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…     

Spin-invariant valuations on the octonionic plane

The dimensions of the spaces of k-homogeneous Spin(9)-invariant valuations on the octonionic plane are computed using results from the theory of differential forms on contact manifolds as well as octonionic geometry and representation theory. Moreover, a valuation on Riemannian manifolds of particular interest is constructed which yields, as a special case, an element of Val₂^Spin (9).     

Examples of 4-manifolds with almost nonpositive curvature

We construct an infinite family of non-homeomorphic 4-manifolds with almost nonpositive sectional curvature whose universal covering space is not contractible. As a consequence, these manifolds do not support metrics with nonpositive sectional curvature. To achieve this, we use a generalization of Bavard's surgery construction, combined with an open book decomposition and knot theory.     

Families of simply connected 4-manifolds with the same Seiberg-Witten invariants

This article presents several new constructions of infinite families of smooth 4-manifolds with the property that any two manifolds in the same family are homeomorphic. While the construction gives strong evidence that any two of these manifolds of are not diffeomorphic, they cannot be distinguished by Seiberg-Witten invariants. Whether these manifolds are, or are not, diffeomorphic seems to be a very difficult question to answer. For one of these constructions, each member of the family is symplectic with the further property that each contains nullhomologous tori with the property that infinitely many log transformations on these tori yield nonsymplectic 4-manifolds. This is detected by calculations of Seiberg-Witten invariants. The surgery in question can be performed on any 4-manifold which contains as a codimension 0 submanifold a punctured surface bundle over a punctured surface and a nontrivial loop in the base which has trivial monodromy. A starting point for another class of examples in this paper is a family of examples which show that the Parshin-Arakelov theorem for holomorphic Lefschetz fibrations is false in the symplectic category. Such families are constructed by means of knot surgery on ellipitic surfaces. It is shown that for a fixed homeomorphism type X (of a simply connected elliptic surface) and a fixed integer g?3, there are infinitely many genus g Lefschetz fibrations on nondiffeomorphic 4-manifolds, all homeomorphic to X.     

Versal unfoldings for linear retarded functional differential equations

We consider parametrized families of linear retarded functional differential equations (RFDEs) projected onto finite-dimensional invariant manifolds, and address the question of versality of the resulting parametrized family of linear ordinary differential equations. A sufficient criterion for versality is given in terms of readily computable quantities. In the case where the unfolding is not versal, we show how to construct a perturbation of the original linear RFDE (in terms of delay differential operators) whose finite-dimensional projection generates a versal unfolding. We illustrate the theory with several examples, and comment on the applicability of these results to bifurcation analyses of nonlinear RFDEs.     

Assembly maps and realization of splitting obstructions

In 1987 Kharshiladze introduced the concept of type for an element in a Wall group, and proved that the elements of first and second type cannot be realized by normal maps of closed manifolds. This approach is sufficiently easy for computing the assembly maps and sometimes very effective. Here we give a geometrical interpretation of this approach by using the Browder–Quinn surgery obstruction groups for filtered manifolds. To understand the obtained relations we give algebraic definitions of the element types which are based on the algebraic surgery theory of Ranicki. Our approach describes a level of indeterminate for the algebraic passing to surgery on a codimension k submanifold of a given Browder–Livesay filtration. Then we study the realization of splitting obstructions by simple homotopy equivalences of closed manifolds, and compute the assembly maps for some classes of groups.     

Inertial manifolds of parabolic differential equations under high-order discretizations

This paper deals with the long-time behaviour of numerical discretizationsof nonlinear parabolic differential equations. For various equationsof mathematical physics, the dynamics are governed by a finite-dimensionalinertial manifold, which attracts solutions at an exponentialrate. We show that Runge-Kutta time and spectral Galerkin spacediscretizations possess inertial manifolds which approximatethe inertial manifold of the continuous problem with the orderof finite-time approximations of smooth solutions. We thus obtainestimates for the distance between the inertial manifolds ofthe partial differential equation and its semi- and full discretizationswhich show the high order of the time discretization and exponentiallyfast convergence of the space discretization. These resultsare obtained by using time analyticity and Gevrey regularityof solutions of the differential equation. As an applicationof the theory, the complex Ginzburg-Landau equation is considered.     

Specific features of families of invariant manifolds of conservative systems

In this paper we demonstrate themethod of the enveloping first integral via an example of a completely integrable system of differential equations. This method allows a researcher to find and investigate singular invariant manifolds for a given family of invariant manifolds of steady motions represented by an initial system of equations. We describe specific properties of branching of the obtained families of singular invariant manifolds.     

Projective Blaschke manifolds

In this paper we define the concept of projective Blaschke manifolds and extend the theory of equiaffine differential geometry to the projective Blaschke manifolds. partially supported by NSFC 10771146 and RFDP(20060610004)     

Essential self-adjointness of powers of generators of hyperbolic equations

Using the theory of hyperbolic equations, simple conditions are given which ensure the essential self-adjointness of all powers of certain formally symmetric differential operators. Typical applications include Dirac and Laplace-Beltrami operators on complete Riemannian manifolds, as well as semibounded operators of Schrödinger type.     

TURNING POINT PROBLEMS FOR THE THIRD ORDER DIFFERENTIAL EQUATIONS EXHIBITING RESONANCE

ＴＵＲＮＩＮＧＰＯＩＮＴＰＲＯＢＬＥＭＳＦＯＲＴＨＥＴＨＩＲＤＯＲＤＥＲＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳＥＸＨＩＢＩＴＩＮＧＲＥＳＯＮＡＮＣＥＸＩＡＯＣＨＥＮＧＹＯＵ（肖成猷）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＥａｓｔＣｈｉ...     

Asymptotic Lagrangian manifolds and the complex WKB method

Lagrangian analysis on dissipative-asymptotic manifolds is considered and applied to the asymptotic theory of partial differential equations. The special case of the theory of asymptotic manifolds with complex germ is presented.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 31, pp. 93–192, 1987.     

The stable manifold theorem for non-linear stochastic systems with memory: II. The local stable manifold theorem

We state and prove a Local Stable Manifold Theorem (Theorem 4.1) for non-linear stochastic differential systems with finite memory (viz. stochastic functional differential equations (sfde's)). We introduce the notion of hyperbolicity for stationary trajectories of sfde's. We then establish the existence of smooth stable and unstable manifolds in a neighborhood of a hyperbolic stationary trajectory. The stable and unstable manifolds are stationary and asymptotically invariant under the stochastic semiflow. The proof uses infinite-dimensional multiplicative ergodic theory techniques developed by D. Ruelle, together with interpolation arguments.     

Bak groups and equivariant surgery

There has been a popular mistake in equivariant surgery theory for almost a decade. This paper introduces Bak groups, presents their geometric role, and corrects the mistake. The form parameters of A. Bak shed light on the connection of involutions on manifolds to surgery on manifolds.Partially supported by Grant-in-Aid for Scientific Research.Dedicated to Professor Shôrô Araki on his sixtieth birthday     

共形对称的K—切触流形

本文建立了共形平坦的K－切触流形的纯量曲率适合的偏微分方程，证得：共形对称的K－切触流形是具常曲率1的Riemann流形，将Okumura和Miyazaawa等人的有关Sasaki流形的结果推广到K－切触流形。     

The spectral method for the flow between two concentric rotating spheres, part II: The full discrete nonlinear Legendre-Galerkin method

The nonlinear Galerkin methods are numerical schemes for evolutionary partial differential equations based on the theory of inertial manifolds and approximate inertial manifolds. In this paper, we consider the flow between two concentric rotating spheres, and combine the Legendre-Galerkin spectral methods in Part I together with the nonlinear Galerkin method, then construct the full discrete nonlinear Legendre-Galerkin spectral scheme, and derive the stability conditions and its error estimate.     

Variational Sequences in Mechanics on Grassmann Fibrations

Extension of the variational sequence theory in mechanics to the first order Grassmann fibrations of 1-dimensional submanifolds is presented. The correspondence with the variational theory of parameter-invariant problems on manifolds is discussed in terms of the theory of jets (slit tangent bundles) and contact elements. In particular, the Helmholtz expressions for parameter-invariant variational problems, measuring local variationality of differential forms and differential equations, are given in the canonical and adapted coordinates. The methods can easily be extended to higher order variational problems.     

Associated families of pluriharmonic maps and isotropy

Like minimal surface immersions in 3-space, pluriharmonic maps into symmetric spaces allow a one-parameter family of isometric deformations rotating the differential (“associated family”); in fact, pluriharmonic maps are characterized by this property. We give a geometric proof of this fact and investigate the “isotropic” case where this family is constant. It turns out that isotropic pluriharmonic maps arise from certain holomorphic maps into flag manifolds. Further, we also consider higher dimensional generalizations of constant mean curvature surfaces which are Kähler submanifolds with parallel (1,1) part of their soecond fundamental form; under certain restrictions there are also characterized by having some kind of (“weak”) associated family. Examples where this family is constant arise from extrinsic Kähler symmetric spaces.