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Quasi-periodic solutions for a class of beam equation system
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 |
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 [
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] |
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. |
[6] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002. |
[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. |
[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. |
[9] |
J. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, Cambridge, 2000.
doi: 10.1017/CBO9780511526404.![]() ![]() ![]() |
[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. |
[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. |
[12] |
A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, John Wiley, New York, 1994. |
[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. |
[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. |
[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. |
[16] |
L. Hsiao, Quasilinear Hyperbolic Systems and Dissipative Mechanisms, World Scientific, London, 1997. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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.![]() ![]() ![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[37] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[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. |
[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. |
[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. |
[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. |
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. |
[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] |
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. |
[6] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002. |
[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. |
[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. |
[9] |
J. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, Cambridge, 2000.
doi: 10.1017/CBO9780511526404.![]() ![]() ![]() |
[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. |
[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. |
[12] |
A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, John Wiley, New York, 1994. |
[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. |
[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. |
[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. |
[16] |
L. Hsiao, Quasilinear Hyperbolic Systems and Dissipative Mechanisms, World Scientific, London, 1997. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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.![]() ![]() ![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[37] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[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. |
[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. |
[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. |
[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. |
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