On random cocycle attractors with autonomous attraction universes

Hongyong Cui; Mirelson M. Freitas; José A. Langa

doi:10.3934/dcdsb.2017142

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2017, Volume 22, Issue 9: 3379-3407. Doi: 10.3934/dcdsb.2017142

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On random cocycle attractors with autonomous attraction universes

1.
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China
2.
Departamento de Matemática, Universidade Federal do Pará, Rua Augusto Corrêa s/n, 66000-000, Belém PA, Brazil
3.
Departamento de Ecuaciones Diferenciales Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160,41080-Sevilla, Spain

Received: October 2016

Revised: February 2017

Published: October 2017

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Abstract

In this paper, for non-autonomous RDS we study cocycle attractors with autonomous attraction universes, i.e. pullback attracting some autonomous random sets, instead of non-autonomous ones. We first compare cocycle attractors with autonomous and non-autonomous attraction universes, and then for autonomous ones we establish some existence criteria and characterization. We also study for cocycle attractors the continuity of sections indexed by non-autonomous symbols to find that the upper semi-continuity is equivalent to uniform compactness of the attractor, while the lower semi-continuity is equivalent to an equi-attracting property under some conditions. Finally, we apply these theoretical results to 2D Navier-Stokes equation with additive white noise and translation bounded non-autonomous forcing.

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Mathematics Subject Classification: Primary:35B40;Secondary:35B41, 37H05, 37H10.

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