Pullback attractors for 2D Navier–Stokes equations with delays and the flattening property

Julia García-Luengo; Pedro Marín-Rubio

doi:10.3934/cpaa.2020094

Article Contents

2020, Volume 19, Issue 4: 2127-2146. Doi: 10.3934/cpaa.2020094

This issue Previous Article Theoretical and numerical results for some bi-objective optimal control problems Next Article PDE problems with concentrating terms near the boundary

Pullback attractors for 2D Navier–Stokes equations with delays and the flattening property

Julia García-Luengo^1, and
Pedro Marín-Rubio^2,

1.
Departamento de Matemática Aplicada a las TIC, Universidad Politécnica de Madrid, C/ Nikola Tesla s/n, 28031, Madrid, Spain
2.
Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, C/ Tarfia s/n, 41012, Sevilla, Spain

Dedicated to Professor Tomás Caraballo on occasion of his Sixtieth Birthday

Received: April 30, 2019

Revised: August 31, 2019

Published: January 2020

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Abstract

This paper treats the existence of pullback attractors for a 2D Navier–Stokes model with finite delay formulated in [Caraballo and Real, J. Differential Equations 205 (2004), 271–297]. Actually, we carry out our study under less restrictive assumptions than in the previous reference. More precisely, we remove a condition on square integrable control of the memory terms, which allows us to consider a bigger class of delay terms. Here we show that the asymptotic compactness of the corresponding processes required to establish the existence of pullback attractors, obtained in [García-Luengo, Marín-Rubio and Real, Adv. Nonlinear Stud. 13 (2013), 331–357] by using an energy method, can be also proved by verifying the flattening property – also known as "Condition (C)". We deal with dynamical systems in suitable phase spaces within two metrics, the $ L^2 $ norm and the $ H^1 $ norm. Moreover, we provide results on the existence of pullback attractors for two possible choices of the attracted universes, namely, the standard one of fixed bounded sets, and secondly, one given by a tempered condition.

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Mathematics Subject Classification: Primary: 35B41, 35Q30, 37L30.

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References

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