American Institute of Mathematical Sciences

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

Anisotropically diffused and damped Navier-Stokes equations

Hermenegildo Borges de Oliveira 1,

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.
Keywords: extinction in a fi nite time., existence, anisotropic diff usion, Navier-Stokes, anisotropic damping.
Mathematics Subject Classification: Primary: 35Q35, 35D30, 35B40; Secondary: 76D05, 76D0.
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).   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[9]

J.-L. Lions., Quelques mèthodes de résolution des problèmes aux limites non liniaires., Dunod, (1969).   Google Scholar

[10]

J. Málek, J. Nečas, M. Rokyta and M.R$\dot u$žička., Weak and measure-valued solutions to evolutionary PDEs., Chapman & Hall, (1996).   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[13]

J. Rákosník., Some remarks to anisotropic Sobolev spaces. II., Beiträge Anal. No. 15 (1980), 15 (1980), 127.   Google Scholar

[14]

M. Troisi., Teoremi di inclusione per spazi di Sobolev non isotropi., Ricerche Mat. 18 (1969), 18 (1969), 3.   Google Scholar

show all references

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).   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[9]

J.-L. Lions., Quelques mèthodes de résolution des problèmes aux limites non liniaires., Dunod, (1969).   Google Scholar

[10]

J. Málek, J. Nečas, M. Rokyta and M.R$\dot u$žička., Weak and measure-valued solutions to evolutionary PDEs., Chapman & Hall, (1996).   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[13]

J. Rákosník., Some remarks to anisotropic Sobolev spaces. II., Beiträge Anal. No. 15 (1980), 15 (1980), 127.   Google Scholar

[14]

M. Troisi., Teoremi di inclusione per spazi di Sobolev non isotropi., Ricerche Mat. 18 (1969), 18 (1969), 3.   Google Scholar

[1]

Xuanji Jia, Zaihong Jiang. An anisotropic regularity criterion for the 3D Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1299-1306. doi: 10.3934/cpaa.2013.12.1299
[2]

Lihuai Du, Ting Zhang. Local and global strong solution to the stochastic 3-D incompressible anisotropic Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4745-4765. doi: 10.3934/dcds.2018209
[3]

Daniel Coutand, Steve Shkoller. Turbulent channel flow in weighted Sobolev spaces using the anisotropic Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations. Communications on Pure & Applied Analysis, 2004, 3 (1) : 1-23. doi: 10.3934/cpaa.2004.3.1
[4]

Patrick Penel, Milan Pokorný. Improvement of some anisotropic regularity criteria for the Navier--Stokes equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1401-1407. doi: 10.3934/dcdss.2013.6.1401
[5]

Xue-Li Song, Yan-Ren Hou. Attractors for the three-dimensional incompressible Navier-Stokes equations with damping. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 239-252. doi: 10.3934/dcds.2011.31.239
[6]

Wojciech M. Zajączkowski. Long time existence of regular solutions to non-homogeneous Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1427-1455. doi: 10.3934/dcdss.2013.6.1427
[7]

Giovanni P. Galdi. Existence and uniqueness of time-periodic solutions to the Navier-Stokes equations in the whole plane. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1237-1257. doi: 10.3934/dcdss.2013.6.1237
[8]

Huicheng Yin, Lin Zhang. The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1063-1102. doi: 10.3934/dcds.2018045
[9]

Daoyuan Fang, Bin Han, Matthias Hieber. Local and global existence results for the Navier-Stokes equations in the rotational framework. Communications on Pure & Applied Analysis, 2015, 14 (2) : 609-622. doi: 10.3934/cpaa.2015.14.609
[10]

Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085
[11]

Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations II: Global existence of small solutions. Evolution Equations & Control Theory, 2012, 1 (1) : 217-234. doi: 10.3934/eect.2012.1.217
[12]

Takeshi Taniguchi. The exponential behavior of Navier-Stokes equations with time delay external force. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 997-1018. doi: 10.3934/dcds.2005.12.997
[13]

Grzegorz Karch, Xiaoxin Zheng. Time-dependent singularities in the Navier-Stokes system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3039-3057. doi: 10.3934/dcds.2015.35.3039
[14]

Yejuan Wang, Tongtong Liang. Mild solutions to the time fractional Navier-Stokes delay differential inclusions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3713-3740. doi: 10.3934/dcdsb.2018312
[15]

Fang Li, Bo You, Yao Xu. Dynamics of weak solutions for the three dimensional Navier-Stokes equations with nonlinear damping. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4267-4284. doi: 10.3934/dcdsb.2018137
[16]

Daniel Pardo, José Valero, Ángel Giménez. Global attractors for weak solutions of the three-dimensional Navier-Stokes equations with damping. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3569-3590. doi: 10.3934/dcdsb.2018279
[17]

Fang Li, Bo You. Pullback exponential attractors for the three dimensional non-autonomous Navier-Stokes equations with nonlinear damping. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 55-80. doi: 10.3934/dcdsb.2019172
[18]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602
[19]

Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149
[20]

Joel Avrin. Global existence and regularity for the Lagrangian averaged Navier-Stokes equations with initial data in $H^{1//2}$. Communications on Pure & Applied Analysis, 2004, 3 (3) : 353-366. doi: 10.3934/cpaa.2004.3.353

 Impact Factor: 

Tools

Metrics

  • PDF downloads (56)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences