# American Institute of Mathematical Sciences

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

## On random cocycle attractors with autonomous attraction universes

 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.

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.

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##### 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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