The Global Attractors for the Dissipative Generalized Hasegawa-Mima Equation

The long time behavior of solutions of the generalized Hasegawa-Mima equation with dissipation term is considered. The existence of global attractors of the periodic initial value problem is proved, and the estimate of the upper bound of the Hausdorff and fractal dimensions for the global attractors is obtained by means of uniform a priori estimates method.     

具有惯性项和阻尼项的Cahn-Hilliard方程的整体吸引子

本文研究具有惯性项和阻尼项的亚三次非线性Cahn-Hilliard方程的初边值问题.在非线性弱正则的条件下,我们建立弱解的适定性,而不考虑非线性项的一阶导数的下界条件.接着利用弱解的渐近紧和能量解的严格Lyapunov函数的存在性,证明在空间(H~2(?)∩H_0~1(?))×L~2(?)上存在整体吸引子.     

Attractors for a Three-Dimensional Thermo-Mechanical Model of Shape Memory Alloys

Abstract In this note, we consider a Frémond model of shape memory alloys. Let us imagine a piece of a shape memory alloy which is fixed on one part of its boundary, and assume that forcing terms, e.g., heat sources and external stress on the remaining part of its boundary, converge to some time-independent functions, in appropriate senses, as time goes to infinity. Under the above assumption, we shall discuss the asymptotic stability for the dynamical system from the viewpoint of the global attractor. More precisely, we generalize the paper [12] dealing with the one-dimensional case. First, we show the existence of the global attractor for the limiting autonomous dynamical system; then we characterize the asymptotic stability for the non-autonomous case by the limiting global attractor. * Project supported by the MIUR-COFIN 2004 research program on “Mathematical Modelling and Analysis of Free Boundary Problems”.     

Regularity of Solutions of the Cahn-Hilliard Equation with Non-constant Mobility

In this paper, we study the regularity of solutions for two-dimensional Cahn-Hilliard equation with non-constant mobility. Basing on the L^p type estimates and Schauder type estimates, we prove the global existence of classical solutions.     

The Inflated Attractors of Non-autonomous Strongly Damped Wave Equations

We investigate the existence and the continuity of the inflated attractors for a class of nonautonomous strongly damped wave equations through differential inclusion.     

Kolmogorov＇s ε-Entropy of Bounded Sets in Discrete Spaces and Attractors of Dissipative Lattice Systems

We obtain an estimate of the upper bound for Kolmogorov＇s ε-entropy for the bounded sets with small ＂tail＂ in discrete spaces, then we present a sufficient condition for the existence of a global attractor for dissipative lattice systems in a reflexive Banach discrete space and establish an upper bound of Kolmogorov＇s ε-entropy of the global attractor for lattice systems.     

Remarks on Maximal Attractors for the Compressible Navier-Stokes Equations of Viscous and Heat Conductive Fluid

This paper is concerned with the existence of maximal attractors in Hi (i = 1, 2,4) for the compressible Navier-Stokes equations for a polytropic viscous heat conductive ideal gas in bounded annular domains Ωn in Rn(n = 2,3).     

Attractors for discrete nonlinear Schrödinger equation with delay

In this paper,we first prove the existence of the global attractor Aν ∈ C（[-ν,0],2）（ν 0） for a weak damping discrete nonlinear Schrdinger equation with delay.Then we consider an upper semi-continuity of Aν as ν → 0＋.     

Asymptotic Smoothing Effect of Solutions to Davey-Stewartson Systems on the Whole Plane

This paper discusses the Cauchy problem of elliptic-elliptic-type Davey-Stewartson systems with zero-order dissipation on R^2. Making use of the Fourier spectral projector, together with a long time comparison between the solutions to the Davey-Stewartson systems and to an auxiliary problem, we prove that the global attractor in H^1 （R^2） for the addressed Davey-Stewartson systems is a compact subset of H^2（R^2）, and thus reveal the asymptotic smoothing effect of the solutions for the systems.     

Fractal Dimension of Random Attractors for Non-autonomous Fractional Stochastic Ginzburg–Landau Equations

This paper considers the dynamical behavior of solutions for non-autonomous stochastic fractional Ginzburg–Landau equations driven by additive noise with α∈(0, 1). First, we give some conditions for bounding the fractal dimension of a random invariant set of non-autonomous random dynamical system. Second, we derive uniform estimates of solutions and establish the existence and uniqueness of tempered pullback random attractors for the equation in H. At last, we prove the finiteness of fractal dimension of random attractors.     

Maximal Attractors for the <Emphasis Type="Italic">m</Emphasis>-Dimensional Cahn-Hilliard System

In this paper we discuss maximal attractors of the m-dimensional Cahn-Hilliard System in the product spaces (L^2(Ω))^m and (H^2(Ω))^m in terms of D.Henry‘s general theory and from the viewpoint of compactness and absorptivity of semigroups as R. Temam did. After giving the existence and uniqueness of global solutions, we technically restrict our discussion to some subspaces, give estimates with a new graph norm, and obtain the existence of maximal attractors and some properties of them.     

Maximal attractor for an Ostwald ripening model

We consider a family of phase-field models that couples a Cahn-Hilliard with several Allen-Cahn equations. We show that such family of phase-field models possesses a maximal attractor with finite fractal dimension.     

反应扩散方程古典解的最大吸引子

利用能量积分和解析半群的有关估计，一类反应扩散方程非负古典解在连续函数空间的最大吸引子的存在性被证明，且非线性项取为任意阶多项式．     

有阻尼Sine－Gordon方程的全局吸引子的维数

本文通过引入新范数,得到有阻尼Ｓｉｎｅ－Ｇｏｒｄｏｎ方程的Ｄｉｒｉｃｈｌｅｔ问题的全局吸引子的维数的一个估计．结果表明：当“阻尼”与“扩散”同时增大或正弦项系数减小时,吸引子的维数减小．特别地,得到了零维吸引子存在的参数条件．     

Dimension of the global attractor for damped nonlinear wave equations

An estimate on the Hausdorff dimension of the global attractor for damped nonlinear wave equations, in two cases of nonlinear damping and linear damping, with Dirichlet boundary condition is obtained. The gained Hausdorff dimension is bounded and is independent of the concrete form of nonlinear damping term. In the case of linear damping, the gained Hausdorff dimension remains small for large damping, which conforms to the physical intuition.

     

Random Attractors for the Stochastic Kuramoto-Sivashinsky Equation

Abstract

Random systems may be more reasonable by incorporating influence of noise into deterministic systems. The notion of a random attractor is one of the very basic concepts of the theory of random dynamical systems. In this article, we consider the well-known Kuramoto–Sivashinsky equation with stochastic perturbation. Our aim is to attempt to obtain a so-called pull-back random attractor for stochastic Kuramoto–Sivashinsky equation. In particular, the Hausdorff dimension of a random attractor is finite. For simplicity, we always restrict ourselves to odd initial conditions, but the result for all initial conditions is also true.     

有阻尼Sine－Gordon方程的全局吸引子的维数

本文通过引入新范数，得到有阻尼Ｓｉｎｅ－Ｇｏｒｄｏｎ方程的Ｄｉｒｉｃｈｌｅｔ问题的全局吸引子的维数的一个估计．结果表明：当“阻尼”与“扩散”同时增大或正弦项系数减小时，吸引子的维数减小．特别地，得到了零维吸引子存在的参数条件．     

On the Global Attractor of Generalized Ginzburg-Landau Equation in One Spatial Dimension

The target of this paper is the long time behaviour of solutions for a generalized Ginzburg-Landau equation on IR. The authors establish the existence of a global attractor of finite Hausdorff and fractal dimension in a weighted Hilbert space for the equation.     

非线性抛物型方程组的最大吸引子

利用能量积分、Sobolev空间的嵌入定理和不变区域,本文证明了一类具有"自然结构条件"的非线性抛物型方程组的最大吸引子的存在性,并给出了一定条件下解的衰减性估计。