一类推广的控制收敛定理及其应用

本文研究了一类推广的控制收敛定理及其应用,利用对控制函数满足条件的修改,建立了推广的控制收敛定理,获得其在弱奇异性积分算子中的应用,简化了经典结论的证明.     

Flows near stagnation points in non-orthogonally colliding disperse viscous flows

Within the framework of a two-fluid dusty-gas model, a family of steady self-similar flows near a stagnation point formed by two different non-orthogonally colliding viscous incompressible streams, one of which contains solid inertial particles, is studied. The limiting case of non-orthogonal impingement of a viscous disperse medium on a rigid wall is considered separately. The possibility of the formation of multiple particle accumulation zones on the envelopes of particle trajectories and on the contact (in the limiting case, rigid) surface is demonstrated. The local structure of the particle velocity and concentration fields is studied. The threshold values of the governing parameters corresponding to qualitative flow pattern reconstruction are found.     

The shape of an axisymmetric bubble in uniform motion

We consider in a frame fixed to a bubble translating with steady speedU, the inviscid, axisymmetric, irrotational motion of the liquid past it. If all speeds are normalized byU and lengths by {ie437-1}, whereT is the surface tension of the liquid-bubble interface, it can be shown that the unknown bubble shape and field depend on a single parameter {ie437-2} alone, where the pressures are the ones in the bubble and far away respectively. WhenΓ is very large the bubble is almost spherical in shape while for Γ<- Γ^* ≈ -0.315, bubbles whose exteriors are simply connected do not exist. We solve the non-linear, free boundary problem for the whole range Γ^* < Γ < ∞ by the use of an analytical representation for the bubble shape, a surface singularity method to compute potential flows and a generalized Newton’s method to continue inΓ. Apart from providing explicit representations for bubble shapes and detailed numerical values for the bubble parameters, we show that the classical linearized solution for largeΓ is a very good approximation, surprisingly, to as low values of Γ as 2. We also show that Miksiset al [1] is inaccurate over the whole range and in serious error for large and smallΓ. These have been corrected.     

空间轴对称变形界面裂纹尖端的应力奇异性分析

假定位移场,并将位移势函数展开成一系列的特征函数,代入空间轴对称变形的基本方程,推得界面裂纹的特征值理论。把这一理论应用到单向增强的连续纤维复合材料上,建立了纤维断裂、基体开裂等模型下应力奇异性的特征方程,分析了界面裂纹尖端的应力奇异性。     

ANALYSIS OF STRONG SINGULARITIES FOR NONLINEAR HYPERBOLIC EQUATIONS

In this paper we review our some results about the strongly singular (discontinuous,measure, or delta function) problems for nonlinear hyperbolic equations.     

Singular Reduction of Poisson Manifolds

The conditions under which it is possible to reduce a Poisson manifold via a regular foliation have been completely characterized by Marsden and Ratiu. In this Letter we show that this characterization can be generalized in a natural way to the singular case and, as a corollary, we obtain that when the singular distribution is given by the tangent spaces to the orbits created by a Hamiltonian Lie group action, one reproduces the Universal Reduction Procedure of Arms, Cushman, and Gotay.     

Singular behavior of vacuum states for compressible fluids

In this survey paper, we will present the recent work on the study of the compressible fluids with vacuum states by illustrating its interesting and singular behavior through some systems of fluid dynamics, that is, Euler equations, Euler–Poisson equations and Navier–Stokes equations. The main concern is the well-posedness of the problem when vacuum presents and the singular behavior of the solution near the interface separating the vacuum and the gas. Furthermore, the relation of the solutions for the gas dynamics with vacuum to those of the Boltzmann equation will also be discussed. In fact, the results obtained so far for vacuum states are far from being complete and satisfactory. Therefore, this paper can only be served as an introduction to this interesting field which has many open and challenging mathematical problems. Moreover, the problems considered here are limited to the author's interest and knowledge in this area.     

Log Structures on Generalized Semi-Stable Varieties

In this paper, a class of morphisms which have a kind of singularity weaker than normal crossing is considered. We construct the obstruction such that the so-called semi-stable log structures exists if and only if the obstruction vanishes. In the case of no power, if the obstruction vanishes, then the semi-stable log structure is unique up to a unique isomorphism. So we obtain a kind of canonical structure on this family of morphisms.     

边值问题中与分离变量法等价的级数解法

用带奇性分离的Ｆｏｕｒｉｅｒ级数解平面边值问题，改进了不能满足全部边界条件的一般级数解法，得到收敛较好的解析算法。文中证明了级数解法与分离变量法的等价性，由此将分离变量解所得的非线性特征方程转化为多项式方程，为近似算法与渐近分析提供了依据。