Interaction Of The Delta Shock Waves And Asymptotic Limits Of Riemann Solutions In Pressureless Hydrodynamic Model

In this paper,two aspects of basic research are mainly carried out on the homogeneous situation of the pressureless hydrodynamic model.On the one hand,we solve the Riemann problem for the homogeneous situation of the model by using phase plane analysis method.On this basis,the interaction problem between the Dirac shock waves(delta shock waves)and the element waves is studied when the initial condition is made up of three piecewide constant states,and the global perturbation Riemann solution of the model is constructed.Finally,the stability of the model under small perturbation of the initial data is verified.On the other hand,the Riemann problem of the homogeneous situation of the model with the perturbed traffic pressure term is carefully discussed by using the phase plane,and the formation of delta shock wave and vacuum state of the model is further studied emphatically when the pertubed parameter goes to 0.Therfore,the article arranged as following.The first chapter,we briefly describe the relevant background of the pressureless hydrodynamic model and the general arrangement of the entire paper.The second chapter mainly describes the concepts of hyperbolic conservation law,which lays a theoretical foundation for the equations studied in this paper.The third chapter focuses on the Riemann problem of the homogeneous situation of the pressureless hydrodynamic model.More precisely,the Riemann solution contains not only delta shock wave,but also vacuum state.Delta shock wave can be solved explicitly with the aid of the delta Rankine–Hugoniot relations united with the over-compressive delta entropy inequality.In addition,the interaction problem between the delta shock waves and the element waves is further discussed comprehensively.Finally,the stability of the system is verified when the perturbed parameter approaches0.The fourth chapter introduced systematically studied the asymptotic limit behavior of Riemann solution of the homogeneous situation of the pressureless hydrodynamic model with the perturbed traffic pressure term.More specifically,there exist four different combinations between 1-rarefaction/1-shock and 2-rarefaction/2-shock waves for our Riemann solutions of the system.On this basis,we study in detail the asymptotic behavior of the Riemann solutions of the system when the pertubed parameter goes to0.Finally,in order to better verify the formation of delta shock wave and vacuum state when the pertubed parameter goes to 0,we select appropriate initial values and give some representative numerical results.The fifth chapter,we briefly summarize the work of the entire paper and point out the direction of future work.