本計劃討論有關非線性守恆律整體解的存在性以及其性質。本計劃的第一部分主要考慮有關具有大規模時間變動源項的非線性守恆律之弱解。例如，氣體動力學中週期管道流的尤拉方程，關於其柯西問題弱整體解的存在性仍然是未解決的問題。在本計畫中，我們嘗試利用稱為廣義格林方法來達到其目的。我們需要對方程式中的通量項及源項做漸進展開，其中波交互作用的估計必須更謹慎的處理。這個方法可以視為 Dafermos 對非耗散系統結果的推廣。如果可能的話，我們也會試圖將此方法推廣到積分-微分系統，例如，尤拉-泊桑方程。至於本計畫的第二部分，我們運用幾何奇異擾動的技巧去研究幾個由交通流量以及燃燒流體模型導出的非線性守衡律其解的漸進行為。我們考慮改良自 H. M. Zhang 交通流量模型的多通道的交通流量問題。在燃燒流體問題中，我們將其結果推廣至片段光滑的源項狀況。我們並試圖推廣 P. Szmolyan 和 M. Wechselberger 的 canard 狀況至更一般的情形。 In this project we study the global existence and behavior of solutions to the general nonlinear balance laws. In the first topic we deal with the weak solutions of general nonlinear balance laws with large time variation sources. For example, compressible Euler equations with time-periodic area duct in gas dynamics. The global existence of weak solutions for the Cauchy problem of such systems is still remained open. In this project, we will try to apply the so called generalzed Glimm method to achieve this goal. The asymptotic expansion of flux and source term is used. The waves interaction estimates are needed to be re-derived in more careful way.This method can be considered as the extension of Dafermos’s results to the non-dissipative systems. If possible, we will extend the results to the integro-differential systems, for example, Euler-Poisson equations. In the second topic we use the technique of geometric singular perturbations to study the asymptotic behavior of solutions for several nonlinear balance laws arise from traffic flow and combustion flow models. We deal with the traffic flow model with multiple lanes, which can be considered as the modified traffic model derived by H. M. Zhang. In the combustion model, we extend our results to the systems with piecewise smooth sources. We try to extend the results of P. Szmolyan and M. Wechselberger in canard cases to more general systems. 研究期間：9908 ~ 10007
