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姓名 黃博峙(Bo-Chih Huang)  查詢紙本館藏   畢業系所 數學系
論文名稱 透過幾何奇異攝動理論探討非線性平衡律駐波解的存在性
(Geometric Singular Perturbation Approach to Stationary Wave Solutions for Viscous Nonlinear Balance Laws)
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摘要(中) 在這篇論文中,我們考慮黏性管道流氣體通過不連續噴嘴、黏性交通流模型問題、以及大氣流體散逸問題等非線性雙曲平衡律正則化方程解的漸近行為。透過動態系統理論的方法,我們可以將此類穩定態問題轉化成奇異攝動問題。我們藉由分析不同尺度的系統來構造奇異駐波解。根據幾何奇異攝動理論,我們能證明原方程確實存在一個駐波解伴隨在奇異駐波解的附近。對於一些特殊的退化奇異解,我們利用更進階的幾何奇異攝動理論來證明這些退化解在小擾動時仍然保持其解的結構。此外,在黏性管道流問題中,我們提供了一個新的熵條件來確保駐波解的唯一性。而在黏性交通流問題中,我們則是針對駐波解的穩定性做了討論。
摘要(英) In this dissertation we consider the asymptotic behavior of solutions for regularized equations to some nonlinear hyperbolic balance laws arising from the following topics: the viscous gas flow through discontinuous nozzle, viscous traffic flow model, and the atmosphere hydrodynamic escape model. Through the dynamical system theory approach, we can transfer our steady-state problem into a singularly perturbed problem. By analyzing the system in different scales, we are able to construct the singular stationary wave solutions. By using the technique of geometric singular perturbations, we can show there exist true stationary solutions for our problems shadowing the singular stationary wave solutions. For some special degenerate singular solutions, we apply more advanced theory from geometric singular perturbation to prove the persistence of these solutions under the perturbation. Moreover, in the first topics, we introduce a new entropy condition to ensure the uniqueness of the stationary solutions, and in the second topics, we also analyze the stability of stationary wave solutions.
關鍵字(中) ★ 流體散逸問題
★ 交通流
★ 可壓縮尤拉方程
★ 幾何奇異攝動
★ 非線性平衡律
關鍵字(英) ★ Hydrodynamic escape problem
★ Traffic flow
★ Compressible Euler equation
★ Geometric singular perturbation
★ Nonlinear balance laws
論文目次 1 Introduction…………………………………………………1
2 Geometric singular perturbation theory…………………………………………………11
2.1 Background and motivation…………………………………………………11
2.2 Invariant manifold theorems…………………………………………………12
2.3 Geometric singular perturbation theory beyond normal hyperbolicity………16
3 The generalized stationary waves for viscous gas flow through a discontinuous cross section………………………………………………………………………………19
3.1 The inviscid steady-states for the mollified system – with continuous cross section……………………………………………………………………………19
3.1.1 Properties of limiting slow system………………………………………20
3.1.2 Properties of limiting fast system…………………………………………22
3.1.3 Stationary waves for expanding and continuous nozzles…………………23
3.1.4 Stationary waves for contracting and continuous nozzles………………25
3.2 The inviscid steady-states for expanding and discontinuous nozzles…………26
3.3 The inviscid steady-states for contracting and discontinuous nozzles………34
4 The existence and instability of stationary waves for viscous traffic flow models...39
4.1 GSP formulation for steady-state problem……………………………………39
4.2 Properties of limiting systems…………………………………………………40
4.2.1 Limiting slow system……………………………………………………41
4.2.2 Limiting fast system………………………………………………………45
4.2.3 Jump curves of limiting slow orbits………………………………………47
4.3 Transversality of curves intersection…………………………………………48
4.4 Classification of inviscid and viscous stationary waves………………………52
4.5 Instability of supersonic smooth stationary waves……………………………56
4.5.1 Eigenvalue problem for stationary waves………………………………56
4.5.2 Center manifold reduction………………………………………………57
5 The stationary waves for hydrodynamic escape model……………………………61
5.1 GSP formulation for hydrodynamic steady-state problem……………………61
5.2 Properties of limiting systems…………………………………………………63
5.2.1 Limiting slow system……………………………………………………64
5.2.2 Limiting fast system………………………………………………………67
5.3 Transversality of curves intersections…………………………………………69
5.4 Dynamics near T for ν>0
5.5 Classification of singular stationary wave solutions and their profiles withν>0…………………………………………………………………………………72
5.5.1 Case 1. B_hinmathcal{Z}_0^s$ and
B_{har}inmathcal{Z}_0^r………………72
5.5.2 Case 2. B_hinmathcal{Z}_0^s$ and
B_{har}inmathcal{Z}_0^s………………73
5.5.3 Case 3. B_hinmathcal{Z}_0^r$ and
B_{har}inmathcal{Z}_0^r………………75
5.5.4 Case 4. B_hinmathcal{Z}_0^r$ and
B_{har}inmathcal{Z}_0^s………………75
References……………………………………………………………………………83
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指導教授 洪盟凱(John M. Hong) 審核日期 2012-7-5
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