@article{09c7ae142b67431183776a86cd38ea45,

title = "Continuity of pullback and uniform attractors",

abstract = "We study the continuity of pullback and uniform attractors for non-autonomous dynamical systems with respect to perturbations of a parameter. Consider a family of dynamical systems parameterized by λ∈Λ, where Λ is a complete metric space, such that for each λ∈Λ there exists a unique pullback attractor Aλ(t). Using the theory of Baire category we show under natural conditions that there exists a residual set Λ⁎⊆Λ such that for every t∈R the function λ↦Aλ(t) is continuous at each λ∈Λ⁎ with respect to the Hausdorff metric. Similarly, given a family of uniform attractors Aλ, there is a residual set at which the map λ↦Aλ is continuous. We also introduce notions of equi-attraction suitable for pullback and uniform attractors and then show when Λ is compact that the continuity of pullback attractors and uniform attractors with respect to λ is equivalent to pullback equi-attraction and, respectively, uniform equi-attraction. These abstract results are then illustrated in the context of the Lorenz equations and the two-dimensional Navier–Stokes equations.",

keywords = "Pullback attractor, Uniform attractor",

author = "Hoang, {Luan T.} and Olson, {Eric J.} and Robinson, {James C.}",

note = "Funding Information: LTH acknowledges the support by NSF grant DMS-1412796. EJO was supported in part by NSF grant DMS-1418928. JCR was supported by an EPSRC Leadership Fellowship EP/G007470/1, as was the visit of EJO to Warwick while on sabbatical leave from UNR. Funding Information: LTH acknowledges the support by NSF grant DMS-1412796 . EJO was supported in part by NSF grant DMS-1418928 . JCR was supported by an EPSRC Leadership Fellowship EP/G007470/1 , as was the visit of EJO to Warwick while on sabbatical leave from UNR. Appendix A This appendix reproduces the formal a priori estimates for the two-dimensional incompressible Navier–Stokes equations stated as Theorem 6.4 in the main body of the paper. These are essentially the estimates that appear in [27] pages 109–111 for the time-independent case that have been adapted for time-dependent body forces f ∈ L ∞ ( R , H ) . Let u 0 ∈ H , t ≥ s , and set u ( t ) = S f ( t , s ) u 0 . Taking the inner product of (6.17) with u and using the orthogonality property (A.1) 〈 B ( u , v ) , v 〉 = 0 we have (A.2) d d t ‖ u ‖ 2 + ν ‖ ∇ u ‖ 2 ≤ ‖ f ‖ 2 ν λ 1 . Integrating (A.2) in time from s to t yields (A.3) ν ∫ s t ‖ ∇ u ( τ ) ‖ 2 d τ ≤ ‖ u 0 ‖ 2 + ( t − s ) ‖ f ‖ L ∞ ( R , H ) 2 ν λ 1 Noting that ‖ f ‖ L ∞ ( R , H ) = ν 2 λ 1 G obtains (6.19) in Theorem 6.4 . By Poincar{\'e}'s inequality we obtain d d t ‖ u ‖ 2 + ν λ 1 ‖ u ‖ 2 ≤ ‖ f ‖ 2 ν λ 1 . Using Gronwall's inequality yields (A.4) ‖ u ( t ) ‖ 2 ≤ ‖ u 0 ‖ 2 e − ν λ 1 ( t − s ) + ρ 0 2 ( 1 − e − ν λ 1 ( t − s ) ) where ρ 0 = ν G . Therefore, for each bounded subset B of H , there exists a time t 0 ( B ) such that (A.5) ‖ u ( t ) ‖ 2 ≤ 2 ρ 0 2 for all t ≥ t 0 ( B ) . We have, consequently, obtained the first part of (6.20) . Finally, we recall here the needed estimates for ‖ ∇ u ( t ) ‖ . (Again, calculations can be found, for example, in [27] pages 109–111.) Let ν > 0 . Recall from (A.4) that ρ 0 = ν G , and define ρ 0 ′ = ρ 0 + 1 , m 1 = λ 1 ρ 0 2 + ρ ′ 0 2 ν , m 2 = 2 ν λ 1 2 ρ 0 2 , m 3 = 2 c 0 ν 3 ρ ′ 0 2 m 1 , where c 0 is the same as in (6.33) . For R > ρ 0 ′ , denote t 1 ( R ) = 1 + 1 ν λ 1 log R 2 2 ρ 0 + 1 . Then, for ‖ u 0 ‖ ≤ R and all t ≥ t 1 ( R ) , (A.6) ‖ ∇ u ( t + s ) ‖ 2 ≤ ρ ( G ) where ρ ( G ) = ( m 1 + m 2 ) e m 3 . Noting that ρ ( G ) is an increasing function of G such that also depends on ν and λ 1 yields the final part of (6.20) in Theorem 6.4 . Publisher Copyright: {\textcopyright} 2017 Elsevier Inc.",

year = "2018",

month = mar,

day = "15",

doi = "10.1016/j.jde.2017.12.002",

language = "English",

volume = "264",

pages = "4067--4093",

journal = "Journal of Differential Equations",

issn = "0022-0396",

number = "6",

}