在本篇研究中，我們將探討氣體通過擴張的管子在怎樣的條件下會擁有全域的經典解。該問題可以用拉格朗日坐標下的完全可壓縮歐拉方程式的初始邊界值問題來描述，當在此方程式應用黎曼不變量時，這可以被視為一個雙曲平衡律系統。我們將在熵和擴張的管子及初始和邊界數值的適當條件下，證明經典解的全域存在定理。此定理主要依賴於局部存在定理和兩個黎曼不變量的均勻性估計，而後者需要引入廣義的Lax 轉換所得出的Riccati 方程式，並從中推論出全域經典解的存在。

In this study, we investigate the global existence of classical solutions for gas flows through a divergent duct. This problem can be described as an initial-boundary value
problem for the full compressible Euler equations with the geometric source in Lagrangian coordinates, which can be viewed as a hyperbolic system of balance laws when the Riemann invariants are applied to the equations. We prove the global existence theorem for classical solutions under appropriate conditions on entropies, divergent ducts, and initial and boundary values. This theorem mainly depends on the local existence theorem and uniform a priori estimates on two Riemann invariants, which are obtained by introducing generalized Lax transformations.

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