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 2 loc (R; H) (see Definition 3.1), which are translation bounded but not translation compact in L 2 loc (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.

In this paper, firstly we introduce the concept of norm-to-weak continuous cocycle in Banach space and give a technical method to verify this kind of continuity, then we obtain some abstract results for the existence of pullback attractors about this kind of cocycle, using the measure of noncompactness. As an application, we prove the existence of pullback attractors in H 1 0 of the cocycle associated with the solutions for some nonlinear nonautonomous reaction-diffusion equations. The attractor pullback attracts all bounded subsets of H 1 0 in the norm of H 1 0 .

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