# American Institute of Mathematical Sciences

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  December 2018 Early access  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 and 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. [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. [3] D. Bresch, B. 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. [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. [5] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002. [6] A. Cheskidov and C. 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Anal., 166 (2003), 359-376.  doi: 10.1007/s00205-002-0234-5. [12] Y. Jia, X. 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. [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. [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. [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. [16] Q. F. Ma, S. 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. [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. [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. [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. [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. [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. [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. [23] J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Matematica Pura ed Applicata, 146 (1987), 65-96. [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. [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. [26] X. L. Song, F. 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. [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. [28] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, New York, 1977. [29] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. [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. [31] Z. J. Zhang, X. 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. [32] C. K. Zhong, M. 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. [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.

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##### 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. [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. [3] D. Bresch, B. 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. [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. [5] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002. [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. [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. [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. [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. [10] L. Hsiao, Quasilinear Hyperbolic Systems and Dissipative Mechanisms, World Scientific, London, 1997. [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. [12] Y. Jia, X. 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. [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. [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. [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. [16] Q. F. Ma, S. 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. [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. [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. [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. [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. [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. [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. [23] J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Matematica Pura ed Applicata, 146 (1987), 65-96. [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. [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. [26] X. L. Song, F. 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. [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. [28] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, New York, 1977. [29] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. [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. [31] Z. J. Zhang, X. 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. [32] C. K. Zhong, M. 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. [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.
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