Smoothing and finite-dimensionality of uniform attractors in Banach spaces

The aim of this paper is to find an upper bound for the fractal dimension of uniform attractors in Banach spaces. The main technique we employ is essentially based on a compact embedding of some auxiliary Banach space into the phase space and a corresponding smoothing effect between these spaces. Ou...

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Autores: Cui, Hongyong, Carvalho, Alexandre N., Cunha, Arthur C., Langa Rosado, José Antonio
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2021
País:España
Institución:Universidad de Sevilla (US)
Repositorio:idUS. Depósito de Investigación de la Universidad de Sevilla
OAI Identifier:oai:idus.us.es:11441/180466
Acceso en línea:https://hdl.handle.net/11441/180466
https://doi.org/10.1016/j.jde.2021.03.013
Access Level:acceso abierto
Palabra clave:Smoothing property
Fractal dimension
Uniform attractor
Regularity
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spelling Smoothing and finite-dimensionality of uniform attractors in Banach spacesCui, HongyongCarvalho, Alexandre N.Cunha, Arthur C.Langa Rosado, José AntonioSmoothing propertyFractal dimensionUniform attractorRegularityThe aim of this paper is to find an upper bound for the fractal dimension of uniform attractors in Banach spaces. The main technique we employ is essentially based on a compact embedding of some auxiliary Banach space into the phase space and a corresponding smoothing effect between these spaces. Our bounds on the fractal dimension of uniform attractors are given in terms of the dimension of the symbol space and the Kolmogorov entropy number of the embedding. In addition, a dynamical analysis on the symbol space is also given, showing that the finite-dimensionality of the hull of a time-dependent function is fully determined by the tails of the function, which allows us to consider more general non-autonomous terms than quasi-periodic functions. As applications, we show that the uniform attractor of the 2D Navier-Stokes equation is finite-dimensional in H and in V, and that of a reaction-diffusion equation is finite-dimensional in L2 and in Lp, with p > 2.ElsevierEcuaciones Diferenciales y Análisis NuméricoFQM314: Análisis Estocástico de Sistemas Diferenciales2021info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfapplication/pdfhttps://hdl.handle.net/11441/180466https://doi.org/10.1016/j.jde.2021.03.013reponame:idUS. Depósito de Investigación de la Universidad de Sevillainstname:Universidad de Sevilla (US)Inglés10.1016/j.jde.2021.03.013info:eu-repo/semantics/openAccessoai:idus.us.es:11441/1804662026-06-17T12:51:07Z
dc.title.none.fl_str_mv Smoothing and finite-dimensionality of uniform attractors in Banach spaces
title Smoothing and finite-dimensionality of uniform attractors in Banach spaces
spellingShingle Smoothing and finite-dimensionality of uniform attractors in Banach spaces
Cui, Hongyong
Smoothing property
Fractal dimension
Uniform attractor
Regularity
title_short Smoothing and finite-dimensionality of uniform attractors in Banach spaces
title_full Smoothing and finite-dimensionality of uniform attractors in Banach spaces
title_fullStr Smoothing and finite-dimensionality of uniform attractors in Banach spaces
title_full_unstemmed Smoothing and finite-dimensionality of uniform attractors in Banach spaces
title_sort Smoothing and finite-dimensionality of uniform attractors in Banach spaces
dc.creator.none.fl_str_mv Cui, Hongyong
Carvalho, Alexandre N.
Cunha, Arthur C.
Langa Rosado, José Antonio
author Cui, Hongyong
author_facet Cui, Hongyong
Carvalho, Alexandre N.
Cunha, Arthur C.
Langa Rosado, José Antonio
author_role author
author2 Carvalho, Alexandre N.
Cunha, Arthur C.
Langa Rosado, José Antonio
author2_role author
author
author
dc.contributor.none.fl_str_mv Ecuaciones Diferenciales y Análisis Numérico
FQM314: Análisis Estocástico de Sistemas Diferenciales
dc.subject.none.fl_str_mv Smoothing property
Fractal dimension
Uniform attractor
Regularity
topic Smoothing property
Fractal dimension
Uniform attractor
Regularity
description The aim of this paper is to find an upper bound for the fractal dimension of uniform attractors in Banach spaces. The main technique we employ is essentially based on a compact embedding of some auxiliary Banach space into the phase space and a corresponding smoothing effect between these spaces. Our bounds on the fractal dimension of uniform attractors are given in terms of the dimension of the symbol space and the Kolmogorov entropy number of the embedding. In addition, a dynamical analysis on the symbol space is also given, showing that the finite-dimensionality of the hull of a time-dependent function is fully determined by the tails of the function, which allows us to consider more general non-autonomous terms than quasi-periodic functions. As applications, we show that the uniform attractor of the 2D Navier-Stokes equation is finite-dimensional in H and in V, and that of a reaction-diffusion equation is finite-dimensional in L2 and in Lp, with p > 2.
publishDate 2021
dc.date.none.fl_str_mv 2021
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/11441/180466
https://doi.org/10.1016/j.jde.2021.03.013
url https://hdl.handle.net/11441/180466
https://doi.org/10.1016/j.jde.2021.03.013
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv 10.1016/j.jde.2021.03.013
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Elsevier
publisher.none.fl_str_mv Elsevier
dc.source.none.fl_str_mv reponame:idUS. Depósito de Investigación de la Universidad de Sevilla
instname:Universidad de Sevilla (US)
instname_str Universidad de Sevilla (US)
reponame_str idUS. Depósito de Investigación de la Universidad de Sevilla
collection idUS. Depósito de Investigación de la Universidad de Sevilla
repository.name.fl_str_mv
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