This paper is concerned with the robustness of a pullback attractor as the time tends to infinity. A pullback attractor is called forward (resp. backward) compact if the union over the future (resp. the past) is pre-compact. We prove that the forward (resp. backward) compactness is a necessary and sufficient condition such that a pullback attractor is upper semi-continuous to a compact set at positive (resp. negative) infinity, and also obtain the minimal limit-set. We further prove the lower semi-continuity of the pullback attractor and get the maximal limit-set at infinity. Some criteria for such robustness are established when the evolution process is forward or backward omegalimit compact. Those theoretical criteria are applied to prove semi-uniform compactness and robustness at infinity in pullback dynamics for a Ginzburg-Landau equation with variable coefficients and a forward or backward tempered nonlinearity.
This paper analyzes the time-dependence and backward controllability of pullback attractors for the trajectory space generated by a nonautonomous evolution equation without uniqueness. A pullback trajectory attractor is called backward controllable if the norm of its union over the past is controlled by a continuous function, and backward compact if it is backward controllable and pre-compact in the past on the underlying space. We then establish two existence theorems for such a backward compact trajectory attractor, which leads to the existence of a pullback attractor with the backward compactness and backward boundedness in two original phase spaces respectively. An essential criterion is the existence of an increasing, compact and absorbing brochette. Applying to the non-autonomous Jeffreys-Oldroyd equations with a backward controllable force, we obtain a backward compact trajectory attractor, and also a pullback attractor with backward compactness in the negative-exponent Sobolev space and backward boundedness in the Lebesgue space.2000 Mathematics Subject Classification. Primary: 35B40, 35B41; Secondary: 37B55.
This paper concerns the long term behavior of the stochastic two-dimensional g-Navier-Stokes equations with additive noise defined on a sequence of expanding domains, where the ultimate domain is unbounded and of Poincaré type. We prove that the weak continuity is uniform with respect to all expanding cocycles, which yields the equi-asymptotic compactness by using an energy equation method. Finally, we show the existence of a random attractor for the equation on each domain and the upper semi-continuity of random attractors as the bounded domain is expanded to the unbounded ultimate domain.
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