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Fractal dimension of a random invariant set
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UNSPECIFIED (2006) Fractal dimension of a random invariant set. JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES, 85 (2). pp. 269-294. doi:10.1016/j.matpur.2005.08.001 ISSN 0021-7824.
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Official URL: http://dx.doi.org/10.1016/j.matpur.2005.08.001
Abstract
In recent years many deterministic parabolic equations have been shown to possess global attractors which, despite being subsets of an infinite-dimensional phase space, are finite-dimensional objects. Debussche showed how to generalize the deterministic theory to show that the random attractors of the corresponding stochastic equations have finite Hausdorff dimension. However, to deduce a parametrization of a 'finite-dimensional' set by a finite number of coordinates a bound on the fractal (upper box-counting) dimension is required. There are non-trivial problems in extending Debussche's techniques to this case, which can be overcome by careful use of the Poincare recurrence theorem. We prove that under the same conditions as in Debussche's paper and an additional concavity assumption, the fractal dimension enjoys the same bound as the Hausdorff dimension. We apply our theorem to the 2d Navier-Stokes equations with additive noise, and give two results that allow different long-time states to be distinguished by a finite number of observations. (c) 2005 Elsevier SAS. All rights reserved.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Journal or Publication Title: | JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES | ||||
Publisher: | GAUTHIER-VILLARS/EDITIONS ELSEVIER | ||||
ISSN: | 0021-7824 | ||||
Official Date: | February 2006 | ||||
Dates: |
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Volume: | 85 | ||||
Number: | 2 | ||||
Number of Pages: | 26 | ||||
Page Range: | pp. 269-294 | ||||
DOI: | 10.1016/j.matpur.2005.08.001 | ||||
Publication Status: | Published |
Data sourced from Thomson Reuters' Web of Knowledge
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