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

黃博峙(Bo-Chih Huang)

指導教授 ： 洪盟凱

國立中央大學/理學院/數學系/博士(2012年)

摘要

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。

關鍵字

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

並列摘要

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.

並列關鍵字

Hydrodynamic escape problem ； Traffic flow ； Compressible Euler equation ； Geometric singular perturbation ； Nonlinear balance laws

參考文獻

[2] A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J.

[3] S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems.

Ann. of Math. 161 (2005), pp. 223-342.

[4] S. R. Chakravarthy and S. Osher, Numerical experiments with the Osher upwind scheme

[5] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 2nd edition.