The Riemman Problem For Chaplygin Gas In Gas Dynamics

In this article,we study the Riemann problem for Chaplygin gas of conservation laws.In section 2,we introduces some useful concepts for the hyperbolic system firstly. Then some general theories about one-dimensional and two-dimensional hyperbolic systems are introduced in the next part of this section,respectively.In section 3,we discuss the one one-dimensional Riemann problem of Chaplygin gas. Different from polytropic gas,the eigenvalues of Chaplygin gas are linear degenerate.So their elementary waves are contact discontinuities.We divided the phase plane into five regions.More interesting thing is that there appear delta wave in one region.Due to the introduction of a suitable generalized Rankine-Hugoniot relation,delta shock gets a well depiction from velocity,location and weight.And we give the entropy condition about the delta wave.The solutions exhibit some phenomena,such as black hole,in the evolution of universe.In section 4,we discuss the initial data of Riemanna problem with four constant states.There are four jumps,each of them emits three waves:((?) or(?)),J~±,((?) or(?)) at t>0.These elementary waves interaction in the cone which vertex is orign in space (x,y,t).The problem is how the twelve waves interact and match together ultimately. Obviously,it is too complicated to deal with but its key point is the interaction of different elementary waves,so we made a restriction that each jump at infinity emits exactly one elementary wave.According to combinations of the four waves and compatibility,We classified these problems into 14 cases.Six of them are irrotational.By using method of generalized characteristic analysis,we construct supersonic solution for each case except the case of 2J~+ + 2J~-.More interesting thing is that delta waves may appear in some cases.And Dirichlet boundary value problems in subsonic domain are formed for some cases.The boundaries of the domains composed of sonic curves and slip lines,he solutions exhibit some phenomena,such as black hole,in the evolution of universe.In section 5,we discuss axisymmetric solutions for the two-dimensional Chaplygin gas.We usethe axisymmetry and self-similarity assumptions to reduce the initial-value problem of partial differential equations to a infinite boundary-value problem for a system of non-autonomous ordinary differential equations.Singularity points of the system con-sist of two-dimensional manifolds in the four-dimensional phase space.These singularity points correspond to surface of characteristics in physical space-time.A global solution may consist of as many as three nontrivial connecting orbits chained together.Different from polytropic gas,discontinuities exist even though the velocity of radially direction is positive and if the velocity of radially direction is negative,there appear concentration phenomenon.The solutions exhibit some phenomena,such as black hole formation and development,expansion and explosive expansion,in the evolution of universe.