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January  2020, 25(1): 55-80. doi: 10.3934/dcdsb.2019172

## Pullback exponential attractors for the three dimensional non-autonomous 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

* Corresponding author: Bo You

Received  November 2018 Published  July 2019

Fund Project: This work was supported by the National Science Foundation of China Grant (11401459, 11801427, 11871389), the Natural Science Foundation of Shaanxi Province (2018JQ1009, 2018JM1012) and the Fundamental Research Funds for the Central Universities (xjj2018088)

The main objective of this paper is to study the long-time behavior of solutions for the three dimensional non-autonomous Navier-Stokes equations with nonlinear damping for $r>4.$ Inspired by the the methods of $\ell$-trajectories in [27], we will prove the existence of a finite dimensional pullback attractor and a pullback exponential attractor, which gives another way of considering the long-time behavior of the non-autonomous evolutionary equations.

Citation: 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
##### 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. 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.  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] A. N. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: Theoretical results, Commun. Pure Appl. Anal., 12 (2013), 3047-3071.  doi: 10.3934/cpaa.2013.12.3047.  Google Scholar [6] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002.  Google Scholar [7] 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 [8] 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 [9] J. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511526404.  Google Scholar [10] R. Czaja and M. Efendiev, Pullback exponential attractors for nonautonomous equations part Ⅰ: semilinear parabolic equations, J. Math. Anal. Appl., 381 (2011), 748-765.  doi: 10.1016/j.jmaa.2011.03.053.  Google Scholar [11] 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 [12] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, John Wiley, New York, 1994.  Google Scholar [13] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbb{R}^3$, C. R. Acad. Sci. Paris Ser. I Math., 330 (2000), 713-718.  doi: 10.1016/S0764-4442(00)00259-7.  Google Scholar [14] M. Efendiev and S. Zelik, Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 703-730.  doi: 10.1017/S030821050000408X.  Google Scholar [15] 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 [16] L. Hsiao, Quasilinear Hyperbolic Systems and Dissipative Mechanisms, World Scientific, London, 1997.  Google Scholar [17] 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 [18] 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.  Google Scholar [19] 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 [20] 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 [21] 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 [22] J. A. Langa, A. Miranville and J. Real, Pullback exponential attractors, Discrete Contin. Dyn. Syst., 26 (2010), 1329-1357.  doi: 10.3934/dcds.2010.26.1329.  Google Scholar [23] F. Li, B. You and Y. Xu, Dynamics of weak solutions for the three dimensional Navier-Stokes equations with nonlinear damping, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4267-4284.   Google Scholar [24] Y. J. Li and C. K. Zhong, Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations, Appl. Math. Comput., 190 (2007), 1020-1029.  doi: 10.1016/j.amc.2006.11.187.  Google Scholar [25] G. Łukaszewicz and J. C. Robinson, Invariant measures for non-autonomous dissipative dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 4211-4222.  doi: 10.3934/dcds.2014.34.4211.  Google Scholar [26] 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 [27] 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 [28] 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 [29] J. C. Robinson, Infinite-dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge, 2001.  doi: 10.1007/978-94-010-0732-0.  Google Scholar [30] 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 [31] 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 [32] J. Simon, Compact sets in the space $L^p(0, T;B)$, Annali di Matematica Pura ed Applicata, 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar [33] 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 [34] 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 [35] 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.   Google Scholar [36] 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), Paper No. 145, 15 pp. doi: 10.1186/s13661-016-0654-z.  Google Scholar [37] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar [38] L. Yang, M. H. Yang and P. E. Kloeden, Pullback attractors for non-autonomous quasi-linear parabolic equations with a dynamical boundary condition, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2635-2651.  doi: 10.3934/dcdsb.2012.17.2635.  Google Scholar [39] B. You and C. K. Zhong, Global attractors for $p$-Laplacian equations with dynamic flux boundary conditions, Adv. Nonlinear Stud., 13 (2013), 391-410.  doi: 10.1515/ans-2013-0208.  Google Scholar [40] 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.  Google Scholar [41] 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. 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.  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] A. N. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: Theoretical results, Commun. Pure Appl. Anal., 12 (2013), 3047-3071.  doi: 10.3934/cpaa.2013.12.3047.  Google Scholar [6] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002.  Google Scholar [7] 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 [8] 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 [9] J. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9780511526404.  Google Scholar [10] R. Czaja and M. Efendiev, Pullback exponential attractors for nonautonomous equations part Ⅰ: semilinear parabolic equations, J. Math. Anal. Appl., 381 (2011), 748-765.  doi: 10.1016/j.jmaa.2011.03.053.  Google Scholar [11] 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 [12] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, John Wiley, New York, 1994.  Google Scholar [13] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbb{R}^3$, C. R. Acad. Sci. Paris Ser. I Math., 330 (2000), 713-718.  doi: 10.1016/S0764-4442(00)00259-7.  Google Scholar [14] M. Efendiev and S. Zelik, Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 703-730.  doi: 10.1017/S030821050000408X.  Google Scholar [15] 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 [16] L. Hsiao, Quasilinear Hyperbolic Systems and Dissipative Mechanisms, World Scientific, London, 1997.  Google Scholar [17] 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 [18] 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.  Google Scholar [19] 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 [20] 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 [21] 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 [22] J. A. Langa, A. Miranville and J. Real, Pullback exponential attractors, Discrete Contin. Dyn. Syst., 26 (2010), 1329-1357.  doi: 10.3934/dcds.2010.26.1329.  Google Scholar [23] F. Li, B. You and Y. Xu, Dynamics of weak solutions for the three dimensional Navier-Stokes equations with nonlinear damping, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4267-4284.   Google Scholar [24] Y. J. Li and C. K. Zhong, Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations, Appl. Math. Comput., 190 (2007), 1020-1029.  doi: 10.1016/j.amc.2006.11.187.  Google Scholar [25] G. Łukaszewicz and J. C. Robinson, Invariant measures for non-autonomous dissipative dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 4211-4222.  doi: 10.3934/dcds.2014.34.4211.  Google Scholar [26] 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 [27] 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 [28] 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 [29] J. C. Robinson, Infinite-dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge, 2001.  doi: 10.1007/978-94-010-0732-0.  Google Scholar [30] 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 [31] 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 [32] J. Simon, Compact sets in the space $L^p(0, T;B)$, Annali di Matematica Pura ed Applicata, 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar [33] 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 [34] 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 [35] 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.   Google Scholar [36] 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), Paper No. 145, 15 pp. doi: 10.1186/s13661-016-0654-z.  Google Scholar [37] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar [38] L. Yang, M. H. Yang and P. E. Kloeden, Pullback attractors for non-autonomous quasi-linear parabolic equations with a dynamical boundary condition, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2635-2651.  doi: 10.3934/dcdsb.2012.17.2635.  Google Scholar [39] B. You and C. K. Zhong, Global attractors for $p$-Laplacian equations with dynamic flux boundary conditions, Adv. Nonlinear Stud., 13 (2013), 391-410.  doi: 10.1515/ans-2013-0208.  Google Scholar [40] 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.  Google Scholar [41] 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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