Global Existence of Classical Solutions for Nonisentropic Gas Flows through Divergent Ducts

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王士銘(Shih-Ming Wang) 查詢紙本館藏

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數學系

論文名稱

(Global Existence of Classical Solutions for Nonisentropic Gas Flows through Divergent Ducts)

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至系統瀏覽論文 (2024-9-1以後開放)

摘要(中)

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。

摘要(英)

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.

關鍵字(中)

★ 守恆律
★ 平衡律
★ 歐拉方程
★ 全域存在性

關鍵字(英)

論文目次

摘要i
Abstract ii
Contents iii
Symbols iv
1 Introduction 1
2 Local existence theorem and Riccati equations 4
2.1 Uniform a priori estimates of Riemann invariants. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Riccati equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Existence and uniqueness of the global classical solution 13
3.1 Initial and boundary conditions of global classical solutions. . . . . . . . . . . . . . . . . 13
Bibliography 17

參考文獻

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(2023), 19-23.
[2] A. Douglis, Existence theorem for hyperbolic systems. Comm. Pure Appl. Math., 5 (1952), 119-154.
[3] T. T. Li, Global Solutions for Quasilineaar Hyperbolic Systems, Wiley, New York, 1994.
[4] T. T. Li and W. C. Yu, Boundary Value Problems for Quasilinear Hyperbolic Systems, Duke University Mathematics Series, V. Durham, NC 27706, Duke University, Mathematics
Department. X, 1985.
[5] L. W. Lin, H. X. Liu and T. Yang, Existence of globally bounded continuous solutions for nonisentropic gas dynamics equations, J. Math. Anal. Appl., 209, (1997), 492-506.
[6] T. Nishida, Nonlinear hyperbolic equations and related topics in fluid dynamics, D´epartment de Math´ematique, Universit´e de Paris-Sud, Orsay, 1978, Publications
Math´ematiques d’Orsay, No. 78-02.
[7] M. Yamaguti and T. Nishida, On some global solution for the quasilinear hyperbolic equations, Funkcial. Ekvac., 11 (1968), 51-57.

指導教授

洪盟凱

審核日期

2023-7-3

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