Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type

We establish the existence and stability of multidimensional transonic shocks for the Euler equations for steady potential compressible fluids. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for the velocity, can be written as a second-order, nonlinear equation of mixed elliptic-hyperbolic type for the velocity potential. The transonic shock problem can be formulated into the following free boundary problem: The free boundary is the location of the transonic shock which divides the two regions of smooth flow, and the equation is hyperbolic in the upstream region where the smooth perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem. Our results indicate that there exists a unique solution of the free boundary problem such that the equation is always elliptic in the downstream region and the free boundary is smooth, provided that the hyperbolic phase is close to a uniform flow. We prove that the free boundary is stable under the steady perturbation of the hyperbolic phase. We also establish the existence and stability of multidimensional transonic shocks near spherical or circular transonic shocks.

     

Differentiable conjugacy of the Poincaré type vector fields on

We prove that on , except for those germs of vector fields whose linear parts are conjugated to , any two Poincaré type vector fields are at least conjugated to each other provided their linear approximations have the same eigenvalues and the nonlinear parts are generic.

     

Hardy spaces of spaces of homogeneous type

Let be a space of homogeneous type, and be the generator of a semigroup with Gaussian kernel bounds on . We define the Hardy spaces of for a range of , by means of area integral function associated with the Poisson semigroup of , which is proved to coincide with the usual atomic Hardy spaces on spaces of homogeneous type.

     

West's problem on equivariant hyperspaces and Banach-Mazur compacta

Let be a compact Lie group, a metric -space, and the hyperspace of all nonempty compact subsets of endowed with the Hausdorff metric topology and with the induced action of . We prove that the following three assertions are equivalent: (a) is locally continuum-connected (resp., connected and locally continuum-connected); (b) is a -ANR (resp., a -AR); (c) is an ANR (resp., an AR). This is applied to show that is an ANR (resp., an AR) for each compact (resp., connected) Lie group . If is a finite group, then is a Hilbert cube whenever is a nondegenerate Peano continuum. Let be the hyperspace of all centrally symmetric, compact, convex bodies , , for which the ordinary Euclidean unit ball is the ellipsoid of minimal volume containing , and let be the complement of the unique -fixed point in . We prove that: (1) for each closed subgroup , is a Hilbert cube manifold; (2) for each closed subgroup acting non-transitively on , the -orbit space and the -fixed point set are Hilbert cubes. As an application we establish new topological models for tha Banach-Mazur compacta and prove that and have the same -homotopy type.

     

齐型空间上交换子的紧性

设(x，d，μ)是齐型空间．本文证明了分数次积分算子Iα与VMO函数构成的交换子I^ba是L^p(X)到L^q(X)的紧算子，其中α∈(0，1)，1＜p＜q＜∞且1/q＝1/p—α。     

心瓣流体动力学研究进展

心瓣流体动力学是近年来才形成的一门新兴边缘学科分支。本文比较全面地介绍了该学科的主要研究内容、研究特点、发生发展的历史和现状,以及笔者及其同事在这方面的工作,提出并论述了对学科发展有重大影响的几个富有挑战性的问题,从而预示了学科的发展方向      

BEPCⅡ大功率波导真空阀门的研制

 大功率波导真空阀门是BEPCⅡ直线加速器微波系统的重点改进项目,作为微波大功率传输和真空密封双重功能的超高真空微波器件,机械结构比较复杂,制造工艺难度大。经过各方努力,已经研究完成16台大功率波导真空阀门,并在2004年全部安装在加速器上使用。目前进入到波导真空阀门的脉冲功率为30~50 MW,高功率运行工作状态良好。对阀门从设计原理、机械结构、微波测试及高功率运行等方面作了较详细的阐述。     

双曲函数型辅助方程构造具5次强非线性项的波方程的新精确孤波解

在辅助方程法的基础上利用两种函数变换和一种双曲函数型辅助方程，通过符号计算系统Mathematica构造了在力学当中一个重要的模型，有5次强非线性项的波方程的新三角函数型和双曲函数型精确孤波解.这种方法寻找其他具5次强非线性项的非线性发展方程的新精确解方面具有普遍意义. 关键词： 双曲函数型辅助方程 函数变换 具5次强非线性项的波方程 精确孤波解     

Some New Hermite–Hadamard and Related Inequalities for Convex Functions via (p,q)-Integral

In this investigation, for convex functions, some new (p,q)–Hermite–Hadamard-type inequalities using the notions of (p,q)π2 derivative and (p,q)π2 integral are obtained. Furthermore, for (p,q)π2-differentiable convex functions, some new (p,q) estimates for midpoint and trapezoidal-type inequalities using the notions of (p,q)π2 integral are offered. It is also shown that the newly proved results for p=1 and q→1− can be converted into some existing results. Finally, we discuss how the special means can be used to address newly discovered inequalities.