无界区域上具线性阻尼的二维 Navier-Stokes 方程的拉回吸引子

该文首先介绍拉回渐近紧非自治动力系统的概念, 给出非自治动力系统拉回吸引子存在定理. 最后证明了无界区域上具线性阻尼的二维Navier-Stokes 方程的拉回吸引子的存在性, 并给出了其Fractal维数估计.     

Pullback attractors for asymptotically compact non-autonomous dynamical systems

First, we introduce the concept of pullback asymptotically compact non-autonomous dynamical system as an extension of the similar concept in the autonomous framework. Our definition is different from that of asymptotic compactness already used in the theory of random and non-autonomous dynamical systems (as developed by Crauel, Flandoli, Kloeden, Schmalfuss, amongst others) which means the existence of a (random or time-dependent) family of compact attracting sets. Next, we prove a result ensuring the existence of a pullback attractor for a non-autonomous dynamical system under the general assumptions of pullback asymptotic compactness and the existence of a pullback absorbing family of sets. This attractor is minimal and, in most practical applications, it is unique. Finally, we illustrate the theory with a 2D Navier–Stokes model in an unbounded domain.     

Pullback attractors for non-autonomous 2D-Navier–Stokes equations in some unbounded domains

In this Note we first introduce the concept of pullback asymptotic compactness. Next, we establish a result ensuring the existence of a pullback attractor for a non-autonomous dynamical system under the general assumptions of pullback asymptotic compactness and the existence of a pullback absorbing family of sets. Finally, we prove the existence of a pullback attractor for a non-autonomous 2D Navier–Stokes model in an unbounded domain, a case in which the theory of uniform attractors does not work since the non-autonomous term is quite general. To cite this article: T. Caraballo et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Asymptotic autonomous attractors for a stochastic lattice model with random viscosity

We study forward asymptotic autonomy of a pullback random attractor for a non-autonomous random lattice system and establish the criteria in terms of convergence, recurrence, forward-pullback absorption and asymptotic smallness of the discrete random dynamical system. By applying the abstract result to both non-autonomous and autonomous stochastic lattice equations with random viscosity, we show the existence of both pullback and global random attractors such that the time-component of the pullback attractor semi-converges to the global attractor as the time-parameter tends to infinity.     

Non-autonomous perturbation of autonomous semilinear differential equations: Continuity of local stable and unstable manifolds

In this paper we prove a result on lower semicontinuity of pullback attractors for dynamical systems given by semilinear differential equations in a Banach space. The situation considered is such that the perturbed dynamical system is non-autonomous whereas the limiting dynamical system is autonomous and has an attractor given as union of unstable manifold of hyperbolic equilibrium points. Starting with a semilinear autonomous equation with a hyperbolic equilibrium solution and introducing a very small non-autonomous perturbation we prove the existence of a hyperbolic global solution for the perturbed equation near this equilibrium. Then we prove that the local unstable and stable manifolds associated to them are given as graphs (roughness of dichotomy plays a fundamental role here). Moreover, we prove the continuity of this local unstable and stable manifolds with respect to the perturbation. With that result we conclude the lower semicontinuity of pullback attractors.     

Existence of backwards-compact pullback attractors for non-autonomous lattice dynamical systems

A pullback attractor is called backward compact if the union of attractors over the past time is pre-compact. We show that this kind of attractor exists for the first-order non-autonomous lattice dynamical system when the external force is backwards tempered and backwards asymptotically tail-null.     

Approximate Forward Attractors of Non-Autonomous Dynamical Systems

In this paper the forward asymptotical behavior of non-autonomous dynamical systems and their attractors are investigated. Under general conditions, the authors show that every neighborhood of pullback attractor has forward attracting property.     

Pullback attractors for a semilinear heat equation in a non-cylindrical domain

The existence and uniqueness of a variational solution satisfying energy equality is proved for a semilinear heat equation in a non-cylindrical domain with homogeneous Dirichlet boundary condition, under the assumption that the spatial domains are bounded and increase with time. In addition, the non-autonomous dynamical system generated by this class of solutions is shown to have a global pullback attractor.     

Asymptotic behavior of non-autonomous Lamé systems with subcritical and critical mixed nonlinearities

This paper deals with the asymptotic behavior of solutions to a class of non-autonomous Lamé systems modeling the physical phenomenon of isotropic elasticity. The main feature of this model is that the nonlinearity can be decomposed into a subcritical part and a critical one. We first show that the system generates a non-autonomous dynamical system, and then prove that the system has a minimal universe pullback attractor. The upper-semicontinuity of these pullback attractors is also established as the perturbation parameter of the external force tends to zero. The quasi-stability ideas developed by Chueshov and Lasiecka (2010, 2008, 2015) are used to prove the pullback asymptotic compactness of the solutions in order to overcome the difficulty caused by the critical growthness of the nonlinearity.     

Equi-attraction and backward compactness of pullback attractors for point-dissipative Ginzburg-Landau equations

A new concept of an equi-attractor is introduced, and defined by the minimal compact set that attracts bounded sets uniformly in the past, for a non-autonomous dynamical system. It is shown that the compact equi-attraction implies the backward compactness of a pullback attractor. Also, an eventually equi-continuous and strongly bounded process has an equi-attractor if and only if it is strongly point dissipative and strongly asymptotically compact. Those results primely strengthen the known existence result of a backward bounded pullback attractor in the literature. Finally, the theoretical criteria are applied to prove the existence of both equi-attractor and backward compact attractor for a Ginzburg-Landau equation with some varying coefficients and a backward tempered external force.     

Attractors for Multi-valued Non-autonomous Dynamical Systems: Relationship,Characterization and Robustness

In this paper we study cocycle attractors, pullback attractors and uniform attractors for multi-valued non-autonomous dynamical systems. We first consider the relationship between the three attractors and find that, under suitable conditions, they imply each other. Then, for generalized dynamical systems, we find that these attractors can be characterized by complete trajectories, which implies that the uniform attractor is lifted invariant, though it has no standard invariance by definition. Finally, we study both upper and lower semi-continuity of these attractors. A weak equi-attraction method is introduced to study the lower semi-continuity, and we show with an example the advantages of this method. A reaction-diffusion system and a scalar ordinary differential inclusion are studied as applications.     

Pullback attractors for a class of extremal solutions of the 3D Navier-Stokes system

In this paper we construct a dynamical process (in general, multivalued) generated by the set of solutions of an optimal control problem for the three-dimensional Navier-Stokes system. We prove the existence of a pullback attractor for such multivalued process. Also, we establish the existence of a uniform global attractor containing the pullback attractor. Moreover, under the unproved assumption that strong globally defined solutions of the three-dimensional Navier-Stokes system exist, which guaranties the existence of a global attractor for the corresponding multivalued semiflow, we show that the pullback attractor of the process coincides with the global attractor of the semiflow.     

Asymptotically finite dimensional pullback behaviour of non-autonomous PDEs

The asymptotic behaviour of general non-autonomous partial differential equations can be described using the concept of pullback attractor. This is, under suitable hypotheses, a time-dependent family of finite-dimensional compact sets. In this work we investigate how this finite-dimensional dynamics on the attractor determines the asymptotic behaviour of non-autonomous PDEs.     

非自治Plate方程时间依赖强拉回吸引子的存在性

结合Plinio等人([Plinio D,Duane G S,Temarn R,Time-dependent attractor for the oscillon equation,Discrete Contin Dyn Syst,2011,29(1):141-167.])提出的时间依赖全局吸引子概念,运用压缩函数的方法,证明了带有时间依赖系数的非自治Plate方程时间依赖拉回吸引子在空间H~4(Ω)∩H~2_0(Ω)×H~2_0(Ω)中的存在性.     

Structure of attractors for skew product semiflows

In this work we study the continuity and structural stability of the uniform attractor associated with non-autonomous perturbations of differential equations. By a careful study of the different definitions of attractor in the non-autonomous framework, we introduce the notion of lifted-invariance on the uniform attractor, which becomes compatible with the dynamics in the global attractor of the associated skew product semiflow, and allows us to describe the internal dynamics and the characterization of the uniform attractors. The associated pullback attractors and their structural stability under perturbations will play a crucial role.     

The stability of attractors for non-autonomous perturbations of gradient-like systems

We study the stability of attractors under non-autonomous perturbations that are uniformly small in time. While in general the pullback attractors for the non-autonomous problems converge towards the autonomous attractor only in the Hausdorff semi-distance (upper semicontinuity), the assumption that the autonomous attractor has a ‘gradient-like’ structure (the union of the unstable manifolds of a finite number of hyperbolic equilibria) implies convergence (i.e. also lower semicontinuity) provided that the local unstable manifolds perturb continuously.We go further when the underlying autonomous system is itself gradient-like, and show that all trajectories converge to one of the hyperbolic trajectories as t→∞. In finite-dimensional systems, in which we can reverse time and apply similar arguments to deduce that all bounded orbits converge to a hyperbolic trajectory as t→−∞, this implies that the ‘gradient-like’ structure of the attractor is also preserved under small non-autonomous perturbations: the pullback attractor is given as the union of the unstable manifolds of a finite number of hyperbolic trajectories.     

Pullback attractors for the non-autonomous Benjamin-Bona-Mahony equations in H

In this paper, we prove the existence of the pullback attractor for the non-autonomous Benjamin-Bona-Mahony equations in H² by establishing the pullback uniformly asymptotical compactness.     

Upper semicontinuous dependence of pullback attractors on time scales

Dynamical equations on time scales typically generate a nonautonomous process, even when the vector field function does not depend explicitly on time. Nonautonomous pullback attractors are thus the appropriate generalisation of autonomous attractors to time scale dynamics. The existence of a pullback attractor follows when the process has a pullback absorbing set. Assuming that a dynamical equation over a given time scale which has no rapidly increasing gaps satisfies a certain dissipativity condition, and thus possesses a pullback attractor, and that its solutions depend uniformly on initial data including the time scale, it is shown that the same dynamical equation over nearby time scales also has a pullback attractor, whose component sets converge upper semicontinuously to the corresponding component sets of the pullback attractor of the original system.     

Single Parameter Dissipativity and Attractors in DiscreteTirne Asynchronous Systems ∗

Discrete time nonautonomous dynamical systems generated by nonautonomous difference equations are formulated as discrete time skew—product systems consisting of cocycle state mappings that are driven by discrete time autonomous dynamical systems. Forwards and pullback attractors are two possible generalizations of autonomous attractors to such systems. Their existence follows from appropriate forwards or pullback dissipativity conditions. For discrete time nonautonomous dynamical systems generated by asynchronous systems with frequency updating components such a dissipativity condition is usually known for a single starting parameter value of the driving system. Additional conditions that then ensure the existence of a forwards or pullback attractor for such an asynchronous system are investigated here