Finite-dimensional negatively invariant subsets of Banach spaces
type: Item
instance of: scholarly article
title: Finite-dimensional negatively invariant subsets of Banach spaces
zbMATH Open document ID: 1494.37047
author: Arthur C. Cunha
author: James C. Robinson
author: A. N. Carvalho
author: J. A. Langa
published in: Journal of Mathematical Analysis and Applications
publication date: 2022-02-14T00:00:00Z
review text: The box-counting dimension (also called fractal dimension or limit capacity) of a compact set \(A\), when finite, allows one to prove the finiteness of both the Hausdorff dimension and the topological dimension. Furthermore it can be used to prove that one can injectively project \(A\) into an Euclidean space. In [\textit{R. Mañé}, Lect. Notes Math. 898, 230--242 (1981; Zbl 0544.58014)] we learned that if \(A\) is a compact subset of a Banach space \(X\) that is negatively invariant for a map \(S\), meaning that \(S(A)\supset A\), and for each \(x\in A\) we can decompose the derivative \(DS(x)=C(x)+L(x)\), where \(C(x)\) is a compact linear operator and \(L\) is a linear strict contraction, then the box-counting dimension of \(A\) is finite. The authors begin presenting a rather simple proof of this result and, in particular, they obtain that when \(S\) is a differentiable compact map, then \(A\) has finite box-counting dimension. Furthermore, they obtain bounds for the box-counting dimension of such sets following the works [\textit{J. Málek} et al., Acta Appl. Math. 37, No. 1--2, 83--97 (1994; Zbl 0815.35084); \textit{S. V. Zelik}, Math. Notes 67, No. 2, 248--251 (2000; Zbl 0974.35016); translation from Mat. Zametki 67, No. 2, 304--308 (2000)]. To this end they assume some smoothing properties of \(S\), which in their simplest form require \(S\) to be Lipschitz from \(X\) into another Banach space \(Z\) that is compactly embedded in \(X\). The resulting bounds depend on the Kolmogorov \(\varepsilon\)-entropy of the embedding of \(Z\) into \(X\). They apply their results to an abstract semilinear parabolic equation and a two-dimensional Navier-Stokes equation on a periodic domain.
reviewed by: Matheus Cheque Bortolan
Mathematics Subject Classification ID: 37L30
Mathematics Subject Classification ID: 37C70
Mathematics Subject Classification ID: 37C45
Mathematics Subject Classification ID: 28A12
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 35K58
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 47J35
zbMATH DE Number: 7472982
zbMATH Keywords: global attractors
zbMATH Keywords: box-counting dimension
zbMATH Keywords: negatively invariant sets
zbMATH Keywords: smoothing
MaRDI profile type: Publication
full work available at URL: j.jmaa.2021.125945
OpenAlex ID: W4200217562
cites work: Random pullback exponential attractors: general existence results for random dynamical systems in Banach spaces
cites work: Attractors for infinite-dimensional non-autonomous dynamical systems
cites work: Finite-dimensional global attractors in Banach spaces
cites work: Pullback exponential attractors for evolution processes in Banach spaces: theoretical results
cites work: Attractors for equations of mathematical physics
cites work: Q4499087
cites work: Q3522682
cites work: Global lyapunov exponents, kaplan-yorke formulas and the dimension of the attractors for 2D navier-stokes equations
cites work: Q5690466
cites work: Exponential attractors in Banach spaces.
cites work: Exponential attractors for a nonlinear reaction-diffusion system in
cites work: Q4431283
cites work: Geometric theory of semilinear parabolic equations
cites work: Fractal dimension, attractors, and the Boussinesq approximation in three dimensions
cites work: Negatively invariant sets of compact maps and an extension of a theorem of Cartwright
cites work: Q3333765
cites work: Q3645637
cites work: Infinite-dimensional dynamical systems. An introduction to dissipative parabolic PDEs and the theory of global attractors
cites work: Q3066835
cites work: Infinite-dimensional dynamical systems in mechanics and physics
cites work: Q4169288
DOI: 10.1016/J.JMAA.2021.125945
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description: scientific article; zbMATH DE number 7472982