Local equi-attraction of pullback attractor sections

In this paper, we study a local equi-attraction of pullback attractors for non-autonomous processes. By the local equi-attraction we mean that any local part of sections of a pullback attractor A={A(τ)}τ∈R are pullback attracting at the same rate, i.e., for any bounded (but arbitrarily large) interval I,limt→∞⁡(supτ∈I⁡distX(U(t,τ−t,B),A(τ)))=0,∀B⊂Xbounded, where X is a metric space and U:R+×R×X→X is a process in X. Our technique makes use of the uniform attractor theory to consider in a dynamic way the time parameter τ involved in the pullback attractor. The analysis shows that, roughly, when a system has a uniform attractor, then the pullback attractor can be locally equi-attracting. As an example, the pullback attractor of 2D Navier-Stokes equation is studied, where the joint continuity in initial time and initial data of the solutions plays a key role. In addition, the continuity of the set-valued mapping τ↦A(τ) in more regular spaces is also studied.