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November  2017, 22(9): 3379-3407. doi: 10.3934/dcdsb.2017142

On random cocycle attractors with autonomous attraction universes

Hongyong Cui 1, , Mirelson M. Freitas 2, and  José A. Langa 3,

1. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China
2. 

Departamento de Matemática, Universidade Federal do Pará, Rua Augusto Corrêa s/n, 66000-000, Belém PA, Brazil
3. 

Departamento de Ecuaciones Diferenciales Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160,41080-Sevilla, Spain

Received  October 2016 Revised  February 2017 Published  April 2017

In this paper, for non-autonomous RDS we study cocycle attractors with autonomous attraction universes, i.e. pullback attracting some autonomous random sets, instead of non-autonomous ones. We first compare cocycle attractors with autonomous and non-autonomous attraction universes, and then for autonomous ones we establish some existence criteria and characterization. We also study for cocycle attractors the continuity of sections indexed by non-autonomous symbols to find that the upper semi-continuity is equivalent to uniform compactness of the attractor, while the lower semi-continuity is equivalent to an equi-attracting property under some conditions. Finally, we apply these theoretical results to 2D Navier-Stokes equation with additive white noise and translation bounded non-autonomous forcing.

Keywords: Random cocycle attractor, pullback attractor, basis of attraction, 2D stochastic Navier-Stokes equation.
Mathematics Subject Classification: Primary:35B40;Secondary:35B41, 37H05, 37H10.
Citation: Hongyong Cui, Mirelson M. Freitas, José A. Langa. On random cocycle attractors with autonomous attraction universes. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3379-3407. doi: 10.3934/dcdsb.2017142
References:
[1]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

A. C. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, volume 182, Springer, 2013. doi: 10.1007/978-1-4614-4581-4.

[3]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, volume 49. American Mathematical Society Providence, RI, USA, 2002.

[4]

M. Coti Zelati and P. Kalita, Minimality properties of set-valued processes and their pullback attractors, SIAM Journal on Mathematical Analysis, 47 (2015), 1530-1561.  doi: 10.1137/140978995.

[5]

H. Crauel, Global random attractors are uniquely determined by attracting deterministic compact sets, Annali di Matematica pura ed applicata, 176 (1999), 57-72.  doi: 10.1007/BF02505989.

[6]

H. Crauel, Random Probability Measures on Polish Spaces, volume 11. CRC press, 2003.

[7]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, Journal of Dynamics and Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.

[8]

H. Cui and J. A. Langa, Uniform attractors for non-autonomous random dynamical systems, Journal of Differential Equations, 263 (2017), 1225-1268.  doi: 10.1016/j.jde.2017.03.018.

[9]

H. Cui, J. A. Langa and Y. Li, Regularity and structure of pullback attractors for reaction-diffusion type systems without uniqueness, Nonlinear Analysis: Theory, Methods & Applications, 140 (2016), 208-235.  doi: 10.1016/j.na.2016.03.012.

[10]

H. Cui, J. A. Langa, Y. Li and J. Valero, Attractors for multi-valued non-autonomous dynamical systems: Relationship, characterization and robustness, Set-Valued and Variational Analysis, page in press, (2016), 1-38. doi: 10.1007/s11228-016-0395-2.

[11]

J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors for three-dimensional non-autonomous navier-stokes-voigt equations, Nonlinearity, 25 (2012), 905-930.  doi: 10.1088/0951-7715/25/4/905.

[12]

P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, The Royal Society, 463 (2007), 163-181. doi: 10.1098/rspa.2006.1753.

[13]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Number 176, American Mathematical Soc. , 2011. doi: 10.1090/surv/176.

[14]

D. Li and P. Kloeden, Equi-attraction and the continuous dependence of attractors on parameters, Glasgow Mathematical Journal, 46 (2004), 131-141.  doi: 10.1017/S0017089503001605.

[15]

D. Li and P. Kloeden, Equi-attraction and the continuous dependence of pullback attractors on parameters, Stochastics and Dynamics, 4 (2004), 373-384.  doi: 10.1142/S0219493704001061.

[16]

D. Li and P. Kloeden, Equi-attraction and continuous dependence of strong attractors of set-valued dynamical systems on parameters, Set-Valued Analysis, 13 (2005), 405-416.  doi: 10.1007/s11228-005-2971-8.

[17]

Q. Ma, S. Wang and C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana University Mathematics Journal, 51 (2002), 1541-1559.  doi: 10.1512/iumj.2002.51.2255.

[18]

P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 3956-3963.  doi: 10.1016/j.na.2009.02.065.

[19]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 2nd edition, 1997. doi: 10.1007/978-1-4612-0645-3.

[20]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, Journal of Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.

[21]

B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete and Continuous Dynamical Systems, 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269.

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

A. C. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, volume 182, Springer, 2013. doi: 10.1007/978-1-4614-4581-4.

[3]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, volume 49. American Mathematical Society Providence, RI, USA, 2002.

[4]

M. Coti Zelati and P. Kalita, Minimality properties of set-valued processes and their pullback attractors, SIAM Journal on Mathematical Analysis, 47 (2015), 1530-1561.  doi: 10.1137/140978995.

[5]

H. Crauel, Global random attractors are uniquely determined by attracting deterministic compact sets, Annali di Matematica pura ed applicata, 176 (1999), 57-72.  doi: 10.1007/BF02505989.

[6]

H. Crauel, Random Probability Measures on Polish Spaces, volume 11. CRC press, 2003.

[7]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, Journal of Dynamics and Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.

[8]

H. Cui and J. A. Langa, Uniform attractors for non-autonomous random dynamical systems, Journal of Differential Equations, 263 (2017), 1225-1268.  doi: 10.1016/j.jde.2017.03.018.

[9]

H. Cui, J. A. Langa and Y. Li, Regularity and structure of pullback attractors for reaction-diffusion type systems without uniqueness, Nonlinear Analysis: Theory, Methods & Applications, 140 (2016), 208-235.  doi: 10.1016/j.na.2016.03.012.

[10]

H. Cui, J. A. Langa, Y. Li and J. Valero, Attractors for multi-valued non-autonomous dynamical systems: Relationship, characterization and robustness, Set-Valued and Variational Analysis, page in press, (2016), 1-38. doi: 10.1007/s11228-016-0395-2.

[11]

J. García-Luengo, P. Marín-Rubio and J. Real, Pullback attractors for three-dimensional non-autonomous navier-stokes-voigt equations, Nonlinearity, 25 (2012), 905-930.  doi: 10.1088/0951-7715/25/4/905.

[12]

P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, The Royal Society, 463 (2007), 163-181. doi: 10.1098/rspa.2006.1753.

[13]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Number 176, American Mathematical Soc. , 2011. doi: 10.1090/surv/176.

[14]

D. Li and P. Kloeden, Equi-attraction and the continuous dependence of attractors on parameters, Glasgow Mathematical Journal, 46 (2004), 131-141.  doi: 10.1017/S0017089503001605.

[15]

D. Li and P. Kloeden, Equi-attraction and the continuous dependence of pullback attractors on parameters, Stochastics and Dynamics, 4 (2004), 373-384.  doi: 10.1142/S0219493704001061.

[16]

D. Li and P. Kloeden, Equi-attraction and continuous dependence of strong attractors of set-valued dynamical systems on parameters, Set-Valued Analysis, 13 (2005), 405-416.  doi: 10.1007/s11228-005-2971-8.

[17]

Q. Ma, S. Wang and C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana University Mathematics Journal, 51 (2002), 1541-1559.  doi: 10.1512/iumj.2002.51.2255.

[18]

P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 3956-3963.  doi: 10.1016/j.na.2009.02.065.

[19]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 2nd edition, 1997. doi: 10.1007/978-1-4612-0645-3.

[20]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, Journal of Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.

[21]

B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete and Continuous Dynamical Systems, 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269.

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