Stability Of Riemann Solutions For Two Classes Of Systems Of Conservation Laws In Geometrical Optics

The stability of Riemann solutions for two classes of systems of conservation laws in geometrical optics is studied in this dissertation.With the vanishing viscosity approach and initial data perturbation method,the stability of delta shock,vacuum and complex wave solutions to the decoupled system of conservation laws in geometrical optics is firstly established.Then with the aid of characteristic analysis method,the Riemann problem for the coupled system of conservation laws in geometrical optics is solved,and three kinds of interesting vacuums and delta-shock solutions are obtained.Next,under certain assumptions of initial data,the stability of delta shock solution is proved under viscous perturbations.Further,the stability of the delta shock and vacuum solutions for initial data perturbation is also proved.Chapter 1 presents research status of systems of conservation laws in geometrical optics and the research work of this dissertation.Chapter 2 first briefly recalls the Riemann solutions including delta shock,vacuum and composite wave solutions to the decoupled system of conservation laws in geometrical optics.Then the existence of solutions to the viscous perturbation system is established.Finally,it is shown that the Riemann solutions to the decoupled system are stable under viscous perturbation.Chapter 3 solves the Riemann problem for the coupled system of conservation laws in geometrical optics.Three kinds of Riemann solutions are obtained constructively,in which there are two kinds of solution consisting of contact discontinuities and vacuums,the another is delta-shock solution.Under the generalized RankineHugoniot relations and two kinds of entropy conditions,the existence and uniqueness of delta-shock solution is proved.Furthermore,under certain assumptions of initial data,the existence of solutions to the viscous regularization system is established,and the stability of the delta-shock solution is shown under viscous perturbations.The theoretical analysis is tested accurately by the numerical results.Chapter 4 considers the perturbed Riemann problem for the decoupled system of conservation laws in geometrical optics with three pieces of constant initial data.By studying the interactions among of delta shocks,vacuums,and contact discontinuities,the Riemann solutions including delta shocks,vacuums and composite waves are obtained.Furthermore,it is proved that as the perturbation parameter goes to zero,the constructed solutions to the system with three pieces of constant initial data tend to the Riemann solutions of the system with two piecewise constants.This indicates that all the delta shocks,vacuums,and composite waves solutions are stable under initial data perturbation.The theoretical analysis is consistent with the numerical results.Chapter 5 discusses the perturbed Riemann problem for the coupled system of conservation laws in geometrical optics with three pieces of constant initial data.By studying the interactions among of delta shocks,vacuums and contact discontinuities,the Riemann solutions involving delta shocks and vacuums are obtained.Then,it is proved that as the perturbation parameter tends to zero,the constructed solutions for the system with three pieces of constant initial data converge to the Riemann solutions for the system with two piecewise constants.This shows that the Riemann solutions are structurally stable under the initial data perturbation.The theoretical analysis is in agreement with the numerical results.