Invariant measures for non-autonomous dissipative dynamical systems

Grzegorz Łukaszewicz; James C. Robinson

doi:10.3934/dcds.2014.34.4211

Article Contents

2014, Volume 34, Issue 10: 4211-4222. Doi: 10.3934/dcds.2014.34.4211

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Invariant measures for non-autonomous dissipative dynamical systems

Grzegorz Łukaszewicz^1, and
James C. Robinson^2,

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

Received: November 30, 2008

Revised: October 31, 2009

Published: September 2014

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Abstract

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.

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Mathematics Subject Classification: 35B41, 35D99, 76F20.

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