Interaction Of Rarefaction Waves And Propagation Of Weak Discontinuities For The2-D Euler Equations

This paper is concerned with the expansion problem of a wedge of gas intovacuum for the two-dimensional Euler equations in isothermal fow and the two di-mensional isentropic ir-rotational pseudo-steady Euler equations for the generalizedChaplygin gas. Furthermore, we give the theory of propagation of weak discontinu-ities for compressible Euler system.In Chapter1, we introduce the background and some development of compress-ible Euler equations. Furthermore, we introduce the physical origin of the Chaplygingas.In Chapter2, we introduce some basic concepts of the hyperbolic conservationequations and the defnitions of the1-D and2-D Riemann problems.In Chapter3, we consider the expansion problem of a wedge of gas into vacuumwith small angle for the two-dimensional isothermal Euler equations, obtain thepriori estimates of the solution and prove the global existence of the smooth solutionto the expansion problem. Our results are based on the characteristic decompositionmethod and the bootstrapping.In Chapter4, we consider the expansion problem of a wedge of gas into vacuumfor the generalized Chaplygin gas of the isentropic Euler equations. At frst, wepresent various characteristic forms and the characteristic decompositions of thewave characteristic inclination angles and the sonic speed. Then, we consider theexpansion problem of a wedge of gas into vacuum for the generalized Chaplygingas, and obtain the priori estimates of the solutions to the expansion problems of awedge of the generalized Chaplygin gas into vacuum with small angle and big angle,respectively. At last, we prove the global existence of the smooth solution to theexpansion problem. In Chapter5, by use of the characteristic decomposition method, we give thetheory of propagation of weak discontinuities for compressible Euler system. It isproved that the weak discontinuities spread along characteristic curves for the Eulersystem. In Section1, we present the results for the scalar conservation laws andthe general2×2system. In Section2, we give the theory of propagation of weakdiscontinuities to the Euler system for polytropic gas in Lagrangian coordinates.In Section3, the results for the two-dimensional steady Euler systems are shown,which include isentropic and ir-rotational fow, the isentropic fow and the full Eulersystem. In Section4, we give the results for the pressure-gradient system and thetwo-dimensional Euler systems for pseudo-steady fow, which include the isentropicir-rotational pseudo-steady fow, the isentropic pseudo-steady fow and the full Eulersystem for pseudo-steady fow. In Section5, two examples for the Euler system areshown, which are the rarefaction wave in the Euler system for polytropic gas andthe expansion problem of gas into vacuum in the two-dimensional Euler equationsin isothermal fow.