摘要
本论文研究了两方面内容：其一为相对论Euler方程组中delta波及真空解问题，另一个为多方气体的平面激波正规反射问题．
在第二章，我们首先介绍了关于双曲型守恒律方程组的一些基本概念，继而分别对一维和二维双曲守恒律方程组的一般理论做了简要的介绍，为后两章的讨论作了准备．
第三章研究了相对论Euler方程组．我们首先用特征线方法分两种情况讨论了零压流相对论Euler方程组的Riemann问题，得到它的解有两种情况：一种出现了delta激波；另一种含有真空．之后我们分别对等温流和多方气体详细研究了当压力消失时，相对论Euler方程组中能量守恒和动量守恒方程组的Riemann解中delta激波和真空状态的形成．当压力消失时，只有两种情况发生：一种为包含两个激波的Riemann解，趋于零压流相对论Euler方程组的一个delta激波解，介于两个激波之间的中间状态的密度趋于一个形成这个delta激波的加权δ-测度；另一种为包含两个疏散波的Riemann解，趋于零压流相对论Euler方程组的包含两个接触间断的解，其中介于这两个接触间断之间的中间状态是一个真空状态．这些结果说明零压流相对论Euler方程组的Riemann解中delta激波的出现源于一种集中现象，而其中的真空状态则源于消失压力极限过程中的气穴现象；这二者在相对论流体力学中都具有重要的物理意义．
在第四章，我们研究了多方气体平面激波的正规反射问题．利用广义特征分析方法，通过力学关系的代数方程，得到了多方气体平面激波出现正规反射的一个修正了的临界条件，它可以表示为关于临界入射角的一个关系式．此外，我们发现这个临界角要比角α_0=arcot(1/1-μ~2ρ0/ρ1(1-ρ0/ρ1))~(1/2)小．正规反射中一个重要且很有意义的问题是跨声激波(反射激波后对应的去流是亚声的)到超声激波(反射激波后相应的去流是超声的)的转变．我们得到了去流为声速的条件，它是跨声激波向超声激波转换的临界条件．
We are concerned with delta shocks and vacuum solutions to the relativistic Euler equations and the regular reflection problem of a planar shock for polytropic gases.
In chapter 2, we give some fundamental notations and concepts for the hyperbolic conservation laws, and introduce some general theories for one dimensional and two dimensional hyperbolic conservation laws respectively, which will be useful to the following contents.
The relativistic Euler equations are considered in chapter 3. Using the method of characteristic analysis, we solve the Riemann problem to pressureless relativistic Euler equations firstly. We present two kinds of solutions: the one includes delta shocks; the other involves vacuum states. The formation of delta shocks and vacuum states in the Riemann solutions to the relativistic Euler system of conservation laws of energy and momentum in special relativity for isothermal gases and polytropic gases are identified and analyzed in detail subsequently, as the pressure vanishes. Two cases occur as the pressure vanishes: the one is the Riemann solution involving two shocks, which tends to a delta shock solution to pressureless relativistic Euler equations, and the intermediate density between the two shocks tends to a weightedδ- measure that forms the delta shock; the other one is the Riemann solution involving two rarefaction waves, which tends to a two contact-discontinuity solution to pressureless relativistic Euler equations, whose intermediate state between the two contact discontinuities is a vacuum state. These results show that the delta shocks for pressureless relativistic Euler equations result from a phenomenon of concentration, while the vacuum states result from a phenomenon of cavitation in the process of vanishing pressure limit; both are fundamental and physical in fluid dynamics.
In chapter 4, we study the regular reflection problem of a planar shock for polytropic gases. Utilizing the method of generalized characteristic analysis and algebraic equations of mechanical relations, we obtain a refined criterion for regular reflection of a planar shock for polytropic gas. which is the representation of critical angle of incidence. Furthermore, we give a result that the critical angle, is less than . A fundamental and significative issue in regular reflection is transition from transonic shock (the relative outflow behind the reflected shock wave is subsonic) to supersonic shock (the relative outflow behind the reflected shock wave is supersonic). We obtain a condition for sonic outflow, which is the criterion of transition from transonic shock to supersonic shock.
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