Smooth attractors have zero ``thickness''
type: Item
instance of: scholarly article
title: Smooth attractors have zero ``thickness''
zbMATH Open document ID: 0971.37011
author: James C. Robinson
author: Peter Friz
published in: Journal of Mathematical Analysis and Applications
publication date: 2001-10-28T00:00:00Z
review text: The authors study the problem of finding one-to-one finite dimensional orthogonal projections of the global attractors \(\mathcal A\) belonging to an infinite dimensional Hilbert space \(H\) such that the inverse mappings are Hölder continuous. It is known [see \textit{B. Hunt} and \textit{V. Kaloshin}, Regularity of embeddings of finite-dimensional fractal sets into infinite dimensional spaces, Nonlinearity 12, 1263-1275 (1999; Zbl 0932.28006)] that if the set \(\mathcal A\) has a finite fractal dimension \(d_f(\mathcal A)\) then a typical linear map \(L: H\to\mathbb R^k\) satisfies the condition \[ |u-v|\leq C|Lu-Lv|^\theta, \quad u,v\in\mathcal A, \quad \forall\;\theta\in\left(0 , \frac{k-2d_f(\mathcal A)}{k(1+\tau/2)}\right) \tag{1} \] where \(\tau=\tau(\mathcal A)\) is the so-called `thickness' of the set \(\mathcal A\) which measures how well the set can be approximated by finite dimensional hyperplanes. In the present paper the authors show that (1) holds for the typical orthogonal projection as well and verify that in the particular case when \(H=L^2(\Omega)\) and when the attractor is smooth (\(\mathcal A\subset W_2^s(\Omega)\) for every \(s>0\), where \(W^s_2\) are the usual Sobolev spaces) the `thickness' of the attractor is equal to zero: \(\tau(\mathcal A)=0\). This result shows that the global attractors of many physically relevant PDEs (including the 2D Navier-Stokes equations with smooth external forces in smooth domains) possess one-to-one finite dimensional orthogonal projections the inverses of which have the Hölder exponents arbitrarily close to one. Indeed, (1) implies that \[ \theta\sim 1-\frac{2d_f(\mathcal A)}k\quad \text{if } \tau(\mathcal A)=0 \] which is arbitrarily close to one if \(k\) is large enough.
reviewed by: S. V. Zelik
Mathematics Subject Classification ID: 37C70
Mathematics Subject Classification ID: 37J30
zbMATH DE Number: 1398594
zbMATH Keywords: global attractors
zbMATH Keywords: Mañé projections
zbMATH Keywords: Hölder continuity
MaRDI profile type: Publication
OpenAlex ID: W2117913339
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DOI: 10.1006/JMAA.1999.6569
description: scientific article; zbMATH DE number 1398594