Pullback attractors for generalized evolutionary systems

Alexey Cheskidov; Landon Kavlie

doi:10.3934/dcdsb.2015.20.749

Article Contents

2015, Volume 20, Issue 3: 749-779. Doi: 10.3934/dcdsb.2015.20.749

This issue Previous Article Non-autonomous dynamical systems Next Article Strong uniform attractors for non-autonomous dissipative PDEs with non translation-compact external forces

Pullback attractors for generalized evolutionary systems

Alexey Cheskidov^1, and
Landon Kavlie^1,

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

Received: November 30, 2013

Revised: February 28, 2014

Published: April 2015

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Abstract

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.

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Mathematics Subject Classification: 35Q30, 37B55, 37D45, 37L05, 37N10.

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