Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

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

M. Efendieva1, S. Zelika1 and A. Miranvillea2

a1 Institut für Analysis, Dynamik und Modelierung, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

a2 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.

(Received August 04 2004)

(Accepted February 09 2005)

Footnotes

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