非线性光学中薛定谔型方程的整体吸引子

研究了一类非线性薛定谔型方程,描述了光波在光折射晶体中的传播.首先构造了该模型整体弱的吸引子,然后通过能量方程的精确分析,证明整体弱吸引子实际为系统整体强吸引子.最后给出了整体吸引子的分形维数和Hausdorff维数的上界估计.     

非线性Sobolev-Galpern方程的有限维整体吸引子

本文研究非线性Sobolev-Galpern方程解的渐近性态．首先证明了该方程在H^2(Ω)∩H0^1(Ω)中整体弱吸引子的存在性，然后利用一个能量方程证明了整体弱吸引子实际上是整体强吸引子，建立了整体吸引子的有限维性．     

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.     

Backward behavior of solutions of the Kuramoto-Sivashinsky equation

We prove that any solution of the Kuramoto-Sivashinsky equation either belongs to the global attractor or it cannot be continued to a solution defined for all negative times. This extends a previous result of the first author who proved that solutions which do not belong to the global attractor have superexponential backward growth. A particular consequence of the result is that the global attractor can be characterized as the maximal invariant set.     

Asymptotic behavior of the Newton–Boussinesq equation in a two-dimensional channel

We prove the existence of a global attractor for the Newton–Boussinesq equation defined in a two-dimensional channel. The asymptotic compactness of the equation is derived by the uniform estimates on the tails of solutions. We also establish the regularity of the global attractor.     

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.     

单模光纤中光孤立子传播的长距离行为(英文)

研究描述单模光纤中光孤立子传播的具光纤损耗项的三阶非线性Ｓｃｈｒｏｄｉｎｇｅｒ方程,首先证明了整体解的存在唯一性结果,然后证明其长距离行为由紧的整体吸引子刻画,并给出了吸引子的Ｈａｕｓｄｏｒｆｆ维数和分形维数的上界估计,最后研究了吸引子的正则性．     

Attractors for the Long-short Wave Equations

In the present paper, we study the long time behaviour of solutions for the long-short wave equations with zero order dissipation. We first construct the global weak attractor for this system in H²_{per} × H¹_{per}. And then by exact analysis of two energy equations, we show that the global weak at attractor is actually the global strong attractor in H²_{per}.     

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.     

One-dimensional attractor for a strongly damped lattice system

In this paper, we consider an attractor of a second order strongly damped lattice system with the non-negative definite symmetric coupled operator. We obtain the existence of a global attractor and prove this attractor is homeomorphic to the circle.     

Global attractor for heat convection problem in a micropolar fluid

The article is devoted to describe asymptotics in the heat convection problem for a micropolar fluid in two dimensions. We show the existence and the uniqueness of global in time solutions and then prove the existence of a global attractor for considered model. Next, the Hausdorff dimension of the global attractor is estimated. Copyright © 2006 John Wiley & Sons, Ltd.     

Finite Dimensional Behavior for Forced Nonlinear Sobolev-Galpern Equations

This paper deals with the asymptotic behavior of solutions for the nonlinear Sobolev-Galpernequations.We first show the existence of the global weak attractor in H~2(Ω)∩H_0~1(Ω) for the equations.Andthen by an energy equation we prove that the global weak attractor is actually the global strong attractor.Thefinite-dimensionality of the global attractor is also established.     

带有热记忆的非均匀柔性结构的长时间动力行为

本文研究了带有热效应的非均匀柔性结构方程，并且该热效应符合Coleman-Gurtin定律.利用半群方法，建立了系统的整体适定性.主要结论是该系统的长时间动力行为.本文证明了系统的拟稳定性，整体吸引子的存在性以及整体吸引子具有有限的分形维数.此外，还证明了指数吸引子的存在性.     

Finite dimensional global and exponential attractors for a class of coupled time-dependent Ginzburg-Landau equations

We study a coupled nonlinear evolution system arising from the Ginzburg-Landau theory for atomic Fermi gases near the BCS (Bardeen-Cooper-Schrieffer)-BEC (Bose-Einstein condensation) crossover.First,we prove that the initial boundary value problem generates a strongly continuous semigroup on a suitable phase-space which possesses a global attractor.Then we establish the existence of an exponential attractor.As a consequence,we show that the global attractor is of finite fractal dimension.     

Attractors for partly dissipative lattice dynamic systems in weighted spaces

In this paper, we study the asymptotic behavior of solutions for the partly dissipative lattice dynamical systems in weighted spaces. We first establish the dynamic systems on infinite lattice, and then prove the existence of the global attractor in weighted spaces by the asymptotic compactness of the solutions. It is shown that the global attractors contain traveling waves. The upper semicontinuity of the global attractor is also considered by finite-dimensional approximations of attractors for the lattice systems.     

The Dynamics of Sine-Gordon System with Dirichlet Boundary Condition

We prove the existence of the global attractor of Sine-Gordon system with Dirichlet boundary condition and show the attractor is the unique steady state when the damping constant and the diffusion constant are sufficiently large.     

Large-time behavior of solutions to degenerate damped hyperbolic equations

We investigate the asymptotic behavior of solutions to damped hyperbolic equations involving strongly degenerate differential operators. First we establish the existence of a global attractor for the damped hyperbolic equation under consideration. Then we prove the finite dimensionality of the global attractor.     

Weak solutions and attractors for motion equations for an objective model of viscoelastic medium

We study the boundary value problem for the equations of motion of incompressible viscoelastic medium with an objective constitutive law of Jeffreys kind. We show existence of global weak solutions for any initial data and construct their minimal uniform trajectory attractor and uniform global attractor. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A subdivision algorithm for the computation of unstable manifolds and global attractors

Summary. Each invariant set of a given dynamical system is part of the global attractor. Therefore the global attractor contains all the potentially interesting dynamics, and, in particular, it contains every (global) unstable manifold. For this reason it is of interest to have an algorithm which allows to approximate the global attractor numerically. In this article we develop such an algorithm using a subdivision technique. We prove convergence of this method in a very general setting, and, moreover, we describe the qualitative convergence behavior in the presence of a hyperbolic structure. The algorithm can successfully be applied to dynamical systems of moderate dimension, and we illustrate this fact by several numerical examples. Received May 11, 1995 / Revised version received December 6, 1995     

Finite and Infinite-Dimensional Attractors of Multivalued Reaction-Diffusion Equations

We study the fractal dimension of the global attractor generated by a multivalued reaction-diffusion equation for which there is no uniqueness of solutions. First we give an example of a global attractor having infinite fractal dimension. Then under certain conditions we obtain an estimate of the fractal dimension. This revised version was published online in June 2006 with corrections to the Cover Date.