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May 2018, 23(3): 1297-1324. doi: 10.3934/dcdsb.2018152

## Squeezing and finite dimensionality of cocycle attractors for 2D stochastic Navier-Stokes equation with non-autonomous forcing

 1 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 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

Dedicated to the memory of professor Igor Chueshov

Received  May 2017 Revised  September 2017 Published  February 2018

In this paper, we study the squeezing property and finite dimensionality of cocycle attractors for non-autonomous dynamical systems (NRDS). We show that the generalized random cocycle squeezing property (RCSP) is a sufficient condition to prove a determining modes result and the finite dimensionality of invariant non-autonomous random sets, where the upper bound of the dimension is uniform for all components of the invariant set. We also prove that the RCSP can imply the pullback flattening property in uniformly convex Banach space so that could also contribute to establish the asymptotic compactness of the system. The cocycle attractor for 2D Navier-Stokes equation with additive white noise and translation bounded non-autonomous forcing is studied as an application.

Citation: Hongyong Cui, Mirelson M. Freitas, José A. Langa. Squeezing and finite dimensionality of cocycle attractors for 2D stochastic Navier-Stokes equation with non-autonomous forcing. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1297-1324. doi: 10.3934/dcdsb.2018152
##### References:
 [1] L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. [2] L. Arnold and I. Chueshov, Order-preserving random dynamical systems: Equilibria, attractors, applications, Dynam. Stability Systems, 13 (1998), 265-280. doi: 10.1080/02681119808806264. [3] T. Caraballo, I. Chueshov and J. A. Langa, Existence of invariant manifolds for coupled parabolic and hyperbolic stochastic partial differential equations, Nonlinearity, 18 (2005), 747-767. doi: 10.1088/0951-7715/18/2/015. [4] D. Cheban and C. Mammana, Relation between different types of global attractors of set-valued nonautonomous dynamical systems, Set-Valued Analysis, 13 (2005), 291-321. doi: 10.1007/s11228-004-0046-x. [5] V. Chepyzhov and M. Vishik, A Hausdorff dimension estimate for kernel sections of nonautonomous evolution equations, Indiana Univ. Math. J., 42 (1993), 1057-1076. doi: 10.1512/iumj.1993.42.42049. [6] V. Chepyzhov and M. Vishik, Attractors of non-autonomous dynamical systems and their dimension, Journal de Mathématiques Pures et Appliquées, 73 (1994), 279-333. [7] _____, Attractors of non-autonomous evolution equations with translation compact symbols, in Partial Differential Operators and Mathematical Physics, Springer, 1995, 49-60. [8] V. Chepyzhov and M. Vishik, Attractors for nonautonomous Navier-Stokes system and other partial differential equations, in Instability in models connected with fluid flows. Ⅰ, vol. 6 of Int. Math. Ser. (N. Y. ), Springer, New York, 2008,135-265. [9] V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl. (9), 73 (1994), 279-333. [10] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, vol. 49, American Mathematical Society Providence, RI, USA, 2002. [11] _____, Monotone Random Systems Theory and Applications, vol. 1779, Springer Science & Business Media, 2002. [12] _____, Invariant manifolds and nonlinear master-slave synchronization in coupled systems, Appl. Anal., 86 (2007), 269-286. doi: 10.1080/00036810601097629. [13] _____, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differential Equations, 252 (2012), 1229-1262. doi: 10.1016/j.jde.2011.08.022. [14] _____, A squeezing property and its applications to a description of long-time behaviour in the three-dimensional viscous primitive equations, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 711-729. doi: 10.1017/S0308210512001953. [15] _____, Dynamics of Quasi-Stable Dissipative Systems, Universitext, Springer, Cham, 2015. [16] I. Chueshov, M. Eller and I. Lasiecka, Finite dimensionality of the attractor for a semilinear wave equation with nonlinear boundary dissipation, Comm. Partial Differential Equations, 29 (2004), 1847-1876. [17] I. Chueshov and I. Lasiecka, Determining functionals for a class of second order in time evolution equations with applications to von Karman equations, in Analysis and optimization of differential systems (Constanta, 2002), Kluwer Acad. Publ., Boston, MA, 2003,109-122. [18] _____, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc. , 195 (2008), ⅷ+183pp. [19] _____, On global attractor for 2D Kirchhoff-Boussinesq model with supercritical nonlinearity, Comm. Partial Differential Equations, 36 (2011), 67-99. [20] I. Chueshov and A. Rezounenko, Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay, Commun. Pure Appl. Anal., 14 (2015), 1685-1704. [21] I. Chueshov and M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems, Dyn. Syst., 19 (2004), 127-144. [22] I. D. Chueshov, Approximate inertial manifolds of exponential order for semilinear parabolic equations subjected to additive white noise, J. Dynam. Differential Equations, 7 (1995), 549-566. [23] _____, The theory of functionals that uniquely determine the asymptotic dynamics of infinite-dimensional dissipative systems, Uspekhi Mat. Nauk, 53 (1998), 77-124. [24] _____, Vvedenie v Teoriyu Beskonechnomernykh Dissipativnykh Sistem, Universitetskie Lektsii po Sovremennoi Matematike. [University Lectures in Contemporary Mathematics], AKTA, Kharkiv, 1999. [25] 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. [26] H. Crauel, A. Debussche and F. Flandoli, Random attractors, Journal of Dynamics and Differential Equations, 9 (1997), 307-341. [27] H. Crauel, P. E. Kloeden and M. Yang, Random attractors of stochastic reaction-diffusion equations on variable domains, Stochastics & Dynamics, 11 (2011), 301-314. [28] H. Cui, M. M. Freitas and J. A. Langa, On random cocycle attractors with autonomous attraction universes, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3379-3407. [29] H. Cui and J. A. Langa, Uniform attractors for non-autonomous random dynamical systems, Journal of Differential Equations, 263 (2017), 1225-1268. [30] 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. [31] H. Cui, Y. Li and J. Yin, Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles, Nonlinear Analysis: Theory, Methods & Applications, 128 (2015), 303-324. [32] A. Debussche, On the finite dimensionality of random attractors, Stoch. Anal. Appl., 15 (1997), 473-491. [33] A. Eden, C. Foias, B. Nicolaenko and R. Teman, Exponential Attractors for Dissipative Evolution Equations, UK: RAM, Wiley, Chichester, 1994. [34] X. Fan, Attractors for a damped stochastic wave equation of the sine-Gordon type with sublinear multiplicative noise, Stochastic Analysis and Applications, 24 (2006), 767-793. [35] F. Flandoli and J. A. Langa, Determining modes for dissipative random dynamical systems, Stochastics: An International Journal of Probability and Stochastic Processes, 66 (1999), 1-25. [36] C. Foias and R. Teman, Some analytic and geometric properties of the solutions of the navier-stokes equations, J. Math. Pure Appl., 58 (1979), 339-368. [37] T. V. Girya and I. D. Chueshov, Inertial manifolds and stationary measures for stochastically perturbed dissipative dynamical systems, Mat. Sb., 186 (1995), 29-46. [38] 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, vol. 463, The Royal Society, 2007,163-181. [39] P. E. Kloeden, C. Pötzsche and M. Rasmussen, Limitations of pullback attractors for processes, Journal of Difference Equations and Applications, 18 (2012), 693-701. [40] J. A. Langa, G.Ł ukaszewicz and J. Real, Finite fractal dimension of pullback attractors for non-autonomous 2D Navier-Stokes equations in some unbounded domains, Nonlinear Anal., 66 (2007), 735-749. [41] J. A. Langa, J. C. Robinson, A. Rodriguez-Bernal, A. Suárez and A. Vidal-López, Existence and nonexistence of unbounded forwards attractor for a class of non-autonomous reaction diffusion equations, Discrete and Continuous Dynamical Systems. Series A, 18 (2007), 483-497. [42] Q. Ma, S. Wang and C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559. [43] G. Ochs, Weak Random Attractors, Citeseer, 1999. [44] J. C. Robinson, Infinite-dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, vol. 28, Cambridge University Press, 2001. [45] _____, Dimensions, Embeddings, and Attractors, Cambridge University Press Cambridge, 2010. [46] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis North-Holland Pub. Co., Amsterdam, 1979. [47] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 2nd ed., 1997. [48] M. I. Vishik and V. V. Chepyzhov, Attractors of nonautonomous dynamical systems and an estimate for their dimension, Mat. Zametki, 51 (1992), 141-143. [49] P. Walters, Introduction to Ergodic Theory, Springer-Verlag, New York, 1982. [50] B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583. [51] _____, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete and Continuous Dynamical Systems, 34 (2014), 269-300. [52] Y. Wang and J. Wang, Pullback attractors for multi-valued non-compact random dynamical systems generated by reaction-diffusion equations on an unbounded domain, Journal of Differential Equations, 259 (2015), 728-776.

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##### References:
 [1] L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. [2] L. Arnold and I. Chueshov, Order-preserving random dynamical systems: Equilibria, attractors, applications, Dynam. Stability Systems, 13 (1998), 265-280. doi: 10.1080/02681119808806264. [3] T. Caraballo, I. Chueshov and J. A. Langa, Existence of invariant manifolds for coupled parabolic and hyperbolic stochastic partial differential equations, Nonlinearity, 18 (2005), 747-767. doi: 10.1088/0951-7715/18/2/015. [4] D. Cheban and C. Mammana, Relation between different types of global attractors of set-valued nonautonomous dynamical systems, Set-Valued Analysis, 13 (2005), 291-321. doi: 10.1007/s11228-004-0046-x. [5] V. Chepyzhov and M. Vishik, A Hausdorff dimension estimate for kernel sections of nonautonomous evolution equations, Indiana Univ. Math. J., 42 (1993), 1057-1076. doi: 10.1512/iumj.1993.42.42049. [6] V. Chepyzhov and M. Vishik, Attractors of non-autonomous dynamical systems and their dimension, Journal de Mathématiques Pures et Appliquées, 73 (1994), 279-333. [7] _____, Attractors of non-autonomous evolution equations with translation compact symbols, in Partial Differential Operators and Mathematical Physics, Springer, 1995, 49-60. [8] V. Chepyzhov and M. Vishik, Attractors for nonautonomous Navier-Stokes system and other partial differential equations, in Instability in models connected with fluid flows. Ⅰ, vol. 6 of Int. Math. Ser. (N. Y. ), Springer, New York, 2008,135-265. [9] V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl. (9), 73 (1994), 279-333. [10] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, vol. 49, American Mathematical Society Providence, RI, USA, 2002. [11] _____, Monotone Random Systems Theory and Applications, vol. 1779, Springer Science & Business Media, 2002. [12] _____, Invariant manifolds and nonlinear master-slave synchronization in coupled systems, Appl. Anal., 86 (2007), 269-286. doi: 10.1080/00036810601097629. [13] _____, Long-time dynamics of Kirchhoff wave models with strong nonlinear damping, J. Differential Equations, 252 (2012), 1229-1262. doi: 10.1016/j.jde.2011.08.022. [14] _____, A squeezing property and its applications to a description of long-time behaviour in the three-dimensional viscous primitive equations, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 711-729. doi: 10.1017/S0308210512001953. [15] _____, Dynamics of Quasi-Stable Dissipative Systems, Universitext, Springer, Cham, 2015. [16] I. Chueshov, M. Eller and I. Lasiecka, Finite dimensionality of the attractor for a semilinear wave equation with nonlinear boundary dissipation, Comm. Partial Differential Equations, 29 (2004), 1847-1876. [17] I. Chueshov and I. Lasiecka, Determining functionals for a class of second order in time evolution equations with applications to von Karman equations, in Analysis and optimization of differential systems (Constanta, 2002), Kluwer Acad. Publ., Boston, MA, 2003,109-122. [18] _____, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc. , 195 (2008), ⅷ+183pp. [19] _____, On global attractor for 2D Kirchhoff-Boussinesq model with supercritical nonlinearity, Comm. Partial Differential Equations, 36 (2011), 67-99. [20] I. Chueshov and A. Rezounenko, Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay, Commun. Pure Appl. Anal., 14 (2015), 1685-1704. [21] I. Chueshov and M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems, Dyn. Syst., 19 (2004), 127-144. [22] I. D. Chueshov, Approximate inertial manifolds of exponential order for semilinear parabolic equations subjected to additive white noise, J. Dynam. Differential Equations, 7 (1995), 549-566. [23] _____, The theory of functionals that uniquely determine the asymptotic dynamics of infinite-dimensional dissipative systems, Uspekhi Mat. Nauk, 53 (1998), 77-124. [24] _____, Vvedenie v Teoriyu Beskonechnomernykh Dissipativnykh Sistem, Universitetskie Lektsii po Sovremennoi Matematike. [University Lectures in Contemporary Mathematics], AKTA, Kharkiv, 1999. [25] 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. [26] H. Crauel, A. Debussche and F. Flandoli, Random attractors, Journal of Dynamics and Differential Equations, 9 (1997), 307-341. [27] H. Crauel, P. E. Kloeden and M. Yang, Random attractors of stochastic reaction-diffusion equations on variable domains, Stochastics & Dynamics, 11 (2011), 301-314. [28] H. Cui, M. M. Freitas and J. A. Langa, On random cocycle attractors with autonomous attraction universes, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3379-3407. [29] H. Cui and J. A. Langa, Uniform attractors for non-autonomous random dynamical systems, Journal of Differential Equations, 263 (2017), 1225-1268. [30] 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. [31] H. Cui, Y. Li and J. Yin, Existence and upper semicontinuity of bi-spatial pullback attractors for smoothing cocycles, Nonlinear Analysis: Theory, Methods & Applications, 128 (2015), 303-324. [32] A. Debussche, On the finite dimensionality of random attractors, Stoch. Anal. Appl., 15 (1997), 473-491. [33] A. Eden, C. Foias, B. Nicolaenko and R. Teman, Exponential Attractors for Dissipative Evolution Equations, UK: RAM, Wiley, Chichester, 1994. [34] X. Fan, Attractors for a damped stochastic wave equation of the sine-Gordon type with sublinear multiplicative noise, Stochastic Analysis and Applications, 24 (2006), 767-793. [35] F. Flandoli and J. A. Langa, Determining modes for dissipative random dynamical systems, Stochastics: An International Journal of Probability and Stochastic Processes, 66 (1999), 1-25. [36] C. Foias and R. Teman, Some analytic and geometric properties of the solutions of the navier-stokes equations, J. Math. Pure Appl., 58 (1979), 339-368. [37] T. V. Girya and I. D. Chueshov, Inertial manifolds and stationary measures for stochastically perturbed dissipative dynamical systems, Mat. Sb., 186 (1995), 29-46. [38] 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, vol. 463, The Royal Society, 2007,163-181. [39] P. E. Kloeden, C. Pötzsche and M. Rasmussen, Limitations of pullback attractors for processes, Journal of Difference Equations and Applications, 18 (2012), 693-701. [40] J. A. Langa, G.Ł ukaszewicz and J. Real, Finite fractal dimension of pullback attractors for non-autonomous 2D Navier-Stokes equations in some unbounded domains, Nonlinear Anal., 66 (2007), 735-749. [41] J. A. Langa, J. C. Robinson, A. Rodriguez-Bernal, A. Suárez and A. Vidal-López, Existence and nonexistence of unbounded forwards attractor for a class of non-autonomous reaction diffusion equations, Discrete and Continuous Dynamical Systems. Series A, 18 (2007), 483-497. [42] Q. Ma, S. Wang and C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559. [43] G. Ochs, Weak Random Attractors, Citeseer, 1999. [44] J. C. Robinson, Infinite-dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, vol. 28, Cambridge University Press, 2001. [45] _____, Dimensions, Embeddings, and Attractors, Cambridge University Press Cambridge, 2010. [46] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis North-Holland Pub. Co., Amsterdam, 1979. [47] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 2nd ed., 1997. [48] M. I. Vishik and V. V. Chepyzhov, Attractors of nonautonomous dynamical systems and an estimate for their dimension, Mat. Zametki, 51 (1992), 141-143. [49] P. Walters, Introduction to Ergodic Theory, Springer-Verlag, New York, 1982. [50] B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583. [51] _____, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete and Continuous Dynamical Systems, 34 (2014), 269-300. [52] Y. Wang and J. Wang, Pullback attractors for multi-valued non-compact random dynamical systems generated by reaction-diffusion equations on an unbounded domain, Journal of Differential Equations, 259 (2015), 728-776.
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