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Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems*

Published online by Cambridge University Press:  12 July 2007

M. Efendiev
Affiliation:
Institut für Analysis, Dynamik und Modelierung, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
S. Zelik
Affiliation:
Institut für Analysis, Dynamik und Modelierung, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
A. Miranville
Affiliation:
Laboratoire de Mathématiques et Applications, Université de Poitiers, UMR 6086–CNRS, SP2MI, Boulevard Marie et Pierre Curie, 86962 Chasseneuil Futuroscope Cedex, France(miranv@math.univ-poitiers.fr)

Abstract

We suggest in this paper a new explicit algorithm allowing us to construct exponential attractors which are uniformly Hölder continuous with respect to the variation of the dynamical system in some natural large class. Moreover, we extend this construction to non-autonomous dynamical systems (dynamical processes) treating in that case the exponential attractor as a uniformly exponentially attracting, finite-dimensional and time-dependent set in the phase space. In particular, this result shows that, for a wide class of non-autonomous equations of mathematical physics, the limit dynamics remains finite dimensional no matter how complicated the dependence of the external forces on time is. We illustrate the main results of this paper on the model example of a non-autonomous reaction–diffusion system in a bounded domain.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2005

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References

* Dedicated to Professor Roger Temam on the occasion of his 65th birthday.