We extend previous results obtained by Rosa (1998 Nonlinear Anal. 32 71-85) on the existence of the global attractor for the two-dimensional Navier-Stokes equations on some unbounded domains. We show that if the forcing term is in the natural space H, then the global attractor is compact not only in the L2 norm but also in the H1 norm, and it attracts all bounded sets in H in the metric of V. The proof is based on the concept of asymptotic compactness and the use of the enstrophy equation. As compared with the work of Rosa, which proved the compactness and the attraction in the L2 norm, the new difficulty comes from the fact that the nonlinear term of the Navier-Stokes equations does not disappear from the enstrophy equation, while it does disappear in the energy equation due to its antisymmetry property.
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