,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
Continuous data assimilation for the three-dimensional Brinkman−Forchheimer-extended Darcy model. Nonlinearity 29 (2016) 1292. | DOI | MR | Zbl
, and ,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
Compact sets in the space . Ann. Mat. Pura Appl. 146 (1987) 65–96. | DOI | MR | Zbl
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