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Flattening, squeezing and the existence of random attractors. (English) Zbl 1133.37323

Summary: The study of qualitative properties of random and stochastic differential equations is now one of the most active fields in the modern theory of dynamical systems. In the deterministic case, the properties of flattening and squeezing in infinite-dimensional autonomous dynamical systems require the existence of a bounded absorbing set and imply the existence of a global attractor. The flattening property involves the behaviour of individual trajectories while the squeezing property involves the difference of trajectories. It is shown here that the flattening property is implied by the squeezing property and is in fact weaker, since the attractor in a system with the flattening property can be infinite-dimensional, whereas it is always finite-dimensional in a system with the squeezing property. The flattening property is then generalized to random dynamical systems, for which it is called the pullback flattening property. It is shown to be weaker than the random squeezing property, but equivalent to pullback asymptotic compactness and pullback limit-set compactness, and thus implies the existence of a random attractor. The results are also valid for deterministic nonautonomous dynamical systems formulated as skew-product flows.

MSC:

37H10 Generation, random and stochastic difference and differential equations
35B41 Attractors
37B55 Topological dynamics of nonautonomous systems
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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References:

[1] Arnold, L. 1998 Random dynamical systems. Berlin, Germany: Springer. · Zbl 0906.34001
[2] Brézis, H. 1983 Analyse fonctionnelle: theorie et applications. Paris, France: Masson. · Zbl 0511.46001
[3] Brzeźniak, Z. & Li, Y. 2002 Asymptotic behaviour of solutions to the 2D stochastic Navier–Stokes equations in unbounded domains–new developments. In <i>Proc. First Sino-German Conf. in Stochastic Analysis</i>. 28 August–3 September, Beijing, China, pp. 78–111. · Zbl 1087.60048
[4] Brzeźniak, Z. & Li, Y. In press. Asymptotic compactness and absorbing sets for 2D stochastic Navier–Stokes equations on some unbounded domains. <i>Trans. Am. Math. Soc.</i>
[5] Chepyzhov, V.V. & Vishik, M.I. 2002 <i>Attractors for equations of mathematical physics</i>, vol. 49. Providence, RI: American Mathematical Society.
[6] Chueshov, I. & Lasiecka, I. 2004 Attractors for second-order evolution equations with a nonlinear damping. <i>J. Dynam. Diff. Equat.</i>&nbsp;<b>16</b>, 469–512, (doi:10.1007/s10884-004-4289-x). · Zbl 1072.37054
[7] Constantin, P., Foias, C. & Temam, R. 1985 Attractors representing turbulent flows. <i>Mem. Amer. Math. Soc.</i>, vol. 53. Providence, RI: American Mathematical Society.
[8] Crauel, H. 1999 Global random attractors are uniquely determined by attracting deterministic compact sets. <i>Ann. Mat. Pura Appl.</i>&nbsp;<b>176</b>, 57–72. · Zbl 0954.37027
[9] Crauel, H. 2001 Random point attractors versus random set attractors. <i>J. Lond. Math. Soc.</i>&nbsp;<b>63</b>, 413–427. · Zbl 1011.37032
[10] Crauel, H. & Flandoli, F. 1994 Attractors for random dynamical systems. <i>Probab. Theory Relat. Fields</i>&nbsp;<b>100</b>, 365–393, (doi:10.1007/BF01193705). · Zbl 0819.58023
[11] Crauel, H., Debussche, A. & Flandoli, F. 1995 Random attractors. <i>J. Dyn. Diff. Equat.</i>&nbsp;<b>9</b>, 307–341, (doi:10.1007/BF02219225).
[12] Debussche, A. 1997 On the finite dimensionality of random attractors. <i>Stoch. Anal. Appl.</i>&nbsp;<b>15</b>, 473–491. · Zbl 0888.60051
[13] Deimling, K. 1985 Nonlinear functional analysis. Berlin, Germany: Springer. · Zbl 0559.47040
[14] Eden, A., Foias, C., Nicolaenko, B. & Temam, R. 1994 Exponential attractors for dissipative evolution equations. Chichester, UK: RAM, Wiley. · Zbl 0842.58056
[15] Flandoli, F. & Langa, J.A. 1999 Determining modes for dissipative random dynamical systems. <i>Stoch. Stoch. Rep.</i>&nbsp;<b>66</b>, 1–25. · Zbl 0923.35213
[16] Flandoli, F. & Schmalfuss, B. 1996 Random attractors for the 3D stochastic Navier–Stokes equation with multiplicative white noise. <i>Stoch. Stoch. Rep.</i>&nbsp;<b>59</b>, 21–45. · Zbl 0870.60057
[17] Foias, C. & Prodi, G. 1967 Sur le comportement global des solutions non-stationnaires des equations de Navier–Stokes en dimension 2. <i>Rend. Sem. Mat. Univ. Padova</i>&nbsp;<b>39</b>, 1–34.
[18] Foias, C. & Temam, R. 1979 Some analytic and geometric properties of the solutions of the Navier–Stokes equations. <i>J. Math. Pure Appl.</i>&nbsp;<b>58</b>, 339–368. · Zbl 0454.35073
[19] Foias, C., Manley, O. & Temam, R. 1988 Modelling of the interaction of small and large eddies in two dimensional turbulent flows. <i>Math. Mod. Numer. Anal.</i>&nbsp;<b>22</b>, 93–118. · Zbl 0663.76054
[20] Hale, J. 1988 <i>Asymptotic behavior of dissipative systems</i>. Mathematical Surveys and Monographs. Providence, RI: AMS.
[21] Khanmamedov, A.Kh. 2006 Global attractors for von Karman equations with nonlinear interior dissipation. <i>J. Math. Anal. Appl.</i>&nbsp;<b>318</b>, 92–101, (doi:10.1016/j.jmaa.2005.05.031). · Zbl 1092.35067
[22] Ladyzhenskaya, O. 1991 Attractors for semigroups and evolution equations. Cambridge, UK: Cambridge University Press. · Zbl 0755.47049
[23] Langa, J.A. 2003 Finite-dimensional limiting dynamics of random dynamical systems. <i>Dyn. Syst.</i>&nbsp;<b>18</b>, 57–68. · Zbl 1038.37041
[24] Ma, Q., Wang, S. & Zhong, C. 2002 Necessary and sufficient conditions for the existence of global attractors for semigroups and applications. <i>Indiana Univ. Math. J.</i>&nbsp;<b>51</b>, 1541–1559, (doi:10.1512/iumj.2002.51.2255). · Zbl 1028.37047
[25] Robinson, J.C. 2001 Infinite-dimensional dynamical systems. Cambridge, UK: Cambridge University Press. · Zbl 1026.37500
[26] Rosa, R. 1998 The global attractor for the 2D Navier–Stokes flow on some unbounded domains. <i>Nonlin. Anal. TMA</i>&nbsp;<b>32</b>, 71–85, (doi:10.1016/S0362-546X(97)00453-7).
[27] Schmalfuss, B. 1992 Backward cocycle and attractors of stochastic differential equations. <i>International seminar on applied mathematics-nonlinear dynamics: attractor approximation and global behaviour</i> (eds. Reitmann, V. Redrich, T. & Kosch, N.J.), pp. 185–192, Leipzig, Berlin: Teubner
[28] Temam, R. 1988 Infinite-dimensional dynamical systems in mechanics and physics. New York, NY: Springer. · Zbl 0662.35001
[29] Vishik, M.I. 1992 Asymptotic behaviour of solutions of evolutionary equations. Cambridge, UK: Cambridge University Press. · Zbl 0797.35016
[30] Wang, Y. Zhong, C. & Zhou, S. In press. Pullback attractors of nonautonomous dynamical systems. <i>Discrete Conts. Dyn. Systems, Ser. A</i>.
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