# American Institute of Mathematical Sciences

## Global attractors for weak solutions of the three-dimensional Navier-Stokes equations with damping

 1 Miquel Rosselló i Alemany, 51, 7 B, Palma de Mallorca, Spain 2 Universidad Miguel Hernández de Elche, Centro de Investigación Operativa, 03202, Elche (Alicante), Spain

* Corresponding author: jvalero@umh.es

Dedicated to Professor Peter Kloeden on his 70th birthday

Received  March 2018 Revised  June 2018 Published  October 2018

Fund Project: This work has been partially supported by Spanish Ministry of Economy and Competitiveness and FEDER, projects MTM2015-63723-P and MTM2016-74921-P, and by Junta de Andalucía (Spain), project P12-FQM-1492

In this paper we obtain the existence of global attractors for the dynamicalsystems generated by weak solution of the three-dimensional Navier-Stokesequations with damping. We consider two cases, depending on the values of the parameter β controlling the damping term. First, we prove that for β≥4 weaksolutions are unique and establish the existence of the global attractor forthe corresponding semigroup. Second, for 3≤β<4 we define amultivalued dynamical systems and prove the existence of the global attractoras well. Finally, some numerical simulations are performed.

Citation: 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, doi: 10.3934/dcdsb.2018279
##### References:
 [1] D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Commun. Math. Phys., 238 (2003), 211-223. doi: 10.1007/s00220-003-0859-8. [2] X. Cai and Q. Jiu, Weak and strong solutions for the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl., 343 (2008), 799-809. doi: 10.1016/j.jmaa.2008.01.041. [3] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, Rhode Island, 2002. [4] J. H. Ferziger and M. Peric, Computational Methods for Fluid Dynamics, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-642-98037-4. [5] M. Gobbino and M. Sardella, On the connectedness of attractors for dynamical systems, J. Differential Equations, 133 (1997), 1-14. doi: 10.1006/jdeq.1996.3166. [6] O. V. Kapustyan, P. O. Kasyanov and J. Valero, Structure and regularity of the global attractor of a reacction-diffusion equation with non-smooth nonlinear term, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 4155-4182. doi: 10.3934/dcds.2014.34.4155. [7] O. V. Kapustyan, P. O. Kasyanov, J. Valero and M. Z. Zgurovsky, Structure of uniform global attractors for general non-autonomous reaction-diffusion systems, in Continuous and Distributed Systems. Theory and Applications, M.Z.Zgurovsky and V.A. Sadovnichiy eds, 211 (2014), 163-180, Cham, Springer. doi: 10.1007/978-3-319-03146-0_12. [8] F. Huang and R. Pan, Convergence rate for compressible Euler equations with damping and vacuum, Arch. Rational Mech. Anal., 166 (2003), 359-376. doi: 10.1007/s00205-002-0234-5. [9] Y. Kim and K. Li, Time-periodic strong solutions of the 3D Navier-Stokes equations with damping, Electron. J. Differential Equations, 2017 (2017), 1-11. [10] O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, New-York, 1991. doi: 10.1017/CBO9780511569418. [11] J. L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites Non Linéaires, Gauthier-Villar, Paris, 1969. [12] A. A. Linninger, M. Xenos, D. C. Zhu, M. R. Somayaji, S. Kondapali and R. D. Penn, Cerebrospinal fluid flow in the normal and hydrocefalic human brain, IEEE Trans. Biomed. Eng., 54 (2007), 291-302. [13] V. S. Valero and J. Melnik AND, On attractors of multi-valued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111. doi: 10.1023/A:1008608431399. [14] J. Robinson, Infinite-dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2001. doi: 10.1007/978-94-010-0732-0. [15] X. Song and Y. Hou, Attractors for the three-dimensional incompressible Navier-Stokes equations with damping, Discrete Contin. Dyn. Syst. Ser. A, 31 (2011), 239-252. doi: 10.3934/dcds.2011.31.239. [16] X. Song and Y. Hou, Uniform attractors for the three-dimensional Navier-Stokes equations with nonlinear damping, J. Math. Anal. Appl., 422 (2015), 337-351. doi: 10.1016/j.jmaa.2014.08.044. [17] X. Song, F. Liang and J. Wu, Pullback D-attractors for three-dimensional Navier-Stokes equations with nonlinear damping, Bound. Value Probl., 2016 (2016), 15PP. doi: 10.1186/s13661-016-0654-z. [18] R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam, 1979. [19] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, SpringerVerlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8. [20] J. Valero, On locally compact attractors of dynamical systems, J. Math. Anal. Appl., 237 (1999), 43-54. doi: 10.1006/jmaa.1999.6446. [21] H. Versteeg and W. Malalasekra, An Introduction to Computational Fluid Dynamics: The Finite Volume Method, Prentice Hall, New York, 2007. [22] W. Wang and G. Zhou, Remarks on the regularity criterion of the Navier-Stokes equations with nonlinear damping, Math. Probl. Eng., 2015 (2015), Art. ID 310934, 5 pp. doi: 10.1155/2015/310934. [23] Z. Zhang, X. Wu and M. Lu, On the uniqueness of strong solution to the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl., 377 (2011), 414-419. doi: 10.1016/j.jmaa.2010.11.019. [24] X. Zhong, Global well-posedness to the incompressible Navier-Stokes equations with damping, Electron. J. Qual. Theory Differ. Equ., 62 (2017), 1-9. [25] Y. Zhou, Regularity and uniqueness for the 3D incompressible Navier-Stokes equations with damping, Appl. Math. Letters, 25 (2012), 1822-1825. doi: 10.1016/j.aml.2012.02.029.

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##### References:
 [1] D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Commun. Math. Phys., 238 (2003), 211-223. doi: 10.1007/s00220-003-0859-8. [2] X. Cai and Q. Jiu, Weak and strong solutions for the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl., 343 (2008), 799-809. doi: 10.1016/j.jmaa.2008.01.041. [3] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, Rhode Island, 2002. [4] J. H. Ferziger and M. Peric, Computational Methods for Fluid Dynamics, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-642-98037-4. [5] M. Gobbino and M. Sardella, On the connectedness of attractors for dynamical systems, J. Differential Equations, 133 (1997), 1-14. doi: 10.1006/jdeq.1996.3166. [6] O. V. Kapustyan, P. O. Kasyanov and J. Valero, Structure and regularity of the global attractor of a reacction-diffusion equation with non-smooth nonlinear term, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 4155-4182. doi: 10.3934/dcds.2014.34.4155. [7] O. V. Kapustyan, P. O. Kasyanov, J. Valero and M. Z. Zgurovsky, Structure of uniform global attractors for general non-autonomous reaction-diffusion systems, in Continuous and Distributed Systems. Theory and Applications, M.Z.Zgurovsky and V.A. Sadovnichiy eds, 211 (2014), 163-180, Cham, Springer. doi: 10.1007/978-3-319-03146-0_12. [8] F. Huang and R. Pan, Convergence rate for compressible Euler equations with damping and vacuum, Arch. Rational Mech. Anal., 166 (2003), 359-376. doi: 10.1007/s00205-002-0234-5. [9] Y. Kim and K. Li, Time-periodic strong solutions of the 3D Navier-Stokes equations with damping, Electron. J. Differential Equations, 2017 (2017), 1-11. [10] O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, New-York, 1991. doi: 10.1017/CBO9780511569418. [11] J. L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites Non Linéaires, Gauthier-Villar, Paris, 1969. [12] A. A. Linninger, M. Xenos, D. C. Zhu, M. R. Somayaji, S. Kondapali and R. D. Penn, Cerebrospinal fluid flow in the normal and hydrocefalic human brain, IEEE Trans. Biomed. Eng., 54 (2007), 291-302. [13] V. S. Valero and J. Melnik AND, On attractors of multi-valued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111. doi: 10.1023/A:1008608431399. [14] J. Robinson, Infinite-dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2001. doi: 10.1007/978-94-010-0732-0. [15] X. Song and Y. Hou, Attractors for the three-dimensional incompressible Navier-Stokes equations with damping, Discrete Contin. Dyn. Syst. Ser. A, 31 (2011), 239-252. doi: 10.3934/dcds.2011.31.239. [16] X. Song and Y. Hou, Uniform attractors for the three-dimensional Navier-Stokes equations with nonlinear damping, J. Math. Anal. Appl., 422 (2015), 337-351. doi: 10.1016/j.jmaa.2014.08.044. [17] X. Song, F. Liang and J. Wu, Pullback D-attractors for three-dimensional Navier-Stokes equations with nonlinear damping, Bound. Value Probl., 2016 (2016), 15PP. doi: 10.1186/s13661-016-0654-z. [18] R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam, 1979. [19] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, SpringerVerlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8. [20] J. Valero, On locally compact attractors of dynamical systems, J. Math. Anal. Appl., 237 (1999), 43-54. doi: 10.1006/jmaa.1999.6446. [21] H. Versteeg and W. Malalasekra, An Introduction to Computational Fluid Dynamics: The Finite Volume Method, Prentice Hall, New York, 2007. [22] W. Wang and G. Zhou, Remarks on the regularity criterion of the Navier-Stokes equations with nonlinear damping, Math. Probl. Eng., 2015 (2015), Art. ID 310934, 5 pp. doi: 10.1155/2015/310934. [23] Z. Zhang, X. Wu and M. Lu, On the uniqueness of strong solution to the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl., 377 (2011), 414-419. doi: 10.1016/j.jmaa.2010.11.019. [24] X. Zhong, Global well-posedness to the incompressible Navier-Stokes equations with damping, Electron. J. Qual. Theory Differ. Equ., 62 (2017), 1-9. [25] Y. Zhou, Regularity and uniqueness for the 3D incompressible Navier-Stokes equations with damping, Appl. Math. Letters, 25 (2012), 1822-1825. doi: 10.1016/j.aml.2012.02.029.
Flow domain
Flow velocity $u$ for the steady state in the $xy$-section at $z = 0$. The darker areas mean lower fluid flow speed. The initial velocity $u_{0}$ is identically zero. Left panel parameters: $\alpha = 0.2;\, \beta = 1$. Right panel parameters: $\alpha = 0.5;\, \beta = 1$
Flow velocity $u$ for the steady state in the $xy$-section at $z = 0$. The darker areas mean lower fluid flow speed. The initial velocity $u_{0}$ is identically zero. Left panel parameters: $\alpha = 0.2;\, \beta = 2$. Right panel parameters: $\alpha = 0.5;\, \beta = 2$
Flow velocity $u$ for the steady state in the $xy$-section at $z = 0$. The darker areas mean lower fluid flow speed. The initial velocity $u_{0}$ is identically zero. Left panel parameters: $\alpha = 0.2;\, \beta = 4$. Right panel parameters: $\alpha = 0.5;\, \beta = 4$
Flow velocity $u$ in the $xy$-section at $z = 0$ for $\alpha = 0.2$ and $\beta = 1$. The darker areas mean lower fluid flow speed. The initial velocity $u_{0} = (1, 0, 0)$. Left panel: state when $t = 0.1$. Right panel: steady state ($t$ large enough)
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