# 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
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