Dynamics of systems on infinite lattices

The dynamics of infinite-dimensional lattice systems is studied. A necessary and sufficient condition for asymptotic compactness of lattice dynamical systems is introduced. It is shown that a lattice system has a global attractor if and only if it has a bounded absorbing set and is asymptotically null. As an application, it is proved that the lattice reaction-diffusion equation has a global attractor in a weighted l² space, which is compact as well as contains traveling waves. The upper semicontinuity of global attractors is also obtained when the lattice reaction-diffusion equation is approached by finite-dimensional systems.     

A random attractor for a stochastic second order lattice system with random coupled coefficients

In this paper, we mainly focus on the asymptotic behavior of solutions to the second-order stochastic lattice equations with random coupled coefficients and multiplicative white noises in weighted spaces of infinite sequences. We first transfer stochastic lattice equations into random lattice equations and prove the existence and uniqueness of solutions which generate a random dynamical system. Second we consider the existence of a tempered random bounded absorbing set and a random attractor for the system. Then we establish the upper semicontinuity of random attractors as the coefficient of the white noise term tends to zero. Finally we present the corresponding results for the system with additive white noises.     

KGS格点系统的全局吸引子

考虑了对应于Klein-Gordon-Schrdinger方程的格点系统(KGS格点系统)的解的长时间行为．首先通过引入一个加权范数与采用解的“切尾”法,证明了全局吸引子的存在性．在此基础上,采用元素分解法与多面体的球覆盖性质, 得到了此吸引子的Kolmogorov δ-熵的上界的一个估计．最后,我们用有限维的常微分方程的全局吸引子逼近它．     

Attractors for Weak Solutions of a Regularized Model of Viscoelastic Mediums Motion With Memory in Non-Autonomous Case

We study existence of attractors for weak solutions of the regularized model for viscoelastic medium motion with memory in non-autonomous case. We apply the theory of trajectory attractors for non-invariant trajectory spaces and prove the existence of trajectory attractor, global attractor, uniform trajectory attractor, and uniform global attractor for this system.     

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.     

Random attractor of stochastic complex Ginzburg–Landau equation with multiplicative noise on unbounded domain

The existence of a compact random attractor for the stochastic complex Ginzburg–Landau equation with multiplicative noise has been investigated on unbounded domain. The solutions are considered in suitable spaces with weights. According to crucial properties of Ornstein–Uhlenbeck process, using the tail-estimates method, the key uniform a priori estimates for the tail of solutions have been obtained, which give the asymptotic compactness of random attractors. Then the existence of a compact random attractor for the corresponding dynamical system is proved in suitable spaces with weights.     

On global attractors of the 3D Navier-Stokes equations

In view of the possibility that the 3D Navier-Stokes equations (NSE) might not always have regular solutions, we introduce an abstract framework for studying the asymptotic behavior of multi-valued dissipative evolutionary systems with respect to two topologies—weak and strong. Each such system possesses a global attractor in the weak topology, but not necessarily in the strong. In case the latter exists and is weakly closed, it coincides with the weak global attractor. We give a sufficient condition for the existence of the strong global attractor, which is verified for the 3D NSE when all solutions on the weak global attractor are strongly continuous. We also introduce and study a two-parameter family of models for the Navier-Stokes equations, with similar properties and open problems. These models always possess weak global attractors, but on some of them every solution blows up (in a norm stronger than the standard energy one) in finite time.     

Invariant manifolds, global attractors, almost automorphic and almost periodic solutions of non-autonomous differential equations

The paper is devoted to the study of non-autonomous evolution equations: invariant manifolds, compact global attractors, almost periodic and almost automorphic solutions. We study this problem in the framework of general non-autonomous (cocycle) dynamical systems. First, we prove that under some conditions such systems admit an invariant continuous section (an invariant manifold). Then, we obtain the conditions for the existence of a compact global attractor and characterize its structure. Third, we derive a criterion for the existence of almost periodic and almost automorphic solutions of different classes of non-autonomous differential equations (both ODEs (in finite and infinite spaces) and PDEs).     

Morse decompositions of nonautonomous dynamical systems

The global asymptotic behavior of dynamical systems on compact metric spaces can be described via Morse decompositions. Their components, the so-called Morse sets, are obtained as intersections of attractors and repellers of the system. In this paper, new notions of attractor and repeller for nonautonomous dynamical systems are introduced which are designed to establish nonautonomous generalizations of the Morse decomposition. The dynamical properties of these decompositions are discussed, and nonautonomous Lyapunov functions which are constant on the Morse sets are constructed explicitly. Moreover, Morse decompositions of one-dimensional and linear systems are studied.

     

Convergence of discretized attractors for parabolic equations on the line

We show that, for a semilinear parabolic equation on the real line satisfying a dissipativity condition, global attractors of time-space discretizations converge (with respect to the Hausdorff semi-distance) to the attractor of the continuous system as the discretization steps tend to zero. The attractors considered correspond to pairs of function spaces (in the sense of Babin-Vishik) with weighted and locally uniform norms (taken from Mielke-Schneider) used both for the continuous and discrete systems. Bibliography: 13 titles. Dedicated to the memory of Olga Aleksandrovna Ladyzhenskaya Published in Zapiski Nauchnykh Seminarov POMI, Vol. 318, 2004, pp. 14–41.     

Random attractor for stochastic second-order non-autonomous stochastic lattice equations with dispersive term

In this paper, we consider the asymptotic behaviour of solutions to second-order non-autonomous stochastic lattice equations with dispersive term and additive white noises in the space of infinite sequences. We first transfer the stochastic lattice equations into random lattice equations, and prove the existence and uniqueness of solutions that generate a random dynamical system. Second, we prove the existence of a tempered random absorbing set and a random attractor for the system. Finally, we establish the upper semi-continuity of the random attractors as the coefficient of the white noise term tends to zero.     

Dynamical behavior for stochastic lattice systems

Random attractors describe the long term behavior of the random dynamical systems. This paper is devoted to a general first order stochastic lattice dynamical systems (SLDS) with some dissipative nonlinearity. We prove the asymptotic compactness of the random dynamical system and obtain the random attractor, which is a compact random invariant set with tempered bound.     

Upper Semicontinuity and Kolmogorov ε-Entropy of Global Attractor for κ-Dimensional Lattice Dynamical System Corresponding to Klein-Gordon-SchrSdinger Equation

In this paper, we establish the existence of a global attractor for a coupled κ-dimensional lattice dynamical system governed by a discrete version of the Klein-Gordon-SchrSdinger Equation. An estimate of the upper bound of the Kohnogorov ε-entropy of the global attractor is made by a method of element decomposition and the covering property of a polyhedron by balls of radii ε in a finite dimensional space. Finally, a scheme to approximate the global attractor by the global attractors of finite-dimensional ordinary differential systems is presented .     

Attractors and dimension of dissipative lattice systems

In this paper, by using the argument in [Q.F. Ma, S.H. Wang, C.K. Zhong, Necessary and sufficient conditions for the existence of global attractor for semigroup and application, Indiana Univ. Math. J., 51(6) (2002), 1541-1559.], we prove that the condition given in [S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations 200 (2004) 342-368.] for the existence of a global attractor for the semigroup associated with general lattice systems on a discrete Hilbert space is a sufficient and necessary condition. As an application, we consider the existence of a global attractor for a second-order lattice system in a discrete weighted space containing all bounded sequences. Finally, we show that the global attractor for first-order and partly dissipative lattice systems corresponding to (partly dissipative) reaction-diffusion equations and second-order dissipative lattice systems corresponding to the strongly damped wave equations have finite fractal dimension if the derivative of the nonlinear term is small at the origin.     

Approximating topological approach to the existence of attractors in fluid mechanics

The aim of this paper is to demonstrate how the approximating topological method can be effectively combined with the theory of attractors of trajectory spaces in problems of fluid mechanics. First we give an exposition of the theory. Then we consider the model of motion of weak aqueous polymer solutions and prove that it has the minimal trajectory attractor and the global one. Finally we prove that the attractors of approximating problem converge to the attractors of the unperturbed one.     

Attractors for partly dissipative lattice dynamic systems in l2×l2

In this paper, we study the asymptotic behavior of solutions for the partly dissipative lattice dynamical systems in $l^2\times l^2$. We prove the asymptotic compactness of the solutions and then establish the existence of the global attractor in $l^2\times l^2$. The upper semicontinuity of the global attractor is also considered by finite-dimensional approximations of attractors for the lattice systems.     

具有白噪声的随机格点系统的随机吸引子的Kolmogorov熵

基于耗散的随机格点系统解的渐近行为理论,主要运用元素分解法与有限维空间中多面体球覆盖的拓扑性质,研究了具有白噪声的随机Klein-Gordon-Schrdinger格点动力系统的随机吸引子的Kolmogorov熵,并得到它的一个上界.     

Attractors and approximations for lattice dynamical systems

We present a sufficient condition for the existence of a global attractor for general lattice dynamical systems, then consider the existence of attractors and their approximation for second-order and first-order lattice systems which, in particular case, can be regarded as the spatial discretizations of corresponding wave equations and reaction-diffusion equations in R^k.     

Shape and Morse theory of attractors

We study the shape (in the sense of Borsuk) of attractors of continuous semi‐dynamical systems on general metric spaces. We show, in particular, that the natural inclusion of the global attractor into the state space is a shape equivalence. This and other results of the paper are used to develop an elementary Morse theory of an attractor. © 2000 John Wiley & Sons, Inc.     

Dynamics of Thermally-Insulated Nonequilibrium Stefan Problem

We study a two-phase Stefan problem with kinetics. Here we prove existence of a finite-dimensional attractor for the problem without heat losses. Fot the most part we use a more elegant technique of energetic type estimates in appropriately defined weighted Sobolev spaces as opposite to the parabolic potentials of [9]. We demonstrate existence of compact attractors in the Sobolev spaces and prove that the attractor consists of sufficiently regular functions. This allows us to show that the Hausdorff dimension of the attractor is finite.