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This paper deals with the asymptotic compactness of the solutions of the two-dimensional Navier-Stokes equations defined in unbounded channel-like domains. In order to overcome the non-compactness of Sobolev embeddings in unbounded domains, we establish the uniform tail-ends estimates of solutions by showing all the solutions are uniformly small outside a sufficiently large bounded domain for all large time and bounded initial data, which implies the asymptotic compactness and hence the existence of global attractors in the natural energy space. The uniform tail-ends estimates are derived by the scalar stream function of the Navier-Stokes equations which is distinct from the reaction-diffusion equation due to the divergence free condition. [ABSTRACT FROM AUTHOR]

This paper is devoted to the existence of invariant measures and their limiting behavior of the stochastic Schrödinger lattice systems with respect to noise intensity. We prove the set of all invariant measures of the stochastic systems is weakly compact when the noise intensity varies in a bounded interval. We further show any limit of a sequence of invariant measures of the perturbed systems must be an invariant measure of the limiting system. [ABSTRACT FROM AUTHOR]