Riemann Problem For The Bi-Symmetric Class Of The Pressure-Gradient System In The Gas Dynamic

In this paper, we mainly study the characteristic decomposition and Riemannproblem for the pressure-gradient system in the gas dynamic.In chapter1, we introduce some development of compressible Euler systemfirstly. Then some special fluid model and particularly some development of pressure-gradient system are introduced.In chapter2, we introduce some useful concepts for the hyperbolic system firstly.Then some general theories about one-dimensional and two-dimensional hyperbolicsystems are introduced in the next part of this section, respectively.In chapter3, we give the derivation of the pressure-gradient system mathemat-ically by the asymptotic derivation method and present a characteristic decompo-sition of pressure-gradient system in the self-similar plane by direct approach. Thedecompositions of the pressure P and characteristics Λ±are obtained. Furthermore,the velocity (u, v) can be also obtained if the flow come from a constant state. Thesedecompositions allow a proof that any wave adjacent to a constant state is a simplewave for the pressure-gradient equations. In addition, in the polar coordinate, wepresent the characteristic decomposition of the two dimensional pressure-gradientequations.In chapter4, we discuss Riemann problem for the bi-symmetric class of thepressure-gradient system. In section1, we present the Riemann problem and dif-ferent cases of one-dimensional and two-dimensional pressure-gradient system. Andgive the bi-symmetric structure of four rarefactions. In section2, some results ofsimple waves are established, and from chapter3we conclude that any wave adja-cent to a constant state is a simple wave and the convex properties for the crossMach charateristics in a simple wave. In Section3, we discuss the interaction of fourrarefaction waves and get the sufcient condition for hyperbolicity. Then we proveexistence of a local smooth solution to the Goursat problem and to an initial bound- ary value problem. By the characteristic decomposition we have a priori estimateson the norm of P (pressure) in C1,1. Finally by continuity method, we prove thatthere exists a global smooth solution up to the vacuum boundary for interactionsof simple waves problem. In Section4, we reduce the vacuum boundary to a singlepoint by the characteristic decomposition.