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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 |
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.
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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