Abstract
The long-time behavior of dissipative partial differential systems is characterized by the presence of a universal attractor X toward which all trajectories converge. This is the largest bounded set in the phase space of the system on which the backward-in-time initial value problem has bounded solutions. The structure of X may be very complicated even in the case of simple ordinary differential equations: X may be a fractal or parafractal set (i.e., a compact set for which the Hausdorff and fractal dimensions are different). In the case of dissipative partial differential equations, although the phase space (in the function space) is an infinite-dimensional Hilbert space, X has finite fractal dimension (see [CF, CFT]). However, the already complex nature of X is in this case further complicated by the infinite degrees of freedom of the ambient space.
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© 1989 Springer-Verlag New York Inc.
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Constantin, P., Foias, C., Nicolaenko, B., Teman, R. (1989). Presentation of the Approach and of the Main Results. In: Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations. Applied Mathematical Sciences, vol 70. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-3506-4_2
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DOI: https://doi.org/10.1007/978-1-4612-3506-4_2
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-8131-3
Online ISBN: 978-1-4612-3506-4
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