Pullback and forward attractors for a 3D LANS$-\alpha$ model with delay

Tomás Caraballo; Antonio M. Márquez-Durán; José Real

doi:10.3934/dcds.2006.15.559

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2006, Volume 15, Issue 2: 559-578. Doi: 10.3934/dcds.2006.15.559

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Pullback and forward attractors for a 3D LANS$-\alpha$ model with delay

1.
Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla
2.
Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla

Received: January 31, 2005

Revised: August 31, 2005

Published: March 2006

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Abstract

We analyze the asymptotic behaviour of a 3D Lagrangian averaged Navier-Stokes $\alpha$-model (3D LANS$-\alpha$) with delays. In fact, we apply the theory of pullback attractors to ensure the existence of a pullback attractor, and at the same time, we also prove the existence of a uniform (forward) attractor in the sense of Chepyzhov and Vishik. Instead of working directly with the 3D LANS$-\alpha$ model, we establish a general theory for an abstract delay model and then we apply the general results to our particular situation.

Keywords:

3D Lagrangian averaged Navier-Stokes equations,
pullback attractor,
forward attractor,
variable delay,
distributed delay.

Mathematics Subject Classification: Primary: 35R10, 35B40, 47H20, 58F39, 73K70.

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