非線性雙曲型平衡律解的存在性及漸進穩定性

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姓名

周世偉(Shih-Wei Chou) 查詢紙本館藏

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論文名稱

非線性雙曲型平衡律解的存在性及漸進穩定性
(Global Existence and Time Asymptotic Stability of Solutions to Nonlinear Hyperbolic Systems of Balance Laws)

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摘要(中)

在這篇論文裡，我們研究具有隨著時間而震盪的通量項及來源項的非線性平衡律，此系統應用於許多著名的方程，如描述氣體在周期性變化管子內的運動行為，行駛中的車輛切換車道時的動作，淺水波與河床交互作用的關係。而處理此類方程是結合了我們處理通量項及來源項的新方法與一般化的格林方法，不但證明了解的存在性，更進一步找出了熵條件去證明滿足為熵解，利用了推廣的格林方法與拉克斯方法，我們處理波的交互作用能更加精準，亦證明其穩定性。最後也給了一些超音波與次音波間的作用關係，及道路寬所影響的傳遞波之行為。

摘要(英)

We study the Cauchy problem for general nonlinear hyperbolic balance laws assuming time-oscillating fluxes and sources. Such nonlinear balance laws arise in, for instance, the nozzle flows of gas dynamics with time periodic ducts, traffic models incorporating lane changing effects model and shallow water equations with time-dependent river’s bottom. The global existence of weak solutions is established by a new version of the generalized Glimm method which incorporates asymptotic expansions of the fluxes and sources. We prove existence of weak solutions and demonstrate that they are indeed entropy solutions satisfying the entropy inequality. The approximate solutions of the perturbed Riemann problem, the building blocks of the generalized Glimm scheme, are constructed by a modified Lax method, and a generalized version of the wave interaction estimates are provided for the proof of stability. The consistency of the scheme is established by proving the weak convergence of the residuals and source terms, thereby extending the methods introduced in [12].

關鍵字(中)

★ 柯西問題
★ 守恆律
★ 熵解
★ 非線性
★ 格林方法
★ 雙曲型

關鍵字(英)

★ cauchy problem
★ conservation laws
★ entropy solutions
★ nonlinear
★ Glimm scheme.
★ hyperbolic

論文目次

1 Introduction
1.1 Nonlinear Balance Laws
1
1
2 The Cauchy Problem for Systems of Nonlinear Balance Laws 17
2.1 Perturbed Riemann problem and error estimate . . . . . . . . . . . . . . . . . . 17
2.2 Generalized Glimm scheme and wave interaction estimate . . . . . . . . . . . . . 24
2.3 月veak convergence and global existence theorem . . . . . . . . . . . . . . . . . . 31
3 Gas dynamical model of transonic flows 39
3.1 The Generalized Glimm Scheme for Gas dynamical model ............ 39
3.2 Convergence of the GGS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 42
3.2.1 Wave interaction estimates . . . . . . . . . . . . . . . . . . . . . . . . .. 42
3.2.2 The stability of the GGS . . . . . . . . . . . . . . . . . . . . . . . . . .. 46
3.3 Applications...................................... 49
4 Hyperbolic Balance Laws with time-oscillation fluxes and sources 53
4.1 Approximate Riemann solver and its error estimate . . . . . . . . . . . . . . .. 53
4.2 Generalized Glimm scheme and wave interaction estimate . . . . . . . . . . . .. 62
4.3 Weak convergence and global existence theorem . . . . . . . . . . . . . . . . .. 72
5 Traveling Wave Solutions to Nonlinear Balance Laws in Traffic Flows 81
5.1 Tr、aveling wave solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.2 Reformulation of the stability problem . . . . . . . . . . . . . . . . . . . . . .. 92
5.3 Energy estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 95
5.4 Conclusions 102
Bibliography 104

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指導教授

洪盟凱(John M. Hong)

審核日期

2013-5-30

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