The Double Riemann Problem For A Macroscopic Production Model And A System Of Symmetric Hyperbolic Conservation Laws

In this paper,we mainly study a symmetric system of keyfitz-kranzer type and a macroscopic production model,focusing on the perturbed Riemann problem and the construction of its global solution under the appropriate assumptions.Firstly,by analyzing the characteristics of the equation,the possible wave states of the system can be determined.Secondly,all Riemann solutions of the two systems are obtained by the phase plane method.The key point is to find the global solution of the Riemann problem of the system through the method of wave interactions when the initial value is perturbed.Finally,on the basis of the perturbations of initial values,it is determined whether the limit solution of the perturbed Riemann problem is consistent with the solution of the perturbed Riemann problem when the perturbation coefficient tends to zero,and then we judge the stability of the system.On this basis,the paper is organized as follows:In the first chapter,we focus on providing the research background,current situation and contents of keyfitz-kranzer system and macroscopic production model.In the second chapter,we introduce the relevant theorems and definitions applied in this paper.In the third chapter,obviously,the exact solutions of the Riemann problem for a system of keyfitz-kranzer type are obtained in fully explicit forms.Furthermore,the global solutions of the double Riemann problems are also constructed explicitly when the initial data are taken to be three piecewise constant states.During the process of constructing the global solutions,all occurring wave interactions have been dealt with in detail by using the method of characteristics.In addition,it is proved rigorously that the Riemann solutions are stable with respect to the specific small perturbations of the initial data.In the fourth chapter,the constructions of global solutions to the double Riemann problem for a macroscopic production model under the equation of state given by the Chaplygin gas are displayed in completely explicit forms when the initial data are taken to be three piecewise constant states by using the method of characteristics.During the process of constructing global solutions,the interactions of elementary waves are discussed in details.Moreover,it is proved rigorously that Riemannsolutions are stable with respect to the specific small perturbations of initial data.In the fifth chapter,the research contents of this paper are summarized and prospected.