American Institute of Mathematical Sciences

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October  2010, 26(4): 1329-1357. doi: 10.3934/dcds.2010.26.1329

Pullback exponential attractors

José A. Langa 1, , Alain Miranville 2, and  José Real 3,

1. 

Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain
2. 

Université de Poitiers, SP2MI, Boulevard Marie et Pierre Curie, 86962 Chasseneuil Futuroscope Cedex, France
3. 

Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla

Received  November 2008 Revised  March 2009 Published  December 2009

In this work, we show how to construct a pullback exponential attractor associated with an infinite dimensional dynamical system, i.e., a family of time dependent compact sets, with finite fractal dimension, which are positively invariant and exponentially attract in the pullback sense every bounded set of the phase space. Our construction is based on the one in Efendiev et al. [11] in which a uniform forwards (and so also pullback) exponential attractor is constructed. We relax the conditions in [11] in order to obtain an unbounded family of exponential attractors for which the uniform convergence fails so that only the pullback attraction is expected. Thus, by proving that global pullback attractors are included in our family of exponential attractors, we generalize the concept of an exponential attractor to the theory of infinite dimensional non-autonomous dynamical systems. We illustrate our results on a 2D Navier-Stokes system in bounded domains.
Keywords: 2D Navier-Stokes equations., non-autonomous PDEs, pullback attractors, infinite-dimensional dynamical systems, exponential attractors.
Mathematics Subject Classification: Primary: 37L55, 93D20; Secondary: 28B10, 34D2.
Citation: José A. Langa, Alain Miranville, José Real. Pullback exponential attractors. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1329-1357. doi: 10.3934/dcds.2010.26.1329
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