Smoothing and finite-dimensionality of uniform attractors in Banach spaces

Autor(es)

Cui, Hongyong ^[1] ; Carvalho, Alexandre N. ^[2] ; Cunha, Arthur C. ^[2] ; Langa, Jose A. ^[3]

Afiliação do(s) autor(es)

^[1] Huazhong Univ Sci & Technol, Sch Math & Stat, Wuhan 430074 - Peoples R China

^[2] Univ Sao Paulo, Inst Ciencias Matemat & Comp, Campus Sao Carlos, Caixa Postal 668, BR-13560970 Sao Carlos, SP - Brazil

^[3] Univ Seville, Dept Ecuac Diferenciales & Anal Numer, Apdo De Correos 1160, E-41080 Seville - Spain

• Tipo de documento: Artigo Científico
• Fonte: Journal of Differential Equations; v. 285, p. 383-428, JUN 5 2021.
• Citações Web of Science: 0

Resumo

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 Hand in V, and that of a reaction-diffusion equation is finite-dimensional in L-2 and in L-p, with p > 2. (C) 2021 Elsevier Inc. All rights reserved. (AU)

• Processo FAPESP: 18/10997-6 - Robusteza de atratores sob perturbações autônomas ou não-autônomas: Estabilidade estrutural
• Beneficiário: Alexandre Nolasco de Carvalho
• Modalidade de apoio: Bolsas no Exterior - Pesquisa
• Processo FAPESP: 18/10634-0 - Estimativas da dimensão fractal de atratores para sistemas dinâmicos autônomos e não-autônomos: aplicações
• Beneficiário: Arthur Cavalcante Cunha
• Modalidade de apoio: Bolsas no Exterior - Estágio de Pesquisa - Doutorado
• Processo FAPESP: 16/26289-5 - Estimativas da Dimensão Fractal para Atratores de Sistemas Dinâmicos Autônomos e Não-Autônomos
• Beneficiário: Arthur Cavalcante Cunha
• Modalidade de apoio: Bolsas no Brasil - Doutorado