Crack Singularities for General Elliptic Systems

We consider general homogeneous Agmon‐Douglis‐Nirenberg elliptic systems with constant coefficients complemented by the same set of boundary conditions on both sides of a crack in a two‐dimensional domain. We prove that the singular functions expressed in polar coordinates (r, θ) near the crack tip all have the form r^{k + 1/2}φ(θ) with k ≥ 0 integer, with the possible exception of a finite number of singularities of the form r^k log r φ(θ). We also prove results about singularities in the case when the boundary conditions on the two sides of the crack are not the same, and in particular in mixed Dirichlet‐Neumann boundary value problems for strongly coercive systems: in the latter case, we prove that the exponents of singularity have the form with real η and integer k. This is valid for general anisotropic elasticity too.     

1∶1内共振悬索的二维奇异性分析

对1∶1内共振悬索系统的二维分岔方程进行了研究．根据奇异性理论得到了3种情况下开折系统的转迁集．转迁集将整个参数空间分成了不同的保持域,得到了各个保持域上的分岔图．     

Formation of Singularities for Quasilinear Hyperbolic Systems with Characteristics with Constant Multiplicity

In this paper we consider the Cauchy problem for quasilinear hyperbolic systems with characteristics with constant multiplicity. Without restriction on characteristics with constant multiplicity(> 1), under the assumptions that there is a genuinely nonlinear simple characteristic and the initial data possess certain decaying properties, the blow-up result is obtained for the C¹ solution to the Cauchy problem.     

On initial value problems related to p-Laplacian and pseudo-Laplacian

Summary We establish some qualitative properties in the sense of weak singularity and super singularity for a certain system of two nonlinear differential equations related to the radially symmetric solution of p-Laplacian and pseudo-Laplacian problems. For the transformed system of differential equations we carry out the classification in the sense of weak singularity, singularity and super singularity. The choice of initial values at the point of singularity for correct settings of Cauchy problem is also considered.     

On the Milnor Number of an Equivariant Singularity

Mathematical Notes -     

Zn—等变的奇点理论

本详细讨论了Ｚｎ－等变奇点理论的基本概念，范式和万有开折的求法，并具有给出了非退化情况下Ｚ３－等变点的范式和万有开折。它们为研究周期参数激励系统的亚谐分叉提供了一种方法。     

Gorenstein Isolated Quotient Singularities of Odd Prime Dimension are Cyclic

In this article, we shall prove that Gorenstein isolated quotient singularities of odd prime dimension are cyclic. In the case where the dimension is bigger than 1 and is not an odd prime number, then there exist Gorenstein isolated noncyclic quotient singularities.     

On the splash singularity for the free-surface of a Navier–Stokes fluid

In fluid dynamics, an interface splash singularity occurs when a locally smooth interface self-intersects in finite time. We prove that for d-dimensional flows, $d=2$ or 3, the free-surface of a viscous water wave, modeled by the incompressible Navier–Stokes equations with moving free-boundary, has a finite-time splash singularity for a large class of specially prepared initial data. In particular, we prove that given a sufficiently smooth initial boundary (which is close to self-intersection) and a divergence-free velocity field designed to push the boundary towards self-intersection, the interface will indeed self-intersect in finite time.     

Nonexistence of self-similar singularities in the viscous magnetohydrodynamics with zero resistivity

We are concerned on the possibility of finite time singularity in a partially viscous magnetohydrodynamic equations in Rn, n=2,3, namely the MHD with positive viscosity and zero resistivity. In the special case of zero magnetic field the system reduces to the Navier-Stokes equations in Rn. In this paper we exclude the scenario of finite time singularity in the form of self-similarity, under suitable integrability conditions on the velocity and the magnetic field. We also prove the nonexistence of asymptotically self-similar singularity. This provides us information on the behavior of solutions near possible singularity of general type as described in Corollary 1.1.     

Kolmogorov condition near hyperbolic singularities of integrable Hamiltonian systems

In this paper we show that, if an integrable Hamiltonian system admits a nondegenerate hyperbolic singularity then it will satisfy the Kolmogorov condegeneracy condition near that singularity (under a mild additional condition, which is trivial if the singularity contains a fixed point).      

具有幂零奇点的七次Hamilton系统Abel积分的零点个数估计

本文研究了具有幂零奇点的七次Hamilton系统的Abel积分的零点个数问题.利用Picard-Fuchs方程法,得到了Abel积分I(h)=∮_(Γh)g(x,y)dx-f(x,y)dy在(0,1/4)上零点个数B(n≤3[(n-1)/4]),其中Γ_h是H(x,y)=x~4+y~4-x~8=h,h∈(0,1/4),所定义的卵形线f(x,y)=∑(1≤4i+4j+1≤n)aijx~(4i+1)y~4j)和g(x,y)=∑(1≤4i+4j+1≤n)bijx~4iy~(4j+1)是x和y的次数不超过n的多项式.     

Local singularities of dynamical systems with shock interactions

Local (qualitative) singularities of a special class of dynamical systems with shock interactions are classified. For the first four of the specified types of local singularities, certain properties of the qualitative structure are described and used to establish the topological equivalence of the corresponding singularities.Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 531–542, October, 1998.     

A refined mixed finite-element method for the stationary Navier Stokes equations with mixed boundary conditions

Serge Nicaise ^{This paper is concerned with the mixed formulation of the Navier–Stokesequations with mixed boundary conditions in 2D polygonal domainsand its numerical approximation. We first describe the regularityof any solution. The problem is then approximated by a mixedfinite-element method where the strain tensor and the antisymmetricgradient tensor, quantities of practical importance, are introducedas new unknowns. An existence result for the finite-elementsolution and convergence results are proved near a nonsingularsolution. Quasi-optimal error estimates are finally presented.}     

具有三重零奇异的时滞微分方程的分支

主要研究三重零奇异的判定和在Rn上零特征根对应的广义特征空间,利用中心流形简化和规范型计算得到参数时滞微分方程的简化形式,对应于文[A note on the triple zero linear degeneracy:Normal forms,dynamical and bifurcation behaviour of an unfolding.Int J Bifur and Chaos,2002,12:2799-2820]中的结果具体分析具有三重零奇异的参数时滞微分方程的分支行为,并给出一例子来阐述得到的结果.     

Pure Log Terminal Blow-Ups

The existence of a pure log terminal blow-up for a log terminal singularity and a criterion for a singularity to be weakly exceptional are proved.     

Universal abelian covers of superisolated singularities

We give explicit examples of Gorenstein surface singularities with integral homology sphere link, which are not complete intersections. Their existence was shown by Luengo–Velasco, Melle–Hernández and Némethi, thereby providing counterexamples to the universal abelian covering conjecture of Neumann and Wahl (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)