p方程组的激波生成与构造

讨论p-方程组的激波生成与构造.精细地刻画了从光滑解到激波产生的转变过程,并给出了在激波生成点附近奇性结构与解的估计.     

一个最值问题的物理模型

题 1　已知ａ >0 ,ｂ >0 ,0 <ｘ <π2 ,求函数ｆ(ｘ) =ａｓｉｎｘ ｂｃｏｓｘ的最小值 .图 1　题 1图解　如图 1,ＡＰＢ为一以 |ＡＢ|=1为直径的半圆 ,设Ａ点有带电为ａ的点电荷 ,Ｂ点有带电为ｂ的点电荷 .对于弧上一点Ｐ ,令∠ＰＡＢ =ｘ ,则ＡＰ =ｓｉｎｘ ,ＰＢ =ｃｏｓｘ ,根据点电荷Ｕ =ｋ Ｑｒ 及电势叠加原理 .则Ｐ点电势ＵＰ=ｋａｓｉｎｘ ｋｂｃｏｓｘ=ｋ(ａｓｉｎｘ ｂｃｏｓｘ) (ｋ为静电常数 ) .这样 ,原问题便转化为在ＡＰＢ上找一点Ｐ0 ,使ＵＰ0 为最小值 .设想有一单位正电荷ｅ从Ｂ点沿圆弧自由运动 ,由其总能量守恒可知 ,当…     

Dirac宇宙学的一个物理模型

本文讨论了在我们的文章[1-2]中提出的广义相对论和Dirac宇宙学的统一理论的各种应用。给出了Friedman-Lemaitre宇宙解,球对称静态解和弱引力场近似下的牛顿方程。本文证明Einstein大爆炸宇宙学中的视界困难可以在我们的统一理论中得到满意的解决。     

一个非线性方程的渐近激波解

该文是利用简捷的方法得到了高精度的非线性问题渐近激波解。     

反射型激波风洞中激波与边界层的相互作用

本文研究了反射型激波风洞中由于非完全反射对激波与壁面边界层相互作用的影响,给出了在反射激波坐标系中计算边界层速度分布、温度分布和马赫数分布的计算方法.算例表明,在计及氮气的平衡真实气体效应的情形下,随着入射激波马赫数M_s的增大,边界层的最小马赫数从壁面处移到边界层内;随着喷管喉道面积的增大,边界层的最小马赫数、反射激波的分叉角α和分叉区后的射流速度均随之减小.计算结果与实验值相比是一致的.     

Burgers方程的激波解与激波位置(英文)

讨论了Burgers方程激波解和位置的转移 .认为 :对该类方程 ,当边值发生微小变化时 ,不仅激波解发生变化 ,而且激波位置将发生较大的变化 ,甚至从内层移到边界 .其激波解也会发生相应的变化 .     

一种在双驱动激波管和激波风洞中实现运动激波与头激波斜相互作用的新方法

本文在讨论和分析了国外现有的运动激波与头激波斜相互作用的两大类实验方案的基础上,提出并实现了在双驱动激波管和激波风洞中形成运动激波与头激波斜相互作用的新方法.这种方法不仅可以获得双波(指运动激波与头激波,下同.)斜相互作用所需要的平面的运动激波,而且可以同时得到双波斜相互作用条件下试验模型表面瞬态压力曲线和流场照片.这种方法还可以用于研究在运动激波前有气流情况下,运动激波在尖劈或尖锥表面规则反射(Regular Reflection)与Mach反射(Mach Reflection)之间的转变.在测试技术方面,本文还提出了一种改进方法,用于测量运动激波的激波Mach数.     

利用衔接法构造奇摄动激波层问题的渐近解

研究了一类具有转向点的奇摄动拟线性边值问题,指出在一定条件下解在转向点t=0呈激波层现象.先用合成展开法构造出问题的形式近似,然后利用衔接法将t=0左、右两边分别具有边界层性质的近似式光滑地衔接起来,从而形成在t=0处具有激波层性质的解,并应用微分不等式理论证明了解的存在性及其渐近性质.     

一个三角问题的构造处理

<正>问题(2006·四川理·11)设a,b,c分别是△ABC的三个内角A,B,C所对的边,则a~2 =b(b+c)是A=2B的( ).(A)充要条件(B)充分而不必要条件(C)必要而不充分条件(D)既不充分又不必要条件     

一个错误命题的反例构造

１９６６年第２期《数学通报》上刊有一篇题为《有一组对边相等和一组对角相等的四边形是平行四边形吗？》（以下简称为《四边形》）的文章,作为数学教师当然知道这是个错误命题,但是文章始终未给出反例的作法．图１其实这个问题并不难解决,下面我们从这个命题的条件分析起．如图１,四边形ＡＢＣＤ中∠Ａ＝∠Ｃ＝α,ＡＤ＝ＢＣ＝ａ,ＡＢ＝ｂ,ＤＣ＝ｃ,ＢＤ＝ｅ,由余弦定理得　　ｅ２＝ａ２＋ｃ２－２ａｃｃｏｓα,　　ｅ２＝ａ２＋ｂ２－２ａｂｃｏｓα．∴　ａ２＋ｃ２－２ａｃｃｏｓα　＝ａ２＋ｂ２－２ａｂｃｏｓα,ｂ２－ｃ…     

Two-Dimensional Shock Waves for a Model Problem

Mathematical Notes - The existence of nonclassical (two-dimensional) shock waves in Riemann’s problem is proved for a modification of the system of shallow water equations.     

Shock Waves in Anisotropic Cylinders

We study small-amplitude longitudinal and torsional shock waves in circular cylinders consisting of an anisotropic medium such that the velocities of the longitudinal and torsional waves are close to each other. Previously, simple waves were considered in the same situation and conditions were found for these waves to overturn and for the corresponding shock waves to form. Here we present the study of shock waves: the shock adiabat and the evolutionary conditions. The results obtained can also be related to shock waves in unbounded media with quadratic nonlinearity.     

Shock Waves in Plane Symmetric Spacetimes

We consider Einstein's equations coupled to the Euler equations in plane symmetry, with compact spatial slices and constant mean curvature time. We show that for a wide variety of equations of state and a large class of initial data, classical solutions break down in finite time. The key mathematical result is a new theorem on the breakdown of solutions of systems of balance laws. We also show that an extension of the solution is possible if the spatial derivatives of the energy density and the velocity are bounded, indicating that the breakdown is really due to the formation of shock waves.     

Shock Waves in Dispersive Eulerian Fluids

The long-time behavior of an initial step resulting in a dispersive shock wave (DSW) for the one-dimensional isentropic Euler equations regularized by generic, third-order dispersion is considered by use of Whitham averaging. Under modest assumptions, the jump conditions (DSW locus and speeds) for admissible, weak DSWs are characterized and found to depend only upon the sign of dispersion (convexity or concavity) and a general pressure law. Two mechanisms leading to the breakdown of this simple wave DSW theory for sufficiently large jumps are identified: a change in the sign of dispersion, leading to gradient catastrophe in the modulation equations, and the loss of genuine nonlinearity in the modulation equations. Large amplitude DSWs are constructed for several particular dispersive fluids with differing pressure laws modeled by the generalized nonlinear Schrödinger equation. These include superfluids (Bose–Einstein condensates and ultracold fermions) and “optical fluids.” Estimates of breaking times for smooth initial data and the long-time behavior of the shock tube problem are presented. Detailed numerical simulations compare favorably with the asymptotic results in the weak to moderate amplitude regimes. Deviations in the large amplitude regime are identified with breakdown of the simple wave DSW theory.     

Formation and Dynamics of Shock Waves in the Degasperis-Procesi Equation

Solutions of the Degasperis-Procesi nonlinear wave equation may develop discontinuities in finite time. As shown by Coclite and Karlsen, there is a uniquely determined entropy weak solution which provides a natural continuation of the solution past such a point. Here we study this phenomenon in detail for solutions involving interacting peakons and antipeakons. We show that a jump discontinuity forms when a peakon collides with an antipeakon, and that the entropy weak solution in this case is described by a "shockpeakon" ansatz reducing the PDE to a system of ODEs for positions, momenta, and shock strengths.     

血液动力学中血管流激波与驻波的相互作用

血液动力学问题是生物力学心血管系统中的重要研究课题.血管内斑块处,血管截面和血管壁的材质发生变化,对血液流动产生重要影响.血液流动中基本波及其相互作用对探究血液流动的规律、生理学意义及与疾病的关系有着重要的意义.本文研究血液动力学血液流动简化数学模型的基本波的相互作用.血管流模型是3×3非严格双曲型方程组.构造性地得到了初值为三段常状态时,血管流问题的解,即解决了激波与驻波的相互作用问题.特别地,给出四种后前激波与驻波的相互作用的结果.     

Some Results on the Stability of Non-classical Shock Waves

Part 1 of this paper establishes the infinite-time stability of a class of over-compressive viscous shock waves; the equations studied here are a mathematical analogue of those of magnetohydrodynamics. Part 2 communicates a rather general short-time stability result for undercompressive shock waves in several space dimensions; technically, this is an easy extension of Majda's corresponding result for Laxian shock waves.     

A Model Equation for Water Waves

Water waves are usually modeled as solutions of Laplace's equation in the fluid domain with appropriate nonlinear boundary conditions. Here we present a simple differential equation on the mean free surface, whose solutions behave like water waves.     

磁流体力学方程组的激波生成

研究磁流体横向流动的一维模型,在解的强间断出现后流场的性质。利用迭代法具体构造了该方程组的强间断—激波以及问题的熵解。同时,利用激波的性质,给出了各物理参量在爆破点附近的奇性估计。