December  2018, 23(10): 4267-4284. doi: 10.3934/dcdsb.2018137

Dynamics of weak solutions for the three dimensional Navier-Stokes equations with nonlinear damping

1. 

School of Mathematics and Statistics, Xidian University, Xi'an 710126, China

2. 

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China

3. 

Department of Mathematics, Nanjing University, Nanjing 210093, China

* Corresponding author

Received  October 2017 Revised  December 2017 Published  April 2018

Fund Project: This work was supported by the National Science Foundation of China Grant (11401459)

The main objective of this paper is to study the existence of a finite dimensional global attractor for the three dimensional Navier-Stokes equations with nonlinear damping for $r>4.$ Motivated by the idea of [1], even though we can obtain the existence of a global attractor for $r≥ 2$ by the multi-valued semi-flow, it is very difficult to provide any information about its fractal dimension. Therefore, we prove the existence of a global attractor in H and provide the upper bound of its fractal dimension by the methods of $\ell$-trajectories in this paper.

Citation: 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
References:
[1]

J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci., 7 (1997), 475-502. doi: 10.1007/s003329900037. Google Scholar

[2]

D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (2003), 211-223. doi: 10.1007/s00220-003-0859-8. Google Scholar

[3]

D. BreschB. Desjardins and C. K. Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems, Comm. Partial Differential Equations, 28 (2003), 843-868. doi: 10.1081/PDE-120020499. Google Scholar

[4]

X. J. Cai and Q. S. 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. Google Scholar

[5]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002. Google Scholar

[6]

A. Cheskidov and C. Foias, On global attractors of the 3D Navier-Stokes equations, J. Differential Equations, 231 (2006), 714-754. doi: 10.1016/j.jde.2006.08.021. Google Scholar

[7]

A. Cheskidov and S. S. Lu, Uniform global attractors for the nonautonomous 3D Navier-Stokes equations, Adv. Math., 267 (2014), 277-306. doi: 10.1016/j.aim.2014.09.005. Google Scholar

[8]

B. Q. Dong and Y. Jia, Stability behaviors of Leray weak solutions to the three-dimensional Navier-Stokes equations, Nonlinear Anal. Real World Appl., 30 (2016), 41-58. doi: 10.1016/j.nonrwa.2015.10.011. Google Scholar

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F. Flandoli and B. Schmalfuß, Weak solutions and attractors for three-dimensional Navier-Stokes equations with nonregular force, J. Dynam. Differential Equations, 11 (1999), 355-398. doi: 10.1023/A:1021937715194. Google Scholar

[10]

L. Hsiao, Quasilinear Hyperbolic Systems and Dissipative Mechanisms, World Scientific, London, 1997. Google Scholar

[11]

F. M. Huang and R. H. Pan, Convergence rate for compressible Euler equations with damping and vacuum, Arch. Ration. Mech. Anal., 166 (2003), 359-376. doi: 10.1007/s00205-002-0234-5. Google Scholar

[12]

Y. JiaX. W. Zhang and B. Q. Dong, The asymptotic behavior of solutions to three-dimensional Navier-Stokes equations with nonlinear damping, Nonlinear Anal. Real World Appl., 12 (2011), 1736-1747. doi: 10.1016/j.nonrwa.2010.11.006. Google Scholar

[13]

Z. H. Jiang, Asymptotic behavior of strong solutions to the 3D Navier-Stokes equations with a nonlinear damping term, Nonlinear Anal., 75 (2012), 5002-5009. doi: 10.1016/j.na.2012.04.014. Google Scholar

[14]

Z. H. Jiang and M. X. Zhu, The large time behavior of solutions to 3D Navier-Stokes equations with nonlinear damping, Math. Methods Appl. Sci., 35 (2012), 97-102. doi: 10.1002/mma.1540. Google Scholar

[15]

A. V. Kapustyan and J. Valero, Weak and strong attractors for the 3D Navier-Stokes system, J. Differential Equations, 240 (2007), 249-278. doi: 10.1016/j.jde.2007.06.008. Google Scholar

[16]

Q. F. MaS. H. Wang and C. K. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255. Google Scholar

[17]

J. Málek and J. Nečas, A finite-dimensional attractor for three-dimensional flow of incompressible fluids, J. Differential Equations, 127 (1996), 498-518. doi: 10.1006/jdeq.1996.0080. Google Scholar

[18]

J. Málek and D. Pražák, Large time behavior via the method of $\ell$-trajectories, J. Differential Equations, 181 (2002), 243-279. doi: 10.1006/jdeq.2001.4087. Google Scholar

[19]

C. Y. Qian, A remark on the global regularity for the 3D Navier-Stokes equations, Appl. Math. Lett., 57 (2016), 126-131. doi: 10.1016/j.aml.2016.01.016. Google Scholar

[20]

J. C. Robinson, Infinite-dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge, 2001. Google Scholar

[21]

R. M. S. Rosa, Asymptotic regularity conditions for the strong convergence towards weak limit sets and weak attractors of the 3D Navier-Stokes equations, J. Differential Equations, 229 (2006), 257-269. doi: 10.1016/j.jde.2006.03.004. Google Scholar

[22]

G. R. Sell, Global attractor for the three dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 8 (1996), 1-33. doi: 10.1007/BF02218613. Google Scholar

[23]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Matematica Pura ed Applicata, 146 (1987), 65-96. Google Scholar

[24]

X. L. Song and Y. R. Hou, Attractors for the three-dimensional incompressible Navier-Stokes equations with damping, Discrete Contin. Dyn. Syst., 31 (2011), 239-252. doi: 10.3934/dcds.2011.31.239. Google Scholar

[25]

X. L. Song and Y. R. Hou, Uniform attractors for three-dimensional Navier-Stokes equations with nonlinear damping, J. Math. Anal. Appl., 422 (2015), 337-351. doi: 10.1016/j.jmaa.2014.08.044. Google Scholar

[26]

X. L. SongF. Liang and J. Su, Exponential attractor for the three dimensional Navier-Stokes equation with nonlinear damping, Journal of Pure and Applied Mathematics: Advances and Applications, 14 (2015), 27-39. Google Scholar

[27]

X. L. Song, F. Liang and J. H. Wu, Pullback $\mathcal{D}$-attractors for three-dimensional Navier-Stokes equations with nonlinear damping, Bound. Value Probl., 2016 (2016), 15pp. Google Scholar

[28]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, New York, 1977. Google Scholar

[29]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. Google Scholar

[30]

B. You and C. K. Zhong, Global attractors for p-Laplacian equations with dynamic flux boundary conditions, Adv. Nonlinear Stud., 13 (2013), 391-410. Google Scholar

[31]

Z. J. ZhangX. L. 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. Google Scholar

[32]

C. K. ZhongM. H. Yang and C. Y. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations, 223 (2006), 367-399. doi: 10.1016/j.jde.2005.06.008. Google Scholar

[33]

Y. Zhou, Regularity and uniqueness for the 3D incompressible Navier-Stokes equations with damping, Appl. Math. Lett., 25 (2012), 1822-1825. doi: 10.1016/j.aml.2012.02.029. Google Scholar

show all references

References:
[1]

J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci., 7 (1997), 475-502. doi: 10.1007/s003329900037. Google Scholar

[2]

D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (2003), 211-223. doi: 10.1007/s00220-003-0859-8. Google Scholar

[3]

D. BreschB. Desjardins and C. K. Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems, Comm. Partial Differential Equations, 28 (2003), 843-868. doi: 10.1081/PDE-120020499. Google Scholar

[4]

X. J. Cai and Q. S. 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. Google Scholar

[5]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002. Google Scholar

[6]

A. Cheskidov and C. Foias, On global attractors of the 3D Navier-Stokes equations, J. Differential Equations, 231 (2006), 714-754. doi: 10.1016/j.jde.2006.08.021. Google Scholar

[7]

A. Cheskidov and S. S. Lu, Uniform global attractors for the nonautonomous 3D Navier-Stokes equations, Adv. Math., 267 (2014), 277-306. doi: 10.1016/j.aim.2014.09.005. Google Scholar

[8]

B. Q. Dong and Y. Jia, Stability behaviors of Leray weak solutions to the three-dimensional Navier-Stokes equations, Nonlinear Anal. Real World Appl., 30 (2016), 41-58. doi: 10.1016/j.nonrwa.2015.10.011. Google Scholar

[9]

F. Flandoli and B. Schmalfuß, Weak solutions and attractors for three-dimensional Navier-Stokes equations with nonregular force, J. Dynam. Differential Equations, 11 (1999), 355-398. doi: 10.1023/A:1021937715194. Google Scholar

[10]

L. Hsiao, Quasilinear Hyperbolic Systems and Dissipative Mechanisms, World Scientific, London, 1997. Google Scholar

[11]

F. M. Huang and R. H. Pan, Convergence rate for compressible Euler equations with damping and vacuum, Arch. Ration. Mech. Anal., 166 (2003), 359-376. doi: 10.1007/s00205-002-0234-5. Google Scholar

[12]

Y. JiaX. W. Zhang and B. Q. Dong, The asymptotic behavior of solutions to three-dimensional Navier-Stokes equations with nonlinear damping, Nonlinear Anal. Real World Appl., 12 (2011), 1736-1747. doi: 10.1016/j.nonrwa.2010.11.006. Google Scholar

[13]

Z. H. Jiang, Asymptotic behavior of strong solutions to the 3D Navier-Stokes equations with a nonlinear damping term, Nonlinear Anal., 75 (2012), 5002-5009. doi: 10.1016/j.na.2012.04.014. Google Scholar

[14]

Z. H. Jiang and M. X. Zhu, The large time behavior of solutions to 3D Navier-Stokes equations with nonlinear damping, Math. Methods Appl. Sci., 35 (2012), 97-102. doi: 10.1002/mma.1540. Google Scholar

[15]

A. V. Kapustyan and J. Valero, Weak and strong attractors for the 3D Navier-Stokes system, J. Differential Equations, 240 (2007), 249-278. doi: 10.1016/j.jde.2007.06.008. Google Scholar

[16]

Q. F. MaS. H. Wang and C. K. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255. Google Scholar

[17]

J. Málek and J. Nečas, A finite-dimensional attractor for three-dimensional flow of incompressible fluids, J. Differential Equations, 127 (1996), 498-518. doi: 10.1006/jdeq.1996.0080. Google Scholar

[18]

J. Málek and D. Pražák, Large time behavior via the method of $\ell$-trajectories, J. Differential Equations, 181 (2002), 243-279. doi: 10.1006/jdeq.2001.4087. Google Scholar

[19]

C. Y. Qian, A remark on the global regularity for the 3D Navier-Stokes equations, Appl. Math. Lett., 57 (2016), 126-131. doi: 10.1016/j.aml.2016.01.016. Google Scholar

[20]

J. C. Robinson, Infinite-dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge, 2001. Google Scholar

[21]

R. M. S. Rosa, Asymptotic regularity conditions for the strong convergence towards weak limit sets and weak attractors of the 3D Navier-Stokes equations, J. Differential Equations, 229 (2006), 257-269. doi: 10.1016/j.jde.2006.03.004. Google Scholar

[22]

G. R. Sell, Global attractor for the three dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 8 (1996), 1-33. doi: 10.1007/BF02218613. Google Scholar

[23]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Matematica Pura ed Applicata, 146 (1987), 65-96. Google Scholar

[24]

X. L. Song and Y. R. Hou, Attractors for the three-dimensional incompressible Navier-Stokes equations with damping, Discrete Contin. Dyn. Syst., 31 (2011), 239-252. doi: 10.3934/dcds.2011.31.239. Google Scholar

[25]

X. L. Song and Y. R. Hou, Uniform attractors for three-dimensional Navier-Stokes equations with nonlinear damping, J. Math. Anal. Appl., 422 (2015), 337-351. doi: 10.1016/j.jmaa.2014.08.044. Google Scholar

[26]

X. L. SongF. Liang and J. Su, Exponential attractor for the three dimensional Navier-Stokes equation with nonlinear damping, Journal of Pure and Applied Mathematics: Advances and Applications, 14 (2015), 27-39. Google Scholar

[27]

X. L. Song, F. Liang and J. H. Wu, Pullback $\mathcal{D}$-attractors for three-dimensional Navier-Stokes equations with nonlinear damping, Bound. Value Probl., 2016 (2016), 15pp. Google Scholar

[28]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, New York, 1977. Google Scholar

[29]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. Google Scholar

[30]

B. You and C. K. Zhong, Global attractors for p-Laplacian equations with dynamic flux boundary conditions, Adv. Nonlinear Stud., 13 (2013), 391-410. Google Scholar

[31]

Z. J. ZhangX. L. 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. Google Scholar

[32]

C. K. ZhongM. H. Yang and C. Y. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations, 223 (2006), 367-399. doi: 10.1016/j.jde.2005.06.008. Google Scholar

[33]

Y. Zhou, Regularity and uniqueness for the 3D incompressible Navier-Stokes equations with damping, Appl. Math. Lett., 25 (2012), 1822-1825. doi: 10.1016/j.aml.2012.02.029. Google Scholar

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