The Weak Solution Property For One Dimensional Hydrodynamics Models

In this paper, we investigate two classes hydrodynamics models-the one dimensional gas dynamic models and the traffic flow models.In chapter 1 and 2, we represent the background of the proposed problem, state the main conclusions and introduce the basic theory of the nonlinear hyperbolic conservation laws.In chapter 3, we compare two Riemann solution constructions of isentropic and non-isentropic gas dynamic equations, with the same initial values. We obtain that when rarefaction waves occur in the non-isentropic solution, the same family rarefaction waves occur in the isentropic solu-tion, however, when advancing or reflected shock waves occur in the isentropic case, the same kind shock waves occur in the non-isentropic case. Moreover, by numerical calculation, it can be found that the wave intensities of the two solutions are quite close. Our conclusion reflects that almost non-isentropic gas dynamic process can be simulated by isentropic gas dynamic process in practice.In chapter 4, a traffic flow model is established based on the "car following" principle with a maximal constraint on the density-velocity relationship. The model develops the Aw-Rascle(A-R) model and amends some "nonphysical" features. Moreover, we construct the solutions of Riemann problem for the model. The Riemann solutions provide a more reasonable invariant region and show the phase transition phenomena.In chapter 5, we study the existence of the invariant region about one-dimensional non-isentropic gas dynamic equations to investigate the uniform boundedness of its weak solution. By consider the Lax-Friedrichs scheme, we prove that there does not exist any bounded invariant region for one-dimensional non-isentropic gas dynamic equations with ideal and polytropic gas state. Besides, we obtain a necessary condition about the state equation to determine the existence of bound invariant region and propose a mathematical example which shows that with special state equation, the bounded invariant region maybe exist.In chapter 6, we state the further research plan of these problems.