The Riemann Problem To The Pressureless Relativistic Euler Equations And Shock Reflection Problems

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.