在本篇論文中，我們探討了在旋轉效應下的非線性雙曲型平衡律，以及討論關於一維平衡律的黎曼問題。此非線性平衡律可以被轉換成沒有源項的系統，但是通量是一個未知及時間的函數。我們利用漸進展開的方法找出此黎曼問題的逼近解。接著，我們拓展此結果去探討伴隨科氏力作用的二維淺水波方程。我們介紹一些轉換方法利用其解對稱的特性，將二維的系統轉換成一維的系統。

In this thesis we study the nonlinear hyperbolic systems of balance laws with rotational effect. The Riemann problem for one-dimensional balance laws is considered. The nonlinear balance laws is transformed into a system without source, but the flux is a function of unknowns and time. The approximate solution of the Riemann problem is constructed by the technique of asymptotic expansion. We extend the results to the two-dimensional shallow water equations with Coriolis force. Some transformations are introduced to transform the two-dimensional system into an one-dimensional system due to the symmetry of solutions.

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