Attractors for nonautonomous 2d Navier-Stokes equations with normal external forces

Songsong Lu; Hongqing Wu; Chengkui Zhong

doi:10.3934/dcds.2005.13.701

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2005, Volume 13, Issue 3: 701-719. Doi: 10.3934/dcds.2005.13.701

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Attractors for nonautonomous 2d Navier-Stokes equations with normal external forces

1.
Department of Mathematics, Lanzhou University, Lanzhou, Gansu, 730000

Received: September 2004

Revised: February 2005

Published online: May 2005

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Abstract

The existence and structure of uniform attractors in $V$ is proved for nonautonomous 2D Navier-stokes equations on bounded domain with a new class of external forces, termed normal in $L_{l o c}^2(\mathbb R; H)$ (see Definition 3.1), which are translation bounded but not translation compact in $L_{l o c}^2(\mathbb R; H)$. To this end, some abstract results are established. First, a characterization on the existence of uniform attractor for a family of processes is presented by the concept of measure of noncompactness as well as a method to verify it. Then, the structure of the uniform attractor is obtained by constructing skew product flow on the extended phase space with weak topology. Finally, the uniform attractor of a process is identified with that of a family of processes with symbols in the closure of the translation family of the original symbol in a Banach space with weak topology.

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Mathematics Subject Classification: 35B40, 35B41, 35Q30, 76D05.

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