臺灣博碩士論文加值系統

在本論文中，我們提供廣義的Glimm scheme來研究具有source項的2×2雙曲守恆律系統初始邊界值問題之整域弱解的存在性。由於source項的結構，我們推廣在[10,13]中所創造的方法來構造黎曼與邊界黎曼兩問題的弱解，而這樣的弱解正好能藉由Glimm scheme來做為構成近似解的要素。藉著修正在[7]的結果及證明residual的弱收斂，我們證實了scheme的相容性與穩定性。此外我們也研究擬線性波方程類之初始邊界值問題的整域Lipschitz連續解的存在性。應用Lax的方法及廣義Glimm方法，我們造出靠近邊界的初始邊界黎曼問題和遠離邊界的擾動離曼問題的近似解，經由證明近似解之residual的弱收斂，我們證實解的導數之整域存在性，進而得到問題的整域Lipschitz連續解的存在性。

In this article we provide a generalized version of Glimm scheme to study the global existence of weak solutions to the initial-boundary value problem of 2 by 2 hyperbolic systems of conservation laws with source terms. Due to the structure of source terms, we extend the methods invented in [10,13] to construct the weak solutions of
Riemann and boundary Riemann problems, which can be dopted as a building block of the approximate solution by Glimm scheme. By modifying the results in [7] and showing the weak convergence of residuals, we establish the stability and consistency of scheme. In addition we investigate the existence of globally Lipschitz continuous solutions to a class of initial-boundary value problem of quasilinear wave equations. Applying the Lax method and generalized Glimm scheme, we construct the approximate solutions of initial-boundary Riemann problem near the boundary and perturbed Riemann problem away the boundary. By showing the weak convergence of residuals for the approximate solutions, we establish the global existence for the
derivatives of solutions and obtain the existence of global
Lipschitz continuous solutions of the problem.

Abstract i
List of Figures iii
1 Introduction 1
2 The Initial Value Problem 9
2.1 Perturbed Riemann problems for (1.1.4) . . . . . . . . . . . . . . . . . 9
2.2 Riemann and perturbed Riemann problems for (1.1.3) . . . . . . . . . . 17
3 Boundary Riemann Problem 22
3.1 Boundary Riemann Problem for (1.1.4) . . . . . . . . . . . . . . . . . . 22
3.2 Boundary Riemann problem for (1.1.3) . . . . . . . . . . . . . . . . . . 28
4 Generalized Glimm Scheme and Stability of Scheme 31
4.1 Generalized Glimm Scheme to (1.1.4) . . . . . . . . . . . . . . . . . . . 31
4.2 Generalized Glimm scheme to (1.1.3) and its stability . . . . . . . . . . 38
5 Weak Convergence of the Residuals 50
5.1 Weak Convergence of the Residuals for (1.1.4) . . . . . . . . . . . . . . 50
5.2 Weak Convergence of the Residuals for (1.1.3) . . . . . . . . . . . . . . 55
6 Application to Quasilinear Wave Equations 62
References 68
ii

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