Riemann Problems For Some Hyperbolic Systems Of Conservation Laws And The Structural Stability Of Riemann Solutions

In this paper, the Riemann problem for the examples of hyperbolic systems of conservation laws is considered. Firstly, let us consider a non-strictly hyperbolic system of conservation laws. In addition, the delta shock wave will be formed during the construction process of Riemann solution when the initial value satisfies certain conditions. Next, the structural stability of Riemann solutions for a strictly hyperbolic system with three piecewise constant states is considered. That is to study the wave interaction problem under all kinds of situations. At last, we consider the limit relations of Riemann solutions for a class of non-strictly hyperbolic systems, from which we can see the formation process of the delta shock wave. The contents studied in the paper have broader physical background and closely connected with the gas dynamics, water wave theory and chromatography equations etc. It has very important in the theoretical analysis and practical application.The paper can be divided into three parts.In Part 1, we study the Riemann problem for a non-strictly hyperbolic system of conservation laws. Firstly, we can construct the global Riemann solutions by phase plane analysis method when the initial data are taken as two smooth constant states. And the graphs of the solutions are shown. Next, the delta shock wave will be formed during the construction process of Riemann solution when the initial value satisfies certain conditions. And the strength of the delta shock wave is calculated.In Part 2, we study the structural stability of Riemann solutions for a strictly hyperbolic system with three piecewise constant states. Firstly, the Riemann problem of the system is considered by self-similar vanishing viscosity method. And next, let us consider the so-called double Riemann problem which is equal to the Riemann problem for the system with three piecewise initial data. In this situation, we study the problem of wave interactions in order to further study the structural stability of Riemann solutions.In Part 3, we study the limit relations of Riemann solutions for a class of non-strictly hyperbolic systems. Firstly, we construct the global Riemann solutions of the class of system by phase plane analysis method. Then, we study the forming process of delta shock wave by taking limit.