在這個計劃裡，我們研讀共振雙曲線型守恆律科西問題弱解的存在性及行為。此類共振雙曲線型守恆律最典型的範例就是管道流中的可壓縮尤拉方程，我們都知道當流體速度充分靠近音速時，此類弱解的存在性仍然尚未解決。故研讀此類問題最主要的困難在於在解裡面的波有相同的波速，當我們使用Glimm差分法時無法去控制逼近解裡的全變動量(total variation)，也就是說我們沒有辦法得到逼近解的收斂性。在這計劃裡我們最主要是要研讀管道流中的可壓縮尤拉方程。此類系統是共振的最主要的原因是我們把管道的截面積當成是一個未知函數，如此一來我們得到一個3x3的系統，其中一個特徵域的速度是零。當波速很靠近音速時，共振現象產生。在這裡我們用另一類方法來建造逼近解，此類逼近解是由擾動黎曼問題的解加上一個線性非守恆系統的解。當我們用此方法時波交互作用時的全變動量(total variation)的估計是非常重要的。我們推廣Glimm文章中波交互作用估計的結果在我們的系統裡，當初始條件充分靠近音速時，可得到逼近解的全變動量(total variation)在所有的時間內都是一致有界。如此一來經由Helly selection principle，我們就可以得到逼近解的收斂性。最後我們引用之前的結果來計算逼近解的餘項，並證明逼近解的餘項趨近於零當網格大小趨近於零，如此一來我們就得到共振系統弱解的廣域存在。 ; In this project we study the existence and behavior of global weak solutions to the Cauchy problem of resonant hyperbolic balance laws. The most interesting case of resonant hyperbolic balance laws is the compressible Euler equations of variable area duct in transonic flow. It is well known that the global existence result for the weak solutions of general resonant balance laws still remained open. The main difficulty of studying this type of systems is that the waves have the same speed so that the total variation of the approximate solutions constructed by the Glimm scheme cannot be controlled, which means that we cannot obtain the convergence of the approximate solutions. We will focus on the compressible Euler equations in transonic flow. The reason that the system is resonant is that we treat the cross section of the duct as an unknown function, it creates a stationary field. Thus, resonance occurs when the speed of wave is close to the sound speed. In this project we will give an alternative way to construct our approximate solutions. The approximate solutions consists of so called perturbed Riemann solutions plus the perturbation solved by a non-conservative linear hyperbolic system. The estimation of the wave interaction plays an important role in this framework. We will extend the results by Glimm for the wave interactions to more general case so that the total variation of approximate solution are uniformly bounded for all time when initial data is sufficiently close to sonic state. Therefore, by Helly selection principle, we can establish the convergence of the approximate solution. In the end, we will apply the results by Hong, or Chang-Hong-Hsu to compute the residual of the Cauchy problem and establish the global existence of the weak solutions to resonant systems. ; 研究期間 9708 ~ 9807
