# American Institute of Mathematical Sciences

October  2014, 34(10): 4211-4222. doi: 10.3934/dcds.2014.34.4211

## Invariant measures for non-autonomous dissipative dynamical systems

 1 University of Warsaw, Institute of Applied Mathematics and Mechanics, Banacha 2, 02-097 Warsaw 2 Mathematics Institute, University of Warwick, Coventry, CV4 7AL., United Kingdom

Received  December 2008 Revised  November 2009 Published  April 2014

Given a non-autonomous process $U(\cdot,\cdot)$ on a complete separable metric space $X$ that has a pullback attractor $A(\cdot)$, we construct a family of invariant Borel probability measures $\{\mu_t\}_{t\in \mathbb{R}}$: the measures satisfy ${\rm supp }\,{\mu_t}\subset A(t)$ for all $t\in \mathbb{R}$ and the invariance property $\mu_t(E)=\mu_\tau(U(t,\tau)^{-1}E)$ for every Borel set $E\in X$. Our construction uses the generalised Banach limit. We then show that a Liouville-type equation holds for the evolution of $\mu_t$ under the process $U(\cdot,\cdot)$ generated by the ordinary differential equation $u_t=F(t,u)$ on a Banach space, and apply our theory to the non-autonomous 2D Navier--Stokes equations on unbounded domains satisfying a Poincaré inequality.
Citation: Grzegorz Łukaszewicz, James C. Robinson. Invariant measures for non-autonomous dissipative dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4211-4222. doi: 10.3934/dcds.2014.34.4211
##### References:
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##### References:
 [1] T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems,, Nonlinear Anal., 64 (2006), 484.  doi: 10.1016/j.na.2005.03.111.  Google Scholar [2] T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains,, C. R. Math. Acad. Sci. Paris, 342 (2006), 263.  doi: 10.1016/j.crma.2005.12.015.  Google Scholar [3] T. Caraballo, P. E. Kloeden and J. Real, Invariant measures and statistical solutions of the globally modified Navier-Stokes equations,, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 761.  doi: 10.3934/dcdsb.2008.10.761.  Google Scholar [4] A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems,, Springer, (2012).  doi: 10.1007/978-1-4614-4581-4.  Google Scholar [5] M. Chekroun and N. E. Glatt-Holtz, Invariant measures for dissipative dynamical systems: Abstract results and applications,, Comm. Math. Phys., 316 (2012), 723.  doi: 10.1007/s00220-012-1515-y.  Google Scholar [6] C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence,, Cambridge University Press, (2001).  doi: 10.1017/CBO9780511546754.  Google Scholar [7] P. E. Kloeden, P. Marín-Rubio and J. Real, Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations,, Commun. Pure Appl. Anal., 8 (2009), 785.  doi: 10.3934/cpaa.2009.8.785.  Google Scholar [8] G. Łukaszewicz, Pullback attractors and statistical solutions for 2-D Navier-Stokes equations,, Discrete Continu. Dyn. Syst. Ser. B, 9 (2008), 643.  doi: 10.3934/dcdsb.2008.9.643.  Google Scholar [9] G. Łukaszewicz, J. Real and J. C. Robinson, Invariant measures for dissipative systems and generalised banach limits,, J. Dynam. Differential Equations, 23 (2011), 225.  doi: 10.1007/s10884-011-9213-6.  Google Scholar [10] V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations,, Princeton University Press, (1960).   Google Scholar [11] J. C. Robinson, Infinite-Dimensional Dynamical Systems,, Cambridge University Press, (2001).  doi: 10.1007/978-94-010-0732-0.  Google Scholar [12] R. Rosa, The global attractor for the 2 D Navier-Stokes flow on some unbounded domains,, Nonlinear Anal., 32 (1998), 71.  doi: 10.1016/S0362-546X(97)00453-7.  Google Scholar [13] R. Temam, Navier-Stokes equations, Theory and Numerical Analysis,, 2nd. ed., (1979).   Google Scholar [14] A. M. van der Vaart and J. A. Wellner, Weak Convergence and Empirical Processes: With Applications to Statistics,, Springer, (2000).   Google Scholar [15] X. Wang, Upper semi-continuity of stationary statistical properties of dissipative systems,, Discrete Contin. Dyn. Syst., 23 (2009), 521.  doi: 10.3934/dcds.2009.23.521.  Google Scholar
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