相对论欧拉方程组的流扰动问题

发布时间：2018-04-20 20:00

本文选题：经典相对论欧拉方程组 + 等熵相对论欧拉方程组　；参考：《云南大学》2016年博士论文

【摘要】：本文研究两类相对论流体力学方程组的流扰动问题.第一类是描述动量守恒和能量守恒的经典相对论欧拉方程组,第二类是描述重子数守恒和动量守恒的等熵相对论欧拉方程组.首先,通过求解带有流扰动的零压相对论欧拉方程组的黎曼问题,发现了两类有趣的U-型拟真空状态解和参数化的狄拉克激波解.进而证明,当流扰动消失时,参数化的狄拉克激波和U-型拟真空状态解收敛到零压相对论欧拉方程组的狄拉克激波和真空状态解.其次,在不同的气体状态方程下,使用特征线分析法和相平面分析法,借助于洛伦兹变换,依次构造性地求解了相应系统的黎曼问题.进一步地,严格证明了,当压力或者流扰动消失时,相对论欧拉方程组的黎曼解收敛到它对应的零压流系统的狄拉克激波和真空解.这表明,零压相对论欧拉方程组的狄拉克激波和真空解对于流扰动是稳定的.第一章介绍相对论流体力学方程组的研究现状和本文的研究工作.第二章讨论基于经典相对论欧拉方程组的零压相对论欧拉方程组的黎曼问题,构造了狄拉克激波解和真空解.第三章考虑经典相对论欧拉方程组的流扰动问题.首先,求解一类纯流扰动的零压相对论欧拉方程组的黎曼问题,获得了倒U-型的拟真空状态解和参数化的狄拉克激波解.随后证明,当流扰动消失时,参数化的狄拉克激波解和倒U-型的拟真空状态解分别收敛到零压相对论欧拉方程组的狄拉克激波解和真空解.其次,求解经典相对论欧拉方程组在包含压力的流扰动下的黎曼问题.当双参数的流扰动消失时,我们严格证明了,包含两个激波的黎曼解趋于零压相对论欧拉方程组的狄拉克激波解；包含两个疏散波的黎曼解趋于零压相对论欧拉方程组的两个接触间断解,并且介于这两个激波之间的非真空状态趋于真空.第四章研究经典相对论修正Chaplygin气体方程组在压力和流扰动分别消失时,黎曼解的极限行为.我们首先求解该系统的黎曼问题,并分析基本波曲线对参数的依赖性.随后证明,当双参数的压力扰动和三参数的流扰动分别消失时,包含两个激波的黎曼解收敛到零压相对论欧拉方程组的狄拉克激波解；包含两个疏散波以及一个非真空中间状态的黎曼解收敛到零压相对论欧拉方程组的真空解.第五章求解基于等熵相对论欧拉方程组的零压相对论欧拉方程组的狄拉克激波和真空状态.第六章研究带有流扰动的等熵相对论欧拉方程组.首先求解一类特殊的纯流扰动的零压相对论欧拉方程组的黎曼问题,构造了U-型的拟真空状态解和参数化的狄拉克激波解.进而证明,当流扰动消失时,U-型的拟真空状态解和参数化的狄拉克激波解分别收敛到对应的零压相对论欧拉方程组的狄拉克激波解和真空解.其次,求解具有流扰动的等熵相对论多方气体欧拉方程组的黎曼问题.进一步地,我们严格证明,当压力和双参数的流扰动消失时,包含两个激波的黎曼解收敛到对应的零压流系统的狄拉克激波解,并且介于这两个激波之间的中间密度趋于一个加权的狄拉克δ-测度即形成狄拉克激波；而包含两个疏散波的黎曼解收敛到零压相对论欧拉方程组的接触间断解,并且它们之间的非真空状态趋于真空.第七章考虑等熵相对论修正Chaplygin气体欧拉方程组的流扰动问题.首先.求解系统的黎曼问题,并构造黎曼解.其次,我们证明,当双参数压力和三参数流扰动分别消失时,包含两个激波的黎曼解趋于相应的零压相对论欧拉方程组的狄拉克激波解；包含两个疏散波的黎曼解趋于零压相对论欧拉方程组的接触间断解,并且介于这两个疏散波之间的非真空状态趋于真空.
[Abstract]:In this paper, we study the flow disturbance of two kinds of relativistic fluid mechanics equations. The first class is the classical relativistic Euler equation describing the conservation of momentum and the conservation of energy. The second is the isentropic relativistic Euler equation describing the conservation of baryon number and the conservation of momentum. First, the zero pressure relativistic Euler equations with flow disturbance are solved. In Riemann's problem, two kinds of interesting U- quasi vacuum state solutions and parameterized Dirac shock wave solutions are found. Further, it is proved that when the flow disturbance disappears, the parameterized Dirac shock and U- quasi vacuum state solutions converge to the zero pressure relativistic Euler equation group of Dirac shock and vacuum state. Second, under different gas state equations, Using the method of characteristic line analysis and phase plane analysis, the Riemann problem of the corresponding system is solved by means of Lorenz transform. Further, it is proved strictly that when the pressure or flow disturbance disappears, the Riemann solution of the relativistic Euler equation converges to the Dirac shock and the vacuum solution of its corresponding zero pressure flow system. The Dirac shock wave and the vacuum solution of the Euler equation in the relativistic system of zero pressure are stable. The first chapter introduces the research status of the relativistic fluid mechanics equations and the research work in this paper. The second chapter discusses the Riemann problem of the zero pressure relativistic Euler equation based on the classical relativistic Euler equation, and constructs the Dirac excitation. In the third chapter, the third chapter considers the flow perturbation problem of the classical relativistic Euler equations. First, the Riemann problem of a class of zero pressure relativistic Eulerian equations of a class of pure flow perturbation is solved. The pseudo vacuum state solution of the inverted U- and the parameterized Dirac shock solution are obtained. Then, it is proved that the parameterized Dirac shock wave when the flow disturbance disappears. The quasi vacuum state solution of the solution and inverted U- converges to the Dirac shock solution and the vacuum solution of the zero pressure relativistic Euler equation group. Secondly, we solve the Riemann problem of the classical relativistic Euler equation under the flow disturbance containing pressure. When the two parameter flow disturbance disappears, we strictly prove that the Riemann solution containing two shock waves tends to zero. The Dirac shock wave solution of the pressure relativistic Euler equation; the Riemann solution containing two evacuation waves tends to two contact discontinuous solutions to the zero pressure relativistic Euler equation, and the non vacuum state between the two shock waves tends to vacuum. The fourth chapter studies the classical relativistic modified Chaplygin gas equations in pressure and flow perturbation respectively. We first solved the Riemann problem of the Riemann solution and analyzed the dependence of the basic wave curve on the parameters. Then, it was proved that when the two parameter pressure disturbance and the three parameter flow disturbance disappeared, the Riemann solution containing two shock waves converged to the Dirac shock solution of the zero pressure relativistic Euler equation; The Riemann solution containing two evacuation waves and a non vacuum intermediate state converges to the vacuum solution of the zero pressure relativistic Euler equation. The fifth chapter solves the Dirac shock and vacuum state of the Euler equation group based on the isentropic relativistic Euler equation. The sixth chapter studies the isentropic relativistic Euler equations with the flow disturbance. First, the Riemann problem of a class of zero pressure relativistic Euler equations of a special kind of pure flow is solved. The quasi vacuum state solution of U- type and the parameterized Dirac shock wave solution are constructed. Then, it is proved that when the flow disturbance disappears, the quasi vacuum state solution of the U- type and the parameterized Dirac shock solution converge to the corresponding zero pressure relativistic Euler square, respectively. The Dirac shock wave solution and the vacuum solution in the process group. Secondly, to solve the Riemann problem of the isentropic Euler equation with the isentropic relativistic gas. Further, we prove that when the pressure and the two parameter flow disturbance disappear, the Riemann solution containing two shock waves converges to the Dirac shock wave solution for the zero pressure flow system. The middle density between the two shock waves tends to a weighted Dirac delta measure to form a Dirac shock wave, and the Riemann solution containing two evacuation waves converges to the contact discontinuous solution of the zero pressure relativistic Euler equation, and the non vacuum state between them tends to vacuum. The seventh chapter considers the isentropic relativity to amend the Chaplygin gas. The Riemann problem of the Euler equation is solved. First, the Riemann problem is solved and the Riemann solution is constructed. Secondly, we prove that when the two parameter pressure and the three parameter flow disturbance disappear respectively, the Riemann solution containing two shock waves tends to the corresponding zero pressure relativistic Eulerian equation of the Dirac shock wave, and the Riemann solution containing two evacuation waves. In the zero pressure relativistic Euler equations, the contact discontinuity solution and the non vacuum state between these two scattered waves tend to vacuum.

【学位授予单位】：云南大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O35

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