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This paper studies the pullback asymptotic behavior of solutions for a non-autonomous incompressible non-Newtonian fluid in two-dimensional (2D) bounded domains. We first prove the existence of pullback attractors A V in space V (has H 2 -regularity, see notation in Section 2) and A H in space H (has L 2 -regularity) for the cocycle corresponding to the solutions of the fluid. Then we verify the regularity of the pullback attractors by showing A V = A H , which implies the pullback asymptotic smoothing effect of the fluid in the sense that the solutions become eventually more regular than the initial data.

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Keywords:Random attractor Stochastic lattice dynamical systems Sine-Gordon equation Random asymptotic nullness Kolmogorov ε-entropyIn this paper, we first present some sufficient conditions for the existence of a global random attractor for general stochastic lattice dynamical systems. These sufficient conditions provide a convenient approach to obtain an upper bound of Kolmogorov ε-entropy for the global random attractor. Then we apply the abstract result to the stochastic lattice sineGordon equation.

In this paper, we study the nonautonomous Klein-Gordon-Schrödinger equations on infinite lattices. We first prove the existence of compact kernel sections for the corresponding process and then obtain an upper bound of the Kolmogorov ε-entropy for these kernel sections. Finally, we establish the upper semicontinuity of the kernel sections.

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We first prove the existence and regularity of the trajectory attractor for a threedimensional system of globally modified Navier-Stokes equations. Then we use the natural translation semigroup and trajectory attractor to construct the trajectory statistical solutions in the trajectory space. In our construction the trajectory statistical solution is an invariant Borel probability measure, which is supported by the trajectory attractor and is invariant under the action of the translation semigroup. As a byproduct of the regularity of the trajectory attractor, we obtain the asymptotic regularity of the trajectory statistical solution in the sense that it is supported by a set in the trajectory space in which all weak solutions are in fact strong solutions.

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