本文選題：經(jīng)典相對(duì)論歐拉方程組 + 等熵相對(duì)論歐拉方程組��； 參考：《云南大學(xué)》2016年博士論文

【摘要】：本文研究?jī)深?lèi)相對(duì)論流體力學(xué)方程組的流擾動(dòng)問(wèn)題.第一類(lèi)是描述動(dòng)量守恒和能量守恒的經(jīng)典相對(duì)論歐拉方程組,第二類(lèi)是描述重子數(shù)守恒和動(dòng)量守恒的等熵相對(duì)論歐拉方程組.首先,通過(guò)求解帶有流擾動(dòng)的零壓相對(duì)論歐拉方程組的黎曼問(wèn)題,發(fā)現(xiàn)了兩類(lèi)有趣的U-型擬真空狀態(tài)解和參數(shù)化的狄拉克激波解.進(jìn)而證明,當(dāng)流擾動(dòng)消失時(shí),參數(shù)化的狄拉克激波和U-型擬真空狀態(tài)解收斂到零壓相對(duì)論歐拉方程組的狄拉克激波和真空狀態(tài)解.其次,在不同的氣體狀態(tài)方程下,使用特征線分析法和相平面分析法,借助于洛倫茲變換,依次構(gòu)造性地求解了相應(yīng)系統(tǒng)的黎曼問(wèn)題.進(jìn)一步地,嚴(yán)格證明了,當(dāng)壓力或者流擾動(dòng)消失時(shí),相對(duì)論歐拉方程組的黎曼解收斂到它對(duì)應(yīng)的零壓流系統(tǒng)的狄拉克激波和真空解.這表明,零壓相對(duì)論歐拉方程組的狄拉克激波和真空解對(duì)于流擾動(dòng)是穩(wěn)定的.第一章介紹相對(duì)論流體力學(xué)方程組的研究現(xiàn)狀和本文的研究工作.第二章討論基于經(jīng)典相對(duì)論歐拉方程組的零壓相對(duì)論歐拉方程組的黎曼問(wèn)題,構(gòu)造了狄拉克激波解和真空解.第三章考慮經(jīng)典相對(duì)論歐拉方程組的流擾動(dòng)問(wèn)題.首先,求解一類(lèi)純流擾動(dòng)的零壓相對(duì)論歐拉方程組的黎曼問(wèn)題,獲得了倒U-型的擬真空狀態(tài)解和參數(shù)化的狄拉克激波解.隨后證明,當(dāng)流擾動(dòng)消失時(shí),參數(shù)化的狄拉克激波解和倒U-型的擬真空狀態(tài)解分別收斂到零壓相對(duì)論歐拉方程組的狄拉克激波解和真空解.其次,求解經(jīng)典相對(duì)論歐拉方程組在包含壓力的流擾動(dòng)下的黎曼問(wèn)題.當(dāng)雙參數(shù)的流擾動(dòng)消失時(shí),我們嚴(yán)格證明了,包含兩個(gè)激波的黎曼解趨于零壓相對(duì)論歐拉方程組的狄拉克激波解；包含兩個(gè)疏散波的黎曼解趨于零壓相對(duì)論歐拉方程組的兩個(gè)接觸間斷解,并且介于這兩個(gè)激波之間的非真空狀態(tài)趨于真空.第四章研究經(jīng)典相對(duì)論修正Chaplygin氣體方程組在壓力和流擾動(dòng)分別消失時(shí),黎曼解的極限行為.我們首先求解該系統(tǒng)的黎曼問(wèn)題,并分析基本波曲線對(duì)參數(shù)的依賴性.隨后證明,當(dāng)雙參數(shù)的壓力擾動(dòng)和三參數(shù)的流擾動(dòng)分別消失時(shí),包含兩個(gè)激波的黎曼解收斂到零壓相對(duì)論歐拉方程組的狄拉克激波解；包含兩個(gè)疏散波以及一個(gè)非真空中間狀態(tài)的黎曼解收斂到零壓相對(duì)論歐拉方程組的真空解.第五章求解基于等熵相對(duì)論歐拉方程組的零壓相對(duì)論歐拉方程組的狄拉克激波和真空狀態(tài).第六章研究帶有流擾動(dòng)的等熵相對(duì)論歐拉方程組.首先求解一類(lèi)特殊的純流擾動(dòng)的零壓相對(duì)論歐拉方程組的黎曼問(wèn)題,構(gòu)造了U-型的擬真空狀態(tài)解和參數(shù)化的狄拉克激波解.進(jìn)而證明,當(dāng)流擾動(dòng)消失時(shí),U-型的擬真空狀態(tài)解和參數(shù)化的狄拉克激波解分別收斂到對(duì)應(yīng)的零壓相對(duì)論歐拉方程組的狄拉克激波解和真空解.其次,求解具有流擾動(dòng)的等熵相對(duì)論多方氣體歐拉方程組的黎曼問(wèn)題.進(jìn)一步地,我們嚴(yán)格證明,當(dāng)壓力和雙參數(shù)的流擾動(dòng)消失時(shí),包含兩個(gè)激波的黎曼解收斂到對(duì)應(yīng)的零壓流系統(tǒng)的狄拉克激波解,并且介于這兩個(gè)激波之間的中間密度趨于一個(gè)加權(quán)的狄拉克δ-測(cè)度即形成狄拉克激波；而包含兩個(gè)疏散波的黎曼解收斂到零壓相對(duì)論歐拉方程組的接觸間斷解,并且它們之間的非真空狀態(tài)趨于真空.第七章考慮等熵相對(duì)論修正Chaplygin氣體歐拉方程組的流擾動(dòng)問(wèn)題.首先.求解系統(tǒng)的黎曼問(wèn)題,并構(gòu)造黎曼解.其次,我們證明,當(dāng)雙參數(shù)壓力和三參數(shù)流擾動(dòng)分別消失時(shí),包含兩個(gè)激波的黎曼解趨于相應(yīng)的零壓相對(duì)論歐拉方程組的狄拉克激波解；包含兩個(gè)疏散波的黎曼解趨于零壓相對(duì)論歐拉方程組的接觸間斷解,并且介于這兩個(gè)疏散波之間的非真空狀態(tài)趨于真空.
[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.

【學(xué)位授予單位】：云南大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類(lèi)號(hào)】：O35

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