本計畫最主要研究兩個議題：第一個議題研究了平衡律的共振雙曲系統合成波的全時存在與行為。完全共振雙曲系統具有此特性：其通量的雅可比矩陣的所有特徵值在整個相域中重合。我們舉出一個弱解（含真空）的例子來說明一些完全共振系統的經典Riemann問題，並表明解此系統Cauchy問題時，將自我相似的Riemann解作為其採取 Glimm方案中所需的建構模塊 (building block) 並不合適。相反地，我們利用正則化的黎曼數據發明了廣義Riemann問題。這種廣義黎曼問題的弱解由合成雙曲波所組成，它們是非線性雙曲波和接觸不連續的組合。這種合成波具有合理的全變差，因此可以應用廣義Glimm方案來建立完全共振系統的弱解的全局存在。對於我們系統的Cauchy問題的廣義Glimm方案，我們加上一個合理的C-F-L條件，為方案的穩定性提供了修正波相互作用的估計。本文的結果表明，解的共振提供了奇異性的效果，不能與黎曼數據的奇異性相結合。這意味著具有合成雙曲波的廣義Riemann解提供了更廣泛的Glimm方案到完全共振系統的構造模塊。第二個議題我們考慮一個多車道交通流模型，它由雙曲線平衡定律系統所決定。 該平衡定律系統是一個2×2非線性雙曲系統且具有不連續的源項。通過新的廣義Glimm方案建立了該多車道模型Cauchy問題的熵解的全局存在性。 Riemann問題的廣義解是透過廣義Glimm方案的構建模塊，以及Lax方法和擾動的發明來建構，該方法利用修正的源項來求解線性化雙曲方程。估計殘差是為了廣義Glimm方案的一致性。波之相互作用估計被提供用於Glimm泛函的衰減和解的漸近行為的結果。 ;In this project we deal with the following two topics. The first one is to study the global in time existence and behavior of composite waves of resonant hyperbolic systems of balance laws. The entirely resonant hyperbolic systems have the property that all the eigenvalues of Jacobian matrix of a flux are coincided in the whole phase domain. We give an example of a weak solution with vacuum for the classical Riemann problem of some entirely resonant system to indicate that the self-similar Riemann solutions are not an appropriate building block of Glimm scheme to the Cauchy problem of such systems. Instead, we invent a generalized Riemann problem with regularized Riemann data. The weak solutions of such generalized Riemann problem consists of constant states separated by the composite hyperbolic wave, which are the combination of nonlinear hyperbolic waves and contact discontinuities. Such composite waves have reasonable values of total variations so that the generalized Glimm scheme can be applied to establish the global existence of weak solutions for the entirely resonant systems. For the generalized Glimm scheme to the Cauchy problem of our system, we impose a justified C-F-L condition. Modified wave interaction estimates are provided for the stability of scheme. The results of this paper indicate that the resonance of solutions provides an effect of singularity which cannot be coupled with the singularity of Riemann data. It means that the generalized Riemann solutions with composite hyperbolic waves provide a more appropriate building block for the generalized Glimm scheme to entirely resonant system.Secondly, we consider a multilane model of traffic flow, which is governed by a hyperbolic system of balance laws. The system of balance laws is given as a 2 by 2 nonlinear hyperbolic system with a discontinuous source term. The global existence of entropy solutions to the Cauchy problem of this multi-lanes model is established by a new version of the generalized Glimm method. The generalized solutions of the Riemann problem, which is the building block of the generalized Glimm scheme, are constructed by Lax's method and an invention of perturbations solving linearized hyperbolic equations with modified source terms. The residuals are estimated for the consistency of the generalized Glimm scheme. The wave interaction estimates are provided for the decay of Glimm functionals and the result of the asymptotic behavior of solutions.
