Finite-Dimensionality of Tempered Random Uniform Attractors

Autor(es)

Cui, Hongyong ^[1] ; 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 Cotreos 1160, Seville 41080 - Spain

• Tipo de documento: Artigo Científico
• Fonte: JOURNAL OF NONLINEAR SCIENCE; v. 32, n. 1 FEB 2022.
• Citações Web of Science: 0

Resumo

Finite-dimensional attractors play an important role in finite-dimensional reduction of PDEs in mathematical modelization and numerical simulations. For non-autonomous random dynamical systems, Cui and Langa (J Differ Equ, 263:1225-1268, 2017) developed a random uniform attractor as a minimal compact random set which provides a certain description of the forward dynamics of the underlying system by forward attraction in probability. In this paper, we study the conditions that ensure a random uniform attractor to have finite fractal dimension. Two main criteria are given, one by a smoothing property and the other by a squeezing property of the system, and neither of the two implies the other. The upper bound of the fractal dimension consists of two parts: the fractal dimension of the symbol space plus a number arising from the smoothing/squeezing property. As an illustrative application, the random uniform attractor of a stochastic reaction-diffusion equation with scalar additive noise is studied, for which the finite-dimensionality in L-2 is established by the squeezing approach and that in H-0(1) by the smoothing framework. In addition, a random absorbing set that absorbs itself after a deterministic period of time is also constructed. (AU)

• 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
• 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