Initial Value Problems For Two Classes Of Hyperbolic Systems Of Conservation Laws Without Convexity

In this dissertation,the Riemann problem and initial value problem with threeconstant states for two classes of hyperbolic systems of conservation laws without convexity.Using the skill of convex hull,with the help of characteristic and phase plane analysis,by the vanishing viscosity method,we constructively obtain all the global solution structures involving delta-shock,vacuum state,contact discontinuity and composite wave.A striking feature is that the composite-wave structure that simultaneously includes delta-shock,vacuum state and contact discontinuity develops in solutions.We also study the interactions among of delta-shock,vacuum and contact discontinuity,and show the corresponding criterions.Therefore,a mathematical theory of the delta-shock for the non-convex hyperbolic system of conservation laws is established,including the stability of delta-shock,vacuum and composite-wave solutions under viscous and initial value perturbations.Chapter 1 introduces the research status of delta-shock and the main content and arrangement of the research work in this dissertation.Chapter 2 solves the Riemann problem for a class of decoupled hyperbolic systems of conservation laws without convexity,and obtains three kinds of Riemann solution structures,including the delta-shock,vacuum and composite-wave solutions.Furthermore,the existence of solutions to the viscous perturbation system is established,and then the stability of Riemann solutions of the system under viscous perturbation is proved.Finally,some numerical results are given to confirm the theoretical analysis.Chapter 3 discusses the Riemann problem with three-constant initial data for a class of decoupled non-convex hyperbolic systems of conservation laws.By studying the interactions among of delta-shocks,vacuums,contact discontinuities and composite waves,fourteen different Riemann solution structures involving the delta-shock,vacuum,contact discontinuity and composite wave and corresponding criteria are obtained.Furthermore,it is shown that when the initial perturbation disappears,the constructed solutions tend to the corresponding Riemann solutions of the system with two-constant states,which indicates that the delta-shock,vacuum and composite-wave solutions of the system are stable under initial values perturbation.The theoretical analysis is consistent with the numerical simulation results.Chapter 4 considers the Riemann problem for a class of coupled hyperbolic systems of conservation laws without convexity.Firstly,three types of Riemann solutions including the delta-shock,vacuum and contact discontinuity are constructed.Under the generalized Rankine-Hugoniot condition and entropy condition,the existence and uniqueness of deltashock solution is verified.Secondly,under certain limited initial data,the existence of solutions to the viscous perturbation system is obtained,and then the stability of the delta-shock solution under viscous perturbation is shown.Finally,the numerical results given are in agreement with theoretical analysis.Chapter 5 studies the Riemann problem with three-constant initial data for a class of coupled non-convex hyperbolic systems of conservation laws.Firstly,by analyzing the interactions among of delta-shocks,vacuums and contact discontinuities,five Riemann solution structures including the delta-shock,vacuum,contact discontinuity and composite wave and corresponding criteria are obtained.Secondly,it is proved that when the initial perturbation disappears,the Riemann solutions under the three-constant states tend to the corresponding Riemann solutions of the system under the two-constant states.It implies that the Riemann solutions of the system are stable under the initial data perturbation.Finally,the corresponding numerical simulation results keep consistent with the theoretical analysis.