# American Institute of Mathematical Sciences

2015, 2015(special): 349-358. doi: 10.3934/proc.2015.0349

## Anisotropically diffused and damped Navier-Stokes equations

 1 FCT - Universidade do Algarve, Campus de Gambelas, 8005-139 Faro, Portugal

Received  September 2014 Revised  February 2015 Published  November 2015

The incompressible Navier-Stokes equations with anisotropic diffusion and anisotropic damping is considered in this work. For the associated initial-boundary value problem, we prove the existence of weak solutions and we establish an energy inequality satisfied by these solutions. We prove also under what conditions the solutions of this problem extinct in a finite time.
Citation: Hermenegildo Borges de Oliveira. Anisotropically diffused and damped Navier-Stokes equations. Conference Publications, 2015, 2015 (special) : 349-358. doi: 10.3934/proc.2015.0349
##### References:
 [1] S.N. Antontsev, J.I. Díaz and S.I. Shmarev., Energy methods for free boundary problems., Progr. Nonlinear Differential Equations Appl. 48, 48 (2002). [2] S.N. Antontsev and H.B. de Oliveira., Analysis of the existence for the steady Navier-Stokes equations with anisotropic diffusion., Adv. Differential Equations 19 (2014) no. 5-6, 19 (2014), 5. [3] S.N. Antontsev and H.B. de Oliveira., Evolution problems of Navier-Stokes type with anisotropic diffusion., \emph{Revista de la Real Academia de Ciencias Exactas, (2015), 1. [4] I. Fragalà, F. Gazzola and B. Kawohl., Existence and nonexistence results for anisotropic quasilinear elliptic equations., Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), 21 (2004), 715. [5] J.K. Djoko, and P.A. Razafimandimby., Analysis of the Brinkman-Forchheimer equations with slip boundary conditions., Appl. Anal. 93 (2014), 93 (2014), 1477. [6] J. Frehse, J. Málek and M. Steinhauer., On analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method., SIAM J. Math. Anal. 34 (2003), 34 (2003), 1064. [7] J. Haškovec and C. Schmeiser., A note on the anisotropic generalizations of the Sobolev and Morrey embedding theorems., Monatsh. Math. 158 (2009), 158 (2009), 71. [8] V. Kalantarov and S. Zelik., Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities., Commun. Pure Appl. Anal. 11 (2012), 11 (2012), 2037. [9] J.-L. Lions., Quelques mèthodes de résolution des problèmes aux limites non liniaires., Dunod, (1969). [10] J. Málek, J. Nečas, M. Rokyta and M.R$\dot u$žička., Weak and measure-valued solutions to evolutionary PDEs., Chapman & Hall, (1996). [11] H.B. de Oliveira., On the influence of an absorption term in incompressible fluid flows., Adv. Math. Fluid Mech. Springer-Verlag (2010), (2010), 409. [12] H.B. de Oliveira., Existence of weak solutions for the generalized Navier-Stokes equations with damping., NoDEA Nonlinear Differential Equations Appl. 20 (2013) no. 3, 20 (2013), 797. [13] J. Rákosník., Some remarks to anisotropic Sobolev spaces. II., Beiträge Anal. No. 15 (1980), 15 (1980), 127. [14] M. Troisi., Teoremi di inclusione per spazi di Sobolev non isotropi., Ricerche Mat. 18 (1969), 18 (1969), 3.

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##### References:
 [1] S.N. Antontsev, J.I. Díaz and S.I. Shmarev., Energy methods for free boundary problems., Progr. Nonlinear Differential Equations Appl. 48, 48 (2002). [2] S.N. Antontsev and H.B. de Oliveira., Analysis of the existence for the steady Navier-Stokes equations with anisotropic diffusion., Adv. Differential Equations 19 (2014) no. 5-6, 19 (2014), 5. [3] S.N. Antontsev and H.B. de Oliveira., Evolution problems of Navier-Stokes type with anisotropic diffusion., \emph{Revista de la Real Academia de Ciencias Exactas, (2015), 1. [4] I. Fragalà, F. Gazzola and B. Kawohl., Existence and nonexistence results for anisotropic quasilinear elliptic equations., Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), 21 (2004), 715. [5] J.K. Djoko, and P.A. Razafimandimby., Analysis of the Brinkman-Forchheimer equations with slip boundary conditions., Appl. Anal. 93 (2014), 93 (2014), 1477. [6] J. Frehse, J. Málek and M. Steinhauer., On analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method., SIAM J. Math. Anal. 34 (2003), 34 (2003), 1064. [7] J. Haškovec and C. Schmeiser., A note on the anisotropic generalizations of the Sobolev and Morrey embedding theorems., Monatsh. Math. 158 (2009), 158 (2009), 71. [8] V. Kalantarov and S. Zelik., Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities., Commun. Pure Appl. Anal. 11 (2012), 11 (2012), 2037. [9] J.-L. Lions., Quelques mèthodes de résolution des problèmes aux limites non liniaires., Dunod, (1969). [10] J. Málek, J. Nečas, M. Rokyta and M.R$\dot u$žička., Weak and measure-valued solutions to evolutionary PDEs., Chapman & Hall, (1996). [11] H.B. de Oliveira., On the influence of an absorption term in incompressible fluid flows., Adv. Math. Fluid Mech. Springer-Verlag (2010), (2010), 409. [12] H.B. de Oliveira., Existence of weak solutions for the generalized Navier-Stokes equations with damping., NoDEA Nonlinear Differential Equations Appl. 20 (2013) no. 3, 20 (2013), 797. [13] J. Rákosník., Some remarks to anisotropic Sobolev spaces. II., Beiträge Anal. No. 15 (1980), 15 (1980), 127. [14] M. Troisi., Teoremi di inclusione per spazi di Sobolev non isotropi., Ricerche Mat. 18 (1969), 18 (1969), 3.
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