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.     

Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise

We study the asymptotic behavior of solutions to the stochastic sine-Gordon lattice equations with multiplicative white noise. We first prove the existence and uniqueness of solutions, and then establish the existence of tempered random bounded absorbing sets and global random attractors.     

A Nine-modes Truncation of the Plane Incompressible Navier-Stokes Equations

In this paper a nine-modes truncation of Navier-Stokes equations for a two-dimensional incompressible fluid on a torus is obtained. The stationary solutions, the existence of attractor and the global stability of the equations are firmly proved. What is more, that the force f acts on the mode ks and k7 respectively produces two systems, which lead to a much richer and varied phenomenon. Numerical simulation is given at last, which shows a stochastic behavior approached through an involved sequence of bifurcations.     

具有非线性热源项的耦合杆系统

主要以经典的算子半群理论为依据,研究了一类具有非线性热效应的耦合杆系统的长时间行为.首先在齐次边界条件和初始条件下,证明了系统解的存在唯一性;其次通过渐近先验估计,证明了系统有界吸收集的存在性;最后利用算子半群的分解技巧,得到了系统全局吸引子的存在性.     

Regularity of the attractor for 3-D complex Ginzburg-Landau equation

In this paper,the existence of global attractor for 3-D complex Ginzburg Landau equation is considered.By a decomposition of solution operator,it is shown that the global attractor A_i in H~i(Ω) is actually equal to a global attractor Aj in H~j(Ω)(i≠j,i,j = 1,2,…m).     

Attractors of derivative complex Ginzburg-Landau equation in unbounded domains

The Ginzburg-Landau-type complex equations are simplified mathematical models for various pattern formation systems in mechanics, physics, and chemistry. In this paper, the derivative complex Ginzburg-Landau (DCGL) equation in an unbounded domain Ω ⊂ ℝ² is studied. We extend the Gagliardo-Nirenberg inequality to the weighted Sobolev spaces introduced by S. V. Zelik. Applied this Gagliardo-Nirenberg inequality of the weighted Sobolev spaces and based on the technique for the semi-linear system of parabolic equations which has been developed by M. A. Efendiev and S. V. Zelik, the global attractor in the corresponding phase space is constructed, the upper bound of its Kolmogorov’s ɛ-entropy is obtained, and the spatial chaos of the attractor for DCGL equation in ℝ² is detailed studied.      

Random attractors for stochastic differential equations driven by two-sided Lévy processes

Abstract

In this paper, the asymptotic behavior of solutions for a nonlinear Marcus stochastic differential equation with multiplicative two-sided Lévy noise is studied. We plan to consider this equation as a random dynamical system. Thus, we have to interpret a Lévy noise as a two-sided metric dynamical system. For that, we have to introduce some fundamental properties of such a noise. So far most studies have only discussed two-sided Lévy processes which are defined by combining two-independent Lévy processes. In this paper, we use another definition of two-sided Lévy process by expanding the probability space. Having this metric dynamical system we will show that the Marcus stochastic differential equation with a particular drift coefficient and multiplicative noise generates a random dynamical system which has a random attractor.     

Random attractor and upper semi-continuity for Zakharov lattice system with multiplicative white noise

In this paper, we study the long-term asymptotic behaviour of solutions to the stochastic Zakharov lattice equation with multiplicative white noise. We first transfer the stochastic lattice equation into a random lattice equation and prove the existence and uniqueness of solutions which generate a random dynamical system. Then we consider the existence of a tempered random bounded absorbing set and a random attractor for the system. Finally we establish the upper semi-continuity of random attractor to the global attractor of the limiting system as the coefficients of the white noise terms tend to zero.     

一类三维拟抛物粘性扩散方程有限差分逼近的长时间行为

研究了一类带有周期边界条件的三维拟抛物粘性扩散方程有限差分解的长时间行为.证明了数值解的存在唯一性,离散系统全局吸引子的存在性,差分格式的长时间稳定性和收敛性.此外,我们给出了上半连续性.     

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.