The existence and the structure of uniform global attractors for nonautonomous Reaction-Diffusion systems without uniqueness
Alexey Cheskidov Songsong Lu
Discrete & Continuous Dynamical Systems - S 2009, 2(1): 55-66 doi: 10.3934/dcdss.2009.2.55
We study the uniform global attractor for a general nonautonomous reaction-diffusion system without uniqueness using a new developed framework of an evolutionary system. We prove the existence and the structure of a weak uniform (with respect to a symbol space) global attractor $\mathcal A$. Moreover, if the external force is normal, we show that this attractor is in fact a strong uniform global attractor. The existence of a uniform (with respect to the initial time) global attractor $\mathcal A^0$ also holds in this case, but its relation to $\mathcal A$ is not yet clear due to the non-uniqueness feature of the system.
keywords: normal symbol. reaction-diffusion system evolutionary system uniform global attractor
Attractors for nonautonomous 2d Navier-Stokes equations with normal external forces
Songsong Lu Hongqing Wu Chengkui Zhong
Discrete & Continuous Dynamical Systems - A 2005, 13(3): 701-719 doi: 10.3934/dcds.2005.13.701
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.
keywords: nonautonomous equation uniform Condition (C) uniformly $\omega$-limit compact process normal function. Uniform attractor Navier-Stokes equations

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