We present the existence of kernel sections (which are all compact, invariant and pullback attracting) of an infinite-dimensional general multi-valued process constructed by the set-valued backward extension of multi-valued semiprocesses. Moreover, the structure of the uniform attractors of a family of multi-valued semiprocesses and the uniform forward attraction of kernel sections of a family of general multi-valued processes are investigated. Finally, we explain our abstract results by considering the mixed wave systems with supercritical exponent and ordinary differential equations.

In this paper, we first establish a set of sufficient and necessary conditions for the existence of globally attractive kernel sections for processes defined on a general Banach space and a weighted space ℓ p ρ of infinite sequences (p ≥ 1), respectively. Then we obtain an upper bound of the Kolmogorov εentropy of kernel sections for processes on the Hilbert space ℓ 2 ρ . As applications, we investigate compact kernel sections for first order, partly dissipative, and second order nonautonomous lattice systems on weighted spaces containing bounded sequences.

There are two results within this paper. The one is the regularity of trajectory attractor and the trajectory asymptotic smoothing effect of the incompressible non-Newtonian fluid on 2D bounded domains, for which the solution to each initial value could be nonunique. The other is the upper semicontinuity of global attractors of the addressed fluid when the spatial domains vary from Ω m to Ω = R × (−L, L), where {Ω m } ∞ m=1 is an expanding sequence of simply connected, bounded and smooth subdomains of Ω such that Ω m → Ω as m → +∞. That is, let A and A m be the global attractors of the fluid corresponding to Ω and Ω m , respectively, we establish that for any neighborhood O(A) of A, the global attractor A m enters O(A) if m is large enough.

The authors construct the trajectory attractor and global attractor for an autonomous two-dimensional non-Newtonian fluid.