Davey-Stewartson方程组的近似惯性流形

本文研究了二维Davey--Stewartson方程组,证明了解的时间解析性和Gevrey类正则性,构造了指数式的近似惯性流形.本文结果表明,如果一个方程的线性主部算子能生成一个解析半群,那么其平坦近似惯性流形(Galerkin近似)和非平坦近似惯性流形具有相同的逼近精度,都可以是指数式的.     

时滞惯性流形及近似时滞惯性流形族

时滞惯性流形是对耗散系统惯性流形、近似惯性流形的最新发展,它基于对 大小涡分量间相互作用更细致的观察,即改变了惯性流形和近似惯性流形方法中大小 涡分量间相互作用为瞬时行为的隐含假定,而认为这种作用与系统的历史相关的．本 文结出了一类耗散系统时滞惯性流形的存在性证明,由于其存在性不需要“谱间隔条 件”保证,因而时滞惯性流形是广泛存在的．随后我们引出了一类离散的时滞惯性流 形,并在此基础上构造了一种近似时滞惯性流形族,分别给出了它们近似时滞惯性流 形的误差估计,结果显示这种新方法为构造稳定和高精度的算法提供了可能．     

弱阻尼KdV方程的近似惯性流形

本文给出了弱阻尼KdV方程近似惯性流形族的一个构造方法，得到的近似惯性流形是一个弱Lipschitz连续的流形，它对吸引子的指数渐近速度比平坦惯性流形相应的速率要高。     

一类非线性双曲动力系统的近似弱惯性流形

引入了近似及半近似弱惯性流形的概念,通过某些变换及逼近技巧,得到了一类非线性双曲动力系统的全局吸引子及半近似弱惯性流形的存在性,并由此结果进而讨论了另一类与量子力学问题相关的系统的弱惯性流形的存在性,最后提出了正则化的问题.     

Cahn－Hiliard方程的近似惯性流形

通过压缩映象原理，本文对Ｃａｈｎ－Ｈｉｌｉａｒｄ方程构造了一簇近似惯性流形，这些近似惯性流形越来越逼近全局吸引子     

Burgers方程的小波近似惯性流形及数值分析

研究Burgers方程小波基下小波近似惯性流形的存在性,并作低阶多分辨分析下的数值分析,数值分析表明小波近似惯性流形更能反映方程的局部行为.     

N-S方程一般近似惯性流形构造和逼近

1．引言为了用有限维常微分方程来研究Navier七lobes（N七）方程的长时间动力学行为,Foias,Sell和Temaml］引入了耗散系统惯性流形的概念．但惯性流形存在的一个相当苛刻的条件一谱间隔条件一是包括N习方程在内的很多耗散系统无法满足的．因此,Foias,Manlea和TemamZ］随后又提出了近似惯性流形的概念．近似惯性流形也是一个光滑的Lipschitz流形,所有原方程的解在时间充分大时,将被吸引进入该流形的一个三邻域中．因其存在性不需要谱间隔条件来保证,从而可证明包括N－S方程在内的一大类耗散系统存在近似惯性流形．利用近似惯性…     

Davey-Stewartson方程组的近似惯性流形

本文研究了二维 Ｄａｖｅｙ－Ｓｔｅｗａｒｔｓｏｎ方程组,证明了解的时间解析性和Ｇｅｖｒｅｙ类正则性,构造了指数式的近似惯性流形．本文结果表明,如果一个方程的线性主部算子能生成一个解析半群,那么其平坦近似惯性流形（Ｇａｌｅｒｋｉｎ近似）和非平坦近似惯性流形具有相同的逼近精度,都可以是指数式的。     

具有拟周期外力的非自治时滞发展方程的近似惯性流形

该文研究了一类具有拟周期外力的非自治时滞发展方程, 通过延伸相平面将非自治系统转化为自治系统, 再证明相应的自治系统的时滞惯性流形的存在性, 并在时滞惯性流形的基础上构造了非自治发展方程的近似惯性流形.     

Cahn—Hilliard方程的近似惯性流形

通过压缩映象原理，本文对Ｃａｈｎ－Ｈｉｌｉａｒｄ方程构造了一簇近似惯性流形，这些近惯性流越来越逼近全局吸引子。     

FULL DISCRETE NONLINEAR GALERKIN METHOD FOR THE NAVIER-STOKES EQUATIONS

This paper deals with the inertial manifold and the approximate inertial manifold concepts of the Navier-Stokes equations with nonhomogeneous boundary conditions and inertial algorithm. Furtheremore, we provide the error estimates of the approximate solutions of the Navier-Stokes Equations.     

On the existence of an inertial manifold for a deconvolution model of the 2D mean Boussinesq equations

We show the existence of an inertial manifold (ie, a globally invariant, exponentially attracting, finite‐dimensional manifold) for the approximate deconvolution model of the 2D mean Boussinesq equations. This model is obtained by means of the Van Cittern approximate deconvolution operators, which is applied to the 2D filtered Boussinesq equations.     

Approximate inertial manifolds for retarded semilinear parabolic equations

The asymptotic behavior of a system of retarded parabolic equations is considered. For any given η>0 we construct an approximate inertial manifold (AIM) which contains all the steady states of the system and has an attractive neighborhood of thickness η. The dependence of AIMs on the delay time is investigated.     

GEVREY CLASS REGULARITY AND APPROXIMATE INERTIAL MANIFOLDS FOR THE NEWTON-BOUSSINESQ EQUATIONS

§１．ＩｎｔｒｏｄｕｃｔｉｏｎＡｐｐｒｏｘｉｍａｔｅｉｎｅｒｔｉａｌｍａｎｉｆｏｌｄｓａｒｅｒｅｌａｔｅｄｔｏｔｈｅｓｔｕｄｙｏｆｌｏｎｇｔｉｍｅｂｅｈａｖｉｏｕｒｏｆｓｏｌｕｔｉｏｎｓｏｆｄｉｓｉｐａｔｉｖｅｐａｒｔｉａｌｄｉｆｅｒｅｎｔｉａｌｅｑ...     

LOCALIZATION AND APPROXIMATION OF ATTRACTORS FOR THE KURAMOTO-SIVASHINSKY EQUATIONS

1Intr0ducti0nThestudy0fglobalattrartorhasagreatdealtodowiththatofthelargetimebehavi0urofthesolutionsofdissipativepartialdtherentialequations.Usually,theattractoristypicallyacomplicated,fractalsubset0fthephasespace.Hellcetheconceptofinertialmallif0ld(IM),apositivelyinvariantfinite-dimensi0nalLipschitzmanifoldthatexponentiallyattractseveryorbit,wasdevelopedandithasbecomeakeytoolinthisareaofstudy.(See,e-g-,[l],[21andreferencestherein.)Themotivatingideabehindtheinertialmanif0ldist0embedtheattrac…     

Inertial and slow manifolds for delay equations with small delays

Yu.A. Ryabov and R.D. Driver proved that delay equations with small delays have Lipschitz inertial manifolds. We prove that these manifolds are smooth. In addition, we show that expansion in the small delay can be used to obtain the dynamical system on the inertial manifold. This justifies “post-Newtonian” approximation for delay equations.     

非线性发展方程的H(o)lder连续惯性流形

引入非线性发展方程的H\"older连续惯性流形的概念,为原来惯性流形概念的推广和修正.惯性流形是有限维不变的Lipschiz流形,是研究发展方程解的长时间性态的合适工具,其缺点是需要谱间隙条件.提出H\"older连续惯性流形也是有限维不变的,但光滑性减弱为H\"older连续,不需要谱间隙条件.该流形与指数吸引子交集具有指数吸引性,无穷维动力系统可在H\"older连续惯性流形上约化为有限维常微分方程组.     

Discrete inertial manifolds

This work is devoted to attractive invariant manifolds for nonautonomous difference equations, occurring in the discretization theory for evolution equations. Such invariant sets provide a discrete counterpart to inertial manifolds of dissipative FDEs and evolutionary PDEs. We discuss their essential properties, like smoothness, the existence of an asymptotic phase, normal hyperbolicity and attractivity in a nonautonomous framework of pullback attraction. As application we show that inertial manifolds of the Allen–Cahn and complex Ginzburg–Landau equation persist under discretization. For the Ginzburg–Landau equation we can also estimate the dimension of the inertial manifold. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global attractors and approximate inertial

An abstract nonautonomous differential equation, u' + Au + F(u) = f ( t ) , is considered using assumptions appropriate for systems of reaction-diffusion equations on multi-dimensional spatial domains. A priori estimates establish the existence of absorbing balls in relevant function spaces, and nonsequently the existence of a global attractor is verified in the associated skew-product flow. It is also shown that approximate inertial manifolds exist for this equation. These are finite-dimensional manifolds which have exponentially attracting neighborhoods under the flow. The dimension of the manifold and the thickness of the attracting neighborhood are inversely related.     

Computational efficiency and approximate inertial manifolds for a Bénard convection system

Summary A computational comparison between classical Galerkin and approximate inertial manifold (AIM) methods is performed for the case of two-dimensional natural convection in a saturated porous material. For prediction of Hopf and torus bifurcations far from convection onset, the improvements of the AIM method over the classical one are small or negligible. Two reasons are given for the lack of distinct improvement. First, the small boundary layer length scale is the source of the instabilities, so it cannot be modeled as a “slave” to the larger scales, as the AIM attempts to do. Second, estimates based on the Gevrey class regularity of solutions to the governing equations show that the classical and AIM methods may be virtually equivalent. It is argued that these two reasons are physical and mathematical reflections of one another.