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Lipschitz deviation and embeddings of global attractors
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de Moura, Eleonora Pinto and Robinson, James C. (2010) Lipschitz deviation and embeddings of global attractors. Nonlinearity, Vol.23 (No.7). pp. 1695-1708. doi:10.1088/0951-7715/23/7/009 ISSN 0951-7715.
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Official URL: http://dx.doi.org/10.1088/0951-7715/23/7/009
Abstract
Hunt and Kaloshin (1999 Nonlinearity 12 1263-75) proved that it is possible to embed a compact subset X of a Hilbert space with upper box-counting dimension d < k into R-N for any N > 2k + 1, using a linear map L whose inverse is Holder continuous with exponent alpha < (N - 2d)/N(1 + tau(X)/2), where tau(X) is the 'thickness exponent' of X. More recently, Ott et al (2006 Ergod. Theory Dyn. Syst. 26 869-91) studied the effect of such embeddings on the Hausdorff dimension of X, and showed that for 'most' linear maps L : H -> R-N, d(H) (L(X)) >= min(N, d(H)(X)/(1 + tau(X)/2)). They also conjectured that 'many of the attractors associated with the evolution equations of mathematical physics have thickness exponent zero'. In this paper we introduce a variant of the thickness exponent, the Lipschitz deviation dev(X): we show that in both of the above results this can be used in place of the thickness exponent, and-appealing to results from the theory of approximate inertial manifolds-we prove that dev(X) = 0 for the attractors of a wide class of semilinear parabolic equations, thus providing a partial answer to the conjecture of Ott, Hunt and Kaloshin. In particular, dev(X) = 0 for the attractor of the 2D Navier-Stokes equations with forcing f is an element of L-2, while current results only guarantee that tau(X) = 0, when f is an element of C-infinity.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics Q Science > QC Physics |
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Nonlinearity | ||||
Publisher: | Institute of Physics Publishing Ltd. | ||||
ISSN: | 0951-7715 | ||||
Official Date: | July 2010 | ||||
Dates: |
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Volume: | Vol.23 | ||||
Number: | No.7 | ||||
Number of Pages: | 14 | ||||
Page Range: | pp. 1695-1708 | ||||
DOI: | 10.1088/0951-7715/23/7/009 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access | ||||
Funder: | CAPES, Engineering and Physical Sciences Research Council (EPSRC) | ||||
Grant number: | EP/G007470/1 (EPSRC) |
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