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On asymptotically autonomous dynamics for multivalued evolution problems
August
2019, 24(8): 3569-3590.
doi: 10.3934/dcdsb.2018279
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 |
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.
Keywords:
Three-dimensional Navier-Stokes equations with damping,
global attractors,
set-valued dynamical systems,
asymptotic behaviour,
turbulence.
Mathematics Subject Classification:
35B40, 35B41, 35K55, 35Q30, 37B25, 58C06.
Citation:
Daniel Pardo, José Valero, Ángel Giménez. Global attractors for weak solutions of the three-dimensional Navier-Stokes equations with damping. Discrete and Continuous Dynamical Systems - B,
2019, 24
(8)
: 3569-3590.
doi: 10.3934/dcdsb.2018279
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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. |
show all references
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. |
Figure 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: $\alpha = 0.2;\, \beta = 1$ . Right panel parameters: $\alpha = 0.5;\, \beta = 1$
Figure 3.
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: $\alpha = 0.5;\, \beta = 2$
Figure 4.
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: $\alpha = 0.5;\, \beta = 4$
Figure 5.
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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