【作者】 郭俐辉；

【导师】 盛万成；

【作者基本信息】 上海大学 ， 计算数学， 2009， 博士

【摘要】 本文研究了气体动力学中Chaplygin气体的Riemann问题。第二章首先介绍了关于双曲守恒律系统的一些基本概念。然后对一维和二维双曲守恒律方程的一般性理论分别给出了介绍。第三章研究了一维Chaplygin气体的Riemann问题。和多方气体不同的是,Chaplygin气体的所有特征都是线性退化的。所以Chaplygin气体的基本波都是接触间断。我们把相平面分为五个区域,有趣的是其中有一个区域会出现delta波。delta波由广义Rankine-Hugoniot关系式的提出,并对delta波从速度,位置和权上分别给出了精确的描述,并且给出了delta波出现的熵条件。这些解展示了宇宙演化过程中的某些现象,例如黑洞。第四章研究了四片常状态初值的两维Chanplygin气体的Riemann问题。初值在原点发出的四条射线上间断,每条初始间断线在t＞0时发射出三道平面基本波((?)或(?))+J+((?)或(?))。这十二道波在(x,y,t)空间中将在以原点为顶点的锥体中相互作用。为了使问题简化而又不失实质,引进假设:每条初始间断线在t＞0时只发射出一道平面基本波。于是,问题就被简化成四道基本波的相互作用。根据四道波的不同组合,将问题分成十四类。其中有六类是无旋的。除了2J~++2J~-以外,利用广义特征分析方法逐个进行讨论,构造出了超音区的解.有趣的是有些情况会出现delta波。亚音区的边界包括音速线和滑移面。这些解也展示了宇宙演化过程中的某些现象,例如黑洞。第五章研究了两维Chaplygin气体具有轴对称性质的解。利用轴对称和自相似假设,我们可以把偏微分方程组的初值问题化为非自治常微分方程组的无穷远边值问题。奇点对应于四维相空间中的两维流形,并且对应于物理空间中特征曲面。全局解至少包含三段非平凡的轨道连接。与多方气体不同的是,即使轴向速度大于零的时候也会出现间断;轴向速度小于零时会出现质量集中的现象。这些解展示了宇宙演化的某些现象,例如黑洞的形成和演化,宇宙的暴涨和膨胀。更多还原

【Abstract】 In this article,we study the Riemann problem for Chaplygin gas of conservation laws.In section 2,we introduces some useful concepts for the hyperbolic system firstly. Then some general theories about one-dimensional and two-dimensional hyperbolic systems are introduced in the next part of this section,respectively.In section 3,we discuss the one one-dimensional Riemann problem of Chaplygin gas. Different from polytropic gas,the eigenvalues of Chaplygin gas are linear degenerate.So their elementary waves are contact discontinuities.We divided the phase plane into five regions.More interesting thing is that there appear delta wave in one region.Due to the introduction of a suitable generalized Rankine-Hugoniot relation,delta shock gets a well depiction from velocity,location and weight.And we give the entropy condition about the delta wave.The solutions exhibit some phenomena,such as black hole,in the evolution of universe.In section 4,we discuss the initial data of Riemanna problem with four constant states.There are four jumps,each of them emits three waves:((?) or(?)),J~±,((?) or(?)) at t>0.These elementary waves interaction in the cone which vertex is orign in space (x,y,t).The problem is how the twelve waves interact and match together ultimately. Obviously,it is too complicated to deal with but its key point is the interaction of different elementary waves,so we made a restriction that each jump at infinity emits exactly one elementary wave.According to combinations of the four waves and compatibility,We classified these problems into 14 cases.Six of them are irrotational.By using method of generalized characteristic analysis,we construct supersonic solution for each case except the case of 2J~+ + 2J~-.More interesting thing is that delta waves may appear in some cases.And Dirichlet boundary value problems in subsonic domain are formed for some cases.The boundaries of the domains composed of sonic curves and slip lines,he solutions exhibit some phenomena,such as black hole,in the evolution of universe.In section 5,we discuss axisymmetric solutions for the two-dimensional Chaplygin gas.We usethe axisymmetry and self-similarity assumptions to reduce the initial-value problem of partial differential equations to a infinite boundary-value problem for a system of non-autonomous ordinary differential equations.Singularity points of the system con-sist of two-dimensional manifolds in the four-dimensional phase space.These singularity points correspond to surface of characteristics in physical space-time.A global solution may consist of as many as three nontrivial connecting orbits chained together.Different from polytropic gas,discontinuities exist even though the velocity of radially direction is positive and if the velocity of radially direction is negative,there appear concentration phenomenon.The solutions exhibit some phenomena,such as black hole formation and development,expansion and explosive expansion,in the evolution of universe.更多还原