在本計畫中，我們探討下列與流體力學相關的主題： 1. 具有 Lipschitz 連續及大規模時間變動因子之通量項與源項的非線性守衡律，其弱解的的整體存在性問題。 2. 幾何奇異擾動在不連續管道中可壓縮流之尤拉方程的應用。 3. 廣義 Buckley-Leverett 方程解的整體存在性及整體行為問題。 4. 具有壓力因子的尤拉-泊桑系統中整體古典解存在的臨界條件問題。關於第一個主題，我們推廣了我們先前的成果，將廣義的格林方法運用在僅具 Lipschitz 連續的通量項及源項的非線性守衡律系統。關於第二的主題，我們將 mollification 和 rescaling 技巧，加上 N. Fenichel 與 P. Szmolyan 發展的架構運用在我們的問題中。關於第三的主題，我們使用對於 BBM 型態方程能量估計法。關於第四個主題，我們推廣了 H. Liu 與 E. Tadmor 的結果並運用在具有壓力因子的尤拉-泊桑系統中，其結果可經由對未知變數做先驗估計來完成。 In this project, we study the following topics arise in fluid dynamics: 1. The global existence of weak solutions to the nonlinear balance laws with Lipschitz continuous flux and source. The flux and source also contain large time oscillating factors. 2. The geometric singular perturbations to compressible Euler equations with discontinuous variable area duct. 3. The global existence and behavior of solutions to the generalized Buckley-Leverett equation. 4. The threshold to the global classical solution of Euler-Poisson system with pressure. For the first topic, we extend the generalized Glimm method in our previous work to Lipschitz continuous flux and source. For the second topic, we apply the mollification and rescaling techniques, together with N. Fenichel and P. Szmolyan’s frameworks to our case. For the third topic, we use the energy estimate method for the BBM type of equations. To the last topic, we extend H. Liu and E. Tadmor’s result to the Euler-Poisson system with pressure. The result can be achieved by a priori estimate for unknowns. 研究期間：10008 ~ 10107
