The Riemann Problem For One-dimensional Nonlinear Nonstrictly Hyperbolic Conservation Laws

In this paper, we are concerned about the Riemann problem for two types of one-dimensional nonlinear nonstrictly hyperbolic conservation laws.In the first two chapter, we state the background of the problems and our results, then introduce some basic concepts and general theories of one-dimensional hyperbolic conservation laws.In the third chapter, we study the Riemann problem for the nonlinear degenerate wave equation. Using the Liu-entropy conditions we analysis the basic waves detailed and construct a global solution of the Riemann problem. Then we construct the approximate solution for the Cauchy problem, verify the consistent estimate of the approximate solution and the Hloc-1compactness of the strong entropies and the entropy flows.In the chapter4, we study the Riemann problem for a system of nonlinear elasticity in a crystalline polymer model. The system contains a hyperbolic-elliptic situation and is hyperbolic degenerated in another region. We learn from the previous methods about the Riemann problem for the system of conservation laws of mixed type(such as the Van der Waals gas equation), use the Liu-entropy condition to analysis the properties of all the basic waves. Finally, we construct the Riemann solutions based on76kinds of different situations of the initial values.