Nonlinear Analysis 66 (2007) 735–749 www.elsevier.com/locate/na Finite fractal dimension of pullback attractors for non-autonomous 2D Navier–Stokes equations in some unbounded domains Jos´ e A. Langa a,∗ , G. Lukaszewicz b , J. Real a a Dpto. Ecuaciones Diferenciales y An´ alisis Num´ erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain b University of Warsaw, Institute of Applied Mathematics and Mechanics, Banacha 2, 02-097 Warsaw, Poland Received 28 August 2005; accepted 14 December 2005 Abstract We study the asymptotic behaviour of non-autonomous 2D Navier–Stokes equations in unbounded domains for which a Poincar´ e inequality holds. In particular, we give sufficient conditions for their pullback attractor to have finite fractal dimension. The existence of pullback attractors in this framework comes from the existence of bounded absorbing sets of pullback asymptotically compact processes [T. Caraballo, G. Lukaszewicz, J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear Anal. 64 (3) (2006) 484–498]. We show that, under suitable conditions, the method of Lyapunov exponents in [P. Constantin, C. Foias, R. Temam, Attractors representing turbulent flows, Mem. Amer. Math. Soc. 53 (1984) [5]] for the dimension of attractors can be developed in this new context. c  2005 Elsevier Ltd. All rights reserved. MSC: 35B41; 35Q35; 37L30 1. Introduction In the study of autonomous infinite dimensional dynamical systems having global attractors, one of the main (and most beautiful) results refers to the finite dimensionality of these maximal invariant compact sets. In the case of dissipative PDEs, for example, this property of the global ∗ Corresponding author. Tel.: +34 95 455 6934; fax: +34 95 455 7982. E-mail address: langa@us.es (J.A. Langa). 0362-546X/$ - see front matter c  2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.12.017