Pullback exponential attractors

Abstract

In this work, we show how to construct a pullback exponentialattractor associated with an infinite dimensional dynamicalsystem, i.e., a family of time dependent compact sets, with finitefractal dimension, which are positively invariant andexponentially attract in the pullback sense every bounded set ofthe phase space. Our construction is based on the one in Efendiev etal. [11] in which a uniform forwards (and so alsopullback) exponential attractor is constructed. We relax theconditions in [11] in order to obtain an unbounded familyof exponential attractors for which the uniform convergence failsso that only the pullback attraction is expected. Thus, by provingthat global pullback attractors are included in our family ofexponential attractors, we generalize the concept of anexponential attractor to the theory of infinite dimensionalnon-autonomous dynamical systems. We illustrate our results on a 2DNavier-Stokes system in bounded domains.