Pullback exponential attractors for the non-autonomous micropolar fluid flows

This paper investigates the pullback asymptotic behaviors for the non-autonomous micropolar fluid flows in 2D bounded domains. We use the energy method, combining with some important properties of the generated processes, to prove the existence of pullback exponential attractors and global pullback attractors and show that they both with finite fractal dimension. Further, we give the relationship between global pullback attractors and pullback exponential attractors.     

Existence and estimation of the Hausdorff dimension of attractors for an epidemic model

We prove the existence of pullback and uniform attractors for the process associated to a non‐autonomous SIR model, with several types of non‐autonomous features. The Hausdorff dimension of the pullback attractor is also estimated. We illustrate some examples of pullback attractors by numerical simulations. Copyright © 2016 John Wiley & Sons, Ltd.     

Finite fractal dimension of pullback attractors for non-autonomous 2D Navier–Stokes equations in some unbounded domains

We study the asymptotic behaviour of non-autonomous 2D Navier–Stokes equations in unbounded domains for which a Poincaré inequality holds. In particular, we give sufficient conditions for their pullback attractor to have finite fractal dimension. The existence of pullback attractors in this framework comes from the existence of bounded absorbing sets of pullback asymptotically compact processes [T. Caraballo, G. ?ukaszewicz, J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear Anal. 64 (3) (2006) 484–498]. We show that, under suitable conditions, the method of Lyapunov exponents in [P. Constantin, C. Foias, R. Temam, Attractors representing turbulent flows, Mem. Amer. Math. Soc. 53 (1984) [5]] for the dimension of attractors can be developed in this new context.     

Pullback attractors of 2D Navier–Stokes equations with weak damping,distributed delay,and continuous delay

In this present paper, the existence of pullback attractors for the 2D Navier–Stokes equation with weak damping, distributed delay, and continuous delay has been considered, by virtue of classical Galerkin's method, we derived the existence and uniqueness of global weak and strong solutions. Using the Aubin–Lions lemma and some energy estimate in the Banach space with delay, we obtained the uniform bounded and existence of uniform pullback absorbing ball for the solution semi‐processes; we concluded the pullback attractors via verifying the pullback asymptotical compactness by the generalized Arzelà–Ascoli theorem. Copyright © 2016 John Wiley & Sons, Ltd.     

Attractors for Differential Equations with Variable Delays

Using the relatively new concept of a pullback attractor, we present some results on the existence of attractors for differential equations with variable delay. We give a variety of examples to which our result applies.     

Asymptotic behavior of the non-autonomous 3D Navier-Stokes problem with coercive force

We construct pullback attractors to the weak solutions of the three-dimensional Dirichlet problem for the incompressible Navier-Stokes equations in the case when the external force may become unbounded as time goes to plus or minus infinity.     

Pullback attractors for a non-autonomous incompressible non-Newtonian fluid

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 AV in space V (has H²-regularity, see notation in Section 2) and AH in space H (has L²-regularity) for the cocycle corresponding to the solutions of the fluid. Then we verify the regularity of the pullback attractors by showing AV=AH, which implies the pullback asymptotic smoothing effect of the fluid in the sense that the solutions become eventually more regular than the initial data.     

Pullback attractors for closed cocycles

In this article, using a result of Pata and Zelik (2007) [45], we derive a general result on the existence of pullback attractors for closed cocycles acting on a Banach space, where the strong continuity is replaced by a much weaker requirement that the cocycle be a closed map. As application, we prove the existence of the pullback attractor of a cocycle associated with the z-weak solutions of a non-autonomous two-dimensional primitive equations of the ocean.     

Pullback Dynamics of 2D Non-autonomous Navier-Stokes Equations with Klein-Voight Damping and Multi-delays

This paper is concerned with the pullback dynamics of 2D non-autonomous Navier-Stokes-Voigt equations with continuous and distributed delays on bounded domain. Under some regular assumptions on initial and delay data, the existence of evolutionary process and the family of pullback attractors for this fluid flow model with Klein-Voight damping are derived. The regular assumption of external force is less than [1].     

Pullback attractors for a two-dimensional Navier-Stokes model in an infinite delay case

We prove the existence of solutions for a Navier-Stokes model in two dimensions with an external force containing infinite delay effects in the weighted space C_γ(H). Then, under additional suitable assumptions, we prove the existence and uniqueness of a stationary solution and the exponential decay of the solutions of the evolutionary problem to this stationary solution. Finally, we study the existence of pullback attractors for the dynamical system associated to the problem under more general assumptions.     

The existence of pullback exponential attractors for nonautonomous dynamical system and applications to nonautonomous reaction diffusion equations

First we establish some sufficient conditions for the existence of pullback exponential attractors by using $\omega-$limit compactness in the framework of process. Then we provide a new method to prove the existence of pullback exponential attractors. As a simple application, we prove the existence of pullback exponential attractors for nonautonomous reaction diffusion equations in $H_0^1$.     

Remarks on nontrivial pullback attractors of the 2-D Navier-Stokes equations with delays

This paper is concerned with some further research on the pullback dynamics for 2-D Navier-Stokes equations with delays. By some new definition of generalized Grashof numbers, we presented some sufficient conditions when the pullback attractors of the 2-D nonautonomous incompressible Navier-Stokes equations with differential continuous delays become a single trajectory, which is a preparation for the fractal dimension of pullback attractors for our problem with constant or variable delays.     

Pullback attractors for the non-autonomous FitzHugh–Nagumo system on unbounded domains

The existence of a pullback attractor is established for the singularly perturbed FitzHugh–Nagumo system defined on the entire space Rⁿ${\mathbb{R}}^n$ when external terms are unbounded in a phase space. The pullback asymptotic compactness of the system is proved by using uniform a priori estimates for far-field values of solutions. Although the limiting system has no global attractor, we show that the pullback attractors for the perturbed system with bounded external terms are uniformly bounded, and hence do not blow up as a small parameter approaches zero.     

Pullback attractors in nonautonomous difference equations

Nonautonomous difference equations are formulated as cocycles which generalize semigroups corresponding to autonomous difference equations. Pullback attractors are the appropriate generalization of autonomous attractors to cocycles. The existence of a pullback attractor follows when the difference equation cocycle has a pullback absorbing set. Results from the literature are outlined, including the construction of a Lyapunov function characterizing pullback attraction, and illustrated with several examples.     

Weak Pullback Attractors of Non-autonomous Difference Inclusions

Weak pullback attractors are defined for non-autonomous difference inclusions and their existence and upper semi continuous convergence under perturbation is established. Unlike strong pullback attractors, invariance and pullback attraction here are required only for (at least) a single trajectory rather than all trajectories at each starting point. The concept is thus useful, in particular, for discrete time control systems.     

Weak pullback attractors of setvalued processes

Weak pullback attractors are defined for nonautonomous setvalued processes and their existence and upper semicontinuous convergence under perturbation is established. Unlike strong pullback attractors, invariance and pullback attraction here are required only for at least one trajectory rather than all trajectories at each starting point. The concept is useful in, for example, continuous time control systems and is related to that of viability.     

Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction–diffusion equations

In this paper, we introduce the concept of norm-to-weak continuous process in a Banach space, and obtain the existence of pullback attractors for this kind of process. Then we give a new method for proving the existence of the pullback attractors. As an application, we obtain the existence of pullback attractors for nonautonomous reaction–diffusion equation in with exponential growth of the external force.     

Pullback permanence in a non-autonomous competitive Lotka-Volterra model

The goal of this work is to study in some detail the asymptotic behaviour of a non-autonomous Lotka-Volterra model, both in the conventional sense (as t→∞) and in the “pullback” sense (starting a fixed initial condition further and further back in time). The non-autonomous terms in our model are chosen such that one species will eventually die out, ruling out any conventional type of permanence. In contrast, we introduce the notion of “pullback permanence” and show that this property is enjoyed by our model. This is not just a mathematical artifice, but rather shows that if we come across an ecology that has been evolving for a very long time we still expect that both species are represented (and their numbers are bounded below), even if the final fate of one of them is less happy. The main tools in the paper are the theory of attractors for non-autonomous differential equations, the sub-supersolution method and the spectral theory for linear elliptic equations.     

The fractal dimension of pullback attractors for the 2D Navier–Stokes equations with delay

This paper is concerned with the bounded fractal and Hausdorff dimension of the pullback attractors for 2D nonautonomous incompressible Navier-Stokes equations with constant delay terms. Using the construction of trace formula with two bases for phase spaces of product flow, the upper boundedness of fractal dimension has been achieved.     

Pullback attractors for a model of weakly concentrated aqueous polymer solution motion with a rheological relation satisfying the objectivity principle

The qualitative dynamics of weak solutions to a nonautonomous model of polymer solution motion (with a rheological relation satisfying the objectivity principle) is studied using the theory of pullback attractors of trajectory spaces. For this purpose, the existence of weak solutions is proved for the model under study, a family of trajectory spaces is defined, trajectory and minimal pullback attractors are introduced, and their existence is proved.