# American Institute of Mathematical Sciences

May  2015, 20(3): 749-779. doi: 10.3934/dcdsb.2015.20.749

## Pullback attractors for generalized evolutionary systems

 1 Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 322 Science and Engineering Offices (M/C 249), 851 S. Morgan Street, Chicago, Illinois 60607-7045, United States, United States

Received  December 2013 Revised  March 2014 Published  January 2015

We give an abstract framework for studying nonautonomous PDEs, called a generalized evolutionary system. In this setting, we define the notion of a pullback attractor. Moreover, we show that the pullback attractor, in the weak sense, must always exist. We then study the structure of these attractors and the existence of a strong pullback attractor. We then apply our framework to both autonomous and nonautonomous evolutionary systems as they first appeared in earlier works by Cheskidov, Foias, and Lu. In this con- text, we compare the pullback attractor to both the global attractor (in the autonomous case) and the uniform attractor (in the nonautonomous case). Finally, we apply our results to the nonautonomous 3D Navier-Stokes equations on a periodic domain with a translationally bounded force. We show that the Leray-Hopf weak solutions form a generalized evolutionary system and must then have a weak pullback attractor.
Citation: Alexey Cheskidov, Landon Kavlie. Pullback attractors for generalized evolutionary systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 749-779. doi: 10.3934/dcdsb.2015.20.749
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