Regularity and asymptotic behavior of solutions of nonautonomous differential equations

The existence of a nonautonomous approximate inertial manifold is shown for problems of the formu + Au + N(t,u)=0, in whichA is a self-adjoint operator with compact resolvent in a Hilbert spaceH. The operatorN(t, u) = G(u) + F(t, u) is nonlinear withG a monotone gradient that is locally Lipschitz fromD(A ^1/2) intoH, andF:⁺×HH a Lipschitz perturbation that is Hölder continuous int. Weak solutions are shown to be uniformly locally Hölder continuous intoD(A) with equicontinuity in families of solutions with ¦u(0)¦ r.A priori estimates of ¦Au(t)¦ are also verified and used in a skew-product flow to show there is a global attractor whose component elements form a equicontinuous family of solutions.     

Some algebraic examples of Frobenius manifolds

Using analytic methods of finite-gap integration, we construct quasihomogeneous algebraic solutions of the WDVV associativity equations and the nonsemisimple Frobenius manifolds associated with them. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 2, pp. 195–206, May, 2007.     

An implicit function theorem for Banach spaces and some applications

We prove a generalized implicit function theorem for Banach spaces, without the usual assumption that the subspaces involved being complemented. Then we apply it to the problem of parametrization of fibers of differentiable maps, the Lie subgroup problem for Banach–Lie groups, as well as Weil’s local rigidity for homomorphisms from finitely generated groups to Banach–Lie groups.      

基于最佳质量网格的薄板问题的非协调流形方法

对于大部分非协调板单元,使用规则网格能得到很好的效果。但是,当网格不规则时,非协调元的数值特性将变得很差,甚至收敛性得不到保证。为解决网格依赖性问题,许多专家学者提出了改造单元,如拟协调元法和广义协调元法,这些方法能解决收敛性问题,但是数值实践证明没有一种单元能在所有情况下都具有良好的数值特性。考虑到流形方法采用两套完全独立的覆盖系统,可以用规则的数学网格来作为数学覆盖进行插值,取得最佳的插值效果,单元收敛性便能得到保证。再结合适用于流形方法的变分提法,建立起流形方法处理非规则物理边界非协调板单元的一般格式。以ACM 薄板单元为例,与ANSYS、拟协调元法和广义协调元法进行了对比,证明本文方法在处理具有曲线边界的薄板弯曲问题时具有收敛快和精度高等优势。     

Application of a non-linear local analysis method for the problem of mixed convection instability

We consider the problem of laminar mixed convection flow between parallel, vertical and uniformly heated plates where the governing dimensionless parameters are the Prandtl, Rayleigh and Reynolds numbers. Using the method based on the centre manifold theorem which was derived from the general theory of dynamical systems, we reduce a three-dimensional simplified model of ordinary differential amplitude equations emanating from the original Navier-Stokes system of the problem in the vicinity of a trivial stationary solution. We have found that when the forcing parameter, the Rayleigh number, increases beyond the critical value Ra_s, the stationary solution is a pitchfork bifurcation point of the system.     

岩石双孔爆破过程的流形元法模拟

在流形元基本理论基础上,引入断裂力学的裂纹产生及扩展判据,应用二阶流形元对岩石双孔爆破过程进行了研究,模拟了双孔同时起爆和毫秒延时起爆两种条件下裂纹的产生和扩展、块体的形成以及漏斗形成过程,分析了爆炸载荷作用下岩石的破坏规律以及起爆条件对该过程的影响,验证了流形元法在研究冲击载荷作用下岩石从连续体到不连续体破坏过程的准确性和有效性,为研究类似问题提供一个新的思路和方法。     

流形元法固定边界约束处理研究

在数值流形方法中,对于材料的固定边界,一般采用罚函数的方法进行处理,即在固定边界上设置刚性弹簧约束其位移来实现固定约束条件的近似满足。罚函数法在理论上不是严格的固定约束处理方法,罚弹簧的布置与弹簧刚度的大小对模拟的效果都会产生影响。基于流形单元上位移函数的组成提出了流形方法固定边界约束处理的新方法,在组成流形单元的物理覆盖上,通过取消相应的覆盖函数在流形单元位移函数中的组成来实现双向固定的约束条件,通过使用只包含单方向位移的覆盖函数使x向固定约束条件和y向固定约束条件得到实现,推导了相应固定约束条件下的流形单元刚度矩阵的数值计算格式。该方法严格满足固定约束的物理意义,简化了固定边界的处理,并经算例证明是有效和准确的,有利于数值流形方法的程序实现和工程应用。     

Monodromy and nonintegrability in complex Hamiltonian systems

This paper investigates the monodromy representation of the normal variational equation along a phase curve of a two-dimensional complex analytic Hamiltonian system. Geometrical conditions are presented which guarantee reducibility, together with additional hypotheses to ensure complete reducibility. Symmetries in the equations are treated in detail. Applications to establishing the nonintegrability of specific systems are presented.     

The attractor of the scalar reaction diffusion equation is a smooth graph

For the scalar reaction diffusion equation with Dirichlet boundary conditions, it is proved that its maximal compact attractor is the graph of a C¹ function from a subset with nonempty interior of a subspace of the state space the dimension of which is equal to the maximal Morse index of the equilibria of the equation.     

An improvement and proof of OGY method

I.IntroductionRecenhycontrollingchaosbecomesaninterestingsubject.In1990,E.Ott,C.GreogiandJ.A.Yorkelllgaveacollceptionofcontrollingchaos,andgaveacontrollingmethodknownasOGYmethod.ThereweremanyresultsthatshowedOGYmethodiseffective.AfterOGY,manymethodsofcontrollingchaosweregiven,mostofthemwerebasedonOGY13'4'5'6'71.SoOGYmethodisthefundamentofcontrollingchaos.ToachieveaproofofOGYisanimportanttaskformathematicstheoryofcontrollingchaos.Thispaperchangesthemethodofchoosingparameter,andprove…