Pullback attractors for the non-autonomous 2D Navier--Stokes equations for minimally regular forcing

Julia García-Luengo; Pedro Marín-Rubio; José Real; James C. Robinson

doi:10.3934/dcds.2014.34.203

Article Contents

2014, Volume 34, Issue 1: 203-227. Doi: 10.3934/dcds.2014.34.203

This issue Previous Article Regularity of pullback attractors and attraction in $H^1$ in arbitrarily large finite intervals for 2D Navier-Stokes equations with infinite delay Next Article Uniform attractor of the non-autonomous discrete Selkov model

Pullback attractors for the non-autonomous 2D Navier--Stokes equations for minimally regular forcing

1.
Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla
2.
Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla
3.
Mathematics Institute, University of Warwick, Coventry CV4 7AL

Received: July 31, 2012

Published: December 2013

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Abstract

This paper treats the existence of pullback attractors for the non-autonomous 2D Navier--Stokes equations in two different spaces, namely $L^2$ and $H^1$. The non-autonomous forcing term is taken in $L^2_{\rm loc}(\mathbb R;H^{-1})$ and $L^2_{\rm loc}(\mathbb R;L^2)$ respectively for these two results: even in the autonomous case it is not straightforward to show the required asymptotic compactness of the flow with this regularity of the forcing term. Here we prove the asymptotic compactness of the corresponding processes by verifying the flattening property -- also known as ``Condition (C)". We also show, using the semigroup method, that a little additional regularity -- $f\in L^p_{\rm loc}(\mathbb R;H^{-1})$ or $f\in L^p_{\rm loc}(\mathbb R;L^2)$ for some $p>2$ -- is enough to ensure the existence of a compact pullback absorbing family (not only asymptotic compactness). Even in the autonomous case the existence of a compact absorbing set for this model is new when $f$ has such limited regularity.

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Mathematics Subject Classification: 35B41, 35B65, 35Q30.

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