American Institute of Mathematical Sciences

Advanced
May  2015, 20(3): 749-779. doi: 10.3934/dcdsb.2015.20.749

Pullback attractors for generalized evolutionary systems

Alexey Cheskidov 1, and  Landon Kavlie 1,

1. 

Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 322 Science and Engineering Offices (M/C 249), 851 S. Morgan Street, Chicago, Illinois 60607-7045, United States, United States

Received  December 2013 Revised  March 2014 Published  January 2015

We give an abstract framework for studying nonautonomous PDEs, called a generalized evolutionary system. In this setting, we define the notion of a pullback attractor. Moreover, we show that the pullback attractor, in the weak sense, must always exist. We then study the structure of these attractors and the existence of a strong pullback attractor. We then apply our framework to both autonomous and nonautonomous evolutionary systems as they first appeared in earlier works by Cheskidov, Foias, and Lu. In this con- text, we compare the pullback attractor to both the global attractor (in the autonomous case) and the uniform attractor (in the nonautonomous case). Finally, we apply our results to the nonautonomous 3D Navier-Stokes equations on a periodic domain with a translationally bounded force. We show that the Leray-Hopf weak solutions form a generalized evolutionary system and must then have a weak pullback attractor.
Keywords: generalized evolutionary systems, Nonautonomous dynamical systems, fluid mechanics, pullback attractors, Navier-Stokes equations..
Mathematics Subject Classification: 35Q30, 37B55, 37D45, 37L05, 37N1.
Citation: Alexey Cheskidov, Landon Kavlie. Pullback attractors for generalized evolutionary systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 749-779. doi: 10.3934/dcdsb.2015.20.749
References:
[1]

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

[2]

T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors of nonautonomous and stochastic multivalued dynamical systems,, Set-Valued Anal., 11 (2003), 153.  doi: 10.1023/A:1022902802385.  Google Scholar

[3]

T. Caraballo, P. Marín-Rubio and J. C. Robinson, A comparison between two theories for multi-valued semiflows and their asymptotic behaviour,, Set-Valued Anal., 11 (2003), 297.  doi: 10.1023/A:1024422619616.  Google Scholar

[4]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems,, Applied Mathematical Sciences, (2013).  doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[5]

D. N. Cheban, Global Attractors of Non-Autonomous Dissipative Dynamical Systems,, Interdisciplinary Mathematical Sciences, (2004).  doi: 10.1142/9789812563088.  Google Scholar

[6]

V. Chepyzhov and M. Vishik, A Hausdorff dimension estimate for kernel sections of nonautonomous evolution equations,, Indiana Univ. Math. J., 42 (1993), 1057.  doi: 10.1512/iumj.1993.42.42049.  Google Scholar

[7]

V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension,, J. Math. Pures Appl. (9), 73 (1994), 279.   Google Scholar

[8]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, American Mathematical Society Colloquium Publications, (2002).   Google Scholar

[9]

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

[10]

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

[11]

A. Cheskidov, Global attractors of evolutionary systems,, J. Dynam. Differential Equations, 21 (2009), 249.  doi: 10.1007/s10884-009-9133-x.  Google Scholar

[12]

A. Cheskidov and S. Lu, The existence and the structure of uniform global attractors for nonautonomous reaction-diffusion systems without uniqueness,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 55.  doi: 10.3934/dcdss.2009.2.55.  Google Scholar

[13]

P. Constantin and C. Foias, Navier-Stokes Equations,, Chicago Lectures in Mathematics. University of Chicago Press, (1988).   Google Scholar

[14]

H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Theory Related Fields, 100 (1994), 365.  doi: 10.1007/BF01193705.  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.  doi: 10.1023/A:1021937715194.  Google Scholar

[16]

C. Foias and R. Temam, The connection between the Navier-Stokes equations, dynamical systems, and turbulence theory,, in Directions in Partial Differential Equations (Madison, (1985), 55.   Google Scholar

[17]

A. Haraux, Systèmes Dynamiques Dissipatifs et Applications,, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], (1991).   Google Scholar

[18]

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

[19]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems,, Mathematical Surveys and Monographs, (2011).  doi: 10.1090/surv/176.  Google Scholar

[20]

P. E. Kloeden and B. Schmalfuß, Nonautonomous systems, cocycle attractors and variable time-step discretization,, Dynamical Numerical Analysis (Atlanta, 14 (1997), 141.  doi: 10.1023/A:1019156812251.  Google Scholar

[21]

P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem,, Chapman & Hall/CRC Research Notes in Mathematics, (2002).  doi: 10.1201/9781420035674.  Google Scholar

[22]

S. Lu, H. Wu and C. Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces,, Discrete Contin. Dyn. Syst., 13 (2005), 701.  doi: 10.3934/dcds.2005.13.701.  Google Scholar

[23]

V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions,, Set-Valued Anal., 6 (1998), 83.  doi: 10.1023/A:1008608431399.  Google Scholar

[24]

J. C. Robinson, Infinite-dimensional Dynamical Systems, An introduction to dissipative parabolic PDEs and the theory of global attractors,, Cambridge Texts in Applied Mathematics, (2001).  doi: 10.1007/978-94-010-0732-0.  Google Scholar

[25]

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.  doi: 10.1016/j.jde.2006.03.004.  Google Scholar

[26]

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

[27]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations,, Applied Mathematical Sciences, (2002).  doi: 10.1007/978-1-4757-5037-9.  Google Scholar

[28]

R. Temam, Navier-Stokes Equations,, Theory and Numerical Analysis, (1984).   Google Scholar

[29]

M. I. Vishik and V. V. Chepyzhov, Trajectory and global attractors of the three-dimensional Navier-Stokes system,, Mat. Zametki, 71 (2002), 194.  doi: 10.1023/A:1014190629738.  Google Scholar

[30]

D. Vorotnikov, Asymptotic behavior of the non-autonomous 3D Navier-Stokes problem with coercive force,, J. Differential Equations, 251 (2011), 2209.  doi: 10.1016/j.jde.2011.07.008.  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.  doi: 10.1007/s003329900037.  Google Scholar

[2]

T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors of nonautonomous and stochastic multivalued dynamical systems,, Set-Valued Anal., 11 (2003), 153.  doi: 10.1023/A:1022902802385.  Google Scholar

[3]

T. Caraballo, P. Marín-Rubio and J. C. Robinson, A comparison between two theories for multi-valued semiflows and their asymptotic behaviour,, Set-Valued Anal., 11 (2003), 297.  doi: 10.1023/A:1024422619616.  Google Scholar

[4]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems,, Applied Mathematical Sciences, (2013).  doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[5]

D. N. Cheban, Global Attractors of Non-Autonomous Dissipative Dynamical Systems,, Interdisciplinary Mathematical Sciences, (2004).  doi: 10.1142/9789812563088.  Google Scholar

[6]

V. Chepyzhov and M. Vishik, A Hausdorff dimension estimate for kernel sections of nonautonomous evolution equations,, Indiana Univ. Math. J., 42 (1993), 1057.  doi: 10.1512/iumj.1993.42.42049.  Google Scholar

[7]

V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension,, J. Math. Pures Appl. (9), 73 (1994), 279.   Google Scholar

[8]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, American Mathematical Society Colloquium Publications, (2002).   Google Scholar

[9]

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

[10]

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

[11]

A. Cheskidov, Global attractors of evolutionary systems,, J. Dynam. Differential Equations, 21 (2009), 249.  doi: 10.1007/s10884-009-9133-x.  Google Scholar

[12]

A. Cheskidov and S. Lu, The existence and the structure of uniform global attractors for nonautonomous reaction-diffusion systems without uniqueness,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 55.  doi: 10.3934/dcdss.2009.2.55.  Google Scholar

[13]

P. Constantin and C. Foias, Navier-Stokes Equations,, Chicago Lectures in Mathematics. University of Chicago Press, (1988).   Google Scholar

[14]

H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Theory Related Fields, 100 (1994), 365.  doi: 10.1007/BF01193705.  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.  doi: 10.1023/A:1021937715194.  Google Scholar

[16]

C. Foias and R. Temam, The connection between the Navier-Stokes equations, dynamical systems, and turbulence theory,, in Directions in Partial Differential Equations (Madison, (1985), 55.   Google Scholar

[17]

A. Haraux, Systèmes Dynamiques Dissipatifs et Applications,, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], (1991).   Google Scholar

[18]

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

[19]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems,, Mathematical Surveys and Monographs, (2011).  doi: 10.1090/surv/176.  Google Scholar

[20]

P. E. Kloeden and B. Schmalfuß, Nonautonomous systems, cocycle attractors and variable time-step discretization,, Dynamical Numerical Analysis (Atlanta, 14 (1997), 141.  doi: 10.1023/A:1019156812251.  Google Scholar

[21]

P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem,, Chapman & Hall/CRC Research Notes in Mathematics, (2002).  doi: 10.1201/9781420035674.  Google Scholar

[22]

S. Lu, H. Wu and C. Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces,, Discrete Contin. Dyn. Syst., 13 (2005), 701.  doi: 10.3934/dcds.2005.13.701.  Google Scholar

[23]

V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions,, Set-Valued Anal., 6 (1998), 83.  doi: 10.1023/A:1008608431399.  Google Scholar

[24]

J. C. Robinson, Infinite-dimensional Dynamical Systems, An introduction to dissipative parabolic PDEs and the theory of global attractors,, Cambridge Texts in Applied Mathematics, (2001).  doi: 10.1007/978-94-010-0732-0.  Google Scholar

[25]

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.  doi: 10.1016/j.jde.2006.03.004.  Google Scholar

[26]

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

[27]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations,, Applied Mathematical Sciences, (2002).  doi: 10.1007/978-1-4757-5037-9.  Google Scholar

[28]

R. Temam, Navier-Stokes Equations,, Theory and Numerical Analysis, (1984).   Google Scholar

[29]

M. I. Vishik and V. V. Chepyzhov, Trajectory and global attractors of the three-dimensional Navier-Stokes system,, Mat. Zametki, 71 (2002), 194.  doi: 10.1023/A:1014190629738.  Google Scholar

[30]

D. Vorotnikov, Asymptotic behavior of the non-autonomous 3D Navier-Stokes problem with coercive force,, J. Differential Equations, 251 (2011), 2209.  doi: 10.1016/j.jde.2011.07.008.  Google Scholar

[1]

Yejuan Wang, Chengkui Zhong, Shengfan Zhou. Pullback attractors of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 587-614. doi: 10.3934/dcds.2006.16.587
[2]

Pedro Marín-Rubio, Antonio M. Márquez-Durán, José Real. Pullback attractors for globally modified Navier-Stokes equations with infinite delays. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 779-796. doi: 10.3934/dcds.2011.31.779
[3]

Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Approximation of attractors of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 215-238. doi: 10.3934/dcdsb.2005.5.215
[4]

Julia García-Luengo, Pedro Marín-Rubio, José Real. Some new regularity results of pullback attractors for 2D Navier-Stokes equations with delays. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1603-1621. doi: 10.3934/cpaa.2015.14.1603
[5]

P.E. Kloeden, José A. Langa, José Real. Pullback V-attractors of the 3-dimensional globally modified Navier-Stokes equations. Communications on Pure & Applied Analysis, 2007, 6 (4) : 937-955. doi: 10.3934/cpaa.2007.6.937
[6]

Grzegorz Łukaszewicz. Pullback attractors and statistical solutions for 2-D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 643-659. doi: 10.3934/dcdsb.2008.9.643
[7]

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
[8]

Songsong Lu, Hongqing Wu, Chengkui Zhong. Attractors for nonautonomous 2d Navier-Stokes equations with normal external forces. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 701-719. doi: 10.3934/dcds.2005.13.701
[9]

Björn Schmalfuss. Attractors for nonautonomous and random dynamical systems perturbed by impulses. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 727-744. doi: 10.3934/dcds.2003.9.727
[10]

David Cheban. Global attractors of nonautonomous quasihomogeneous dynamical systems. Conference Publications, 2001, 2001 (Special) : 96-101. doi: 10.3934/proc.2001.2001.96
[11]

Julia García-Luengo, Pedro Marín-Rubio, José Real. Regularity of pullback attractors and attraction in $H^1$ in arbitrarily large finite intervals for 2D Navier-Stokes equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 181-201. doi: 10.3934/dcds.2014.34.181
[12]

Ioana Moise, Ricardo Rosa, Xiaoming Wang. Attractors for noncompact nonautonomous systems via energy equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 473-496. doi: 10.3934/dcds.2004.10.473
[13]

Xue-Li Song, Yan-Ren Hou. Attractors for the three-dimensional incompressible Navier-Stokes equations with damping. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 239-252. doi: 10.3934/dcds.2011.31.239
[14]

Vena Pearl Bongolan-walsh, David Cheban, Jinqiao Duan. Recurrent motions in the nonautonomous Navier-Stokes system. Discrete & Continuous Dynamical Systems - B, 2003, 3 (2) : 255-262. doi: 10.3934/dcdsb.2003.3.255
[15]

Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Invariant manifolds as pullback attractors of nonautonomous differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 579-596. doi: 10.3934/dcds.2006.15.579
[16]

Jishan Fan, Yasuhide Fukumoto, Yong Zhou. Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations. Kinetic & Related Models, 2013, 6 (3) : 545-556. doi: 10.3934/krm.2013.6.545
[17]

Xuewei Ju, Desheng Li, Jinqiao Duan. Forward attraction of pullback attractors and synchronizing behavior of gradient-like systems with nonautonomous perturbations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1175-1197. doi: 10.3934/dcdsb.2019011
[18]

Julia García-Luengo, Pedro Marín-Rubio, José Real, James C. Robinson. Pullback attractors for the non-autonomous 2D Navier--Stokes equations for minimally regular forcing. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 203-227. doi: 10.3934/dcds.2014.34.203
[19]

Pedro Marín-Rubio, José Real. Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 989-1006. doi: 10.3934/dcds.2010.26.989
[20]

Julia García-Luengo, Pedro Marín-Rubio. Pullback attractors for 2D Navier–Stokes equations with delays and the flattening property. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2127-2146. doi: 10.3934/cpaa.2020094

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (31)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences