有關非線性平衡定律之柯西問題的廣域弱解｜國立中央大學博碩士論文系統

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研究生：	張淵 Yuan Chang
論文名稱：	有關非線性平衡定律之柯西問題的廣域弱解 Global Weak Solutions to the Cauchy Problem ofNonlinear Balance Laws
指導教授：	洪盟凱 John M. Hong
口試委員:
學位類別：	博士 Doctor
系所名稱：	理學院 - 數學系 Department of Mathematics
畢業學年度：	96
語文別：	英文
論文頁數：	77
中文關鍵詞：	廣義化Glimm方法、Cauchy問題、平衡定律之雙曲線系統、擬線性波方程、Lax方法、特徵方法、擾動Riemann問題、Riemann問題、非線性平衡定律、守恒定律
外文關鍵詞：	Conservation laws, Lax''s method, Characteristic method, Perturbed Riemann problems, Riemann problems, Quasilinear wave equations, Hyperbolic systems of balance laws, Cauchy problem, Nonlinear balance laws, Generalized Glimm''s method
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本篇論文分為兩個部分。部分Ⅰ在探討對於具有奇異來源項的純量平衡定律的Riemann問題的Lax型解的存在性與唯一性。部分Ⅱ在探討對於擬線性波方程類的廣域Lipschitz連續解。
在部分Ⅰ我們對於純量非線性平衡定律的Riemann問題給予構造廣義化熵解的新途徑。方程式的來源項為奇異的,因其為δ函數與不連續函數的乘積。將來源項重新公式化地闡述,我們研究對應的擾動Riemann問題。擾動Riemann解的存在性與穩定性被建立,且Riemann問題的廣義化熵解被構造,其為對應的擾動Riemann解的極限。廣義化熵解的自我相似性亦得到,使得Lax方法可被擴展至具有奇異來源項的純量非線性平衡定律。
在部分Ⅱ我們對於擬線性波方程的Cauchy問題類研究廣域Lipschitz連續解的存在性。應用Lax方法與廣義化Glimm方法,我們構造對應的擾動Riemann問題的近似解且建立解的導數的廣域存在性。那麼,經由對於方程式的來源項證明其殘數為弱收歛,可完成廣域Lipschitz連續解的存在性。

This thesis is divided into two parts. The part I is: Existence and Uniqueness of Lax-Type Solutions to the Riemann Problem of Scalar Balance Law with Singular Source Term,and the part II is: Globally Lipschitz Continuous Solutions to a Class of Quasilinear Wave Equations.
In the part I of the thesis we give a new approach of constructing the generalized entropy solutions to the Riemann problem of scalar nonlinear balance laws. The source term of equation is singular in the sense that it is a product of delta function and a discontinuous function. By re-formulating the source term, we study the corresponding perturbed Riemann problem. The existence and stability of perturbed Riemann solutions is established, and the generalized entropy solutions of Riemann problem are constructed as the limit of corresponding perturbed Riemann solutions. The self-similarity of generalized entropy solutions is also obtained so that Lax''s method can be extended to the scalar nonlinear balance laws with singular source terms.
In the part II of the thesis we investigate the existence of globally Lipschitz continuous solutions to a class of Cauchy problem of quasilinear wave equations. Applying the Lax''s method and generalized Glimm''s method, we construct the approximate solutions of the corresponding perturbed Riemann problem and establish the global existence for the derivatives of solutions. Then, the existence of global Lipschitz continuous solutions can be carried out by showing the weak convergence of residuals for the source term of equation.
Keywords. Conservation laws; Nonlinear balance laws; Riemann problems; Perturbed Riemann problems; Characteristic method; Lax''s method; Quasilinear wave equations; Hyperbolic systems of balance laws; Cauchy problem; Generalized Glimm''s method.

1 Introduction 													  1
I Existence and Uniqueness of Lax-Type Solutions to the
Riemann Problem of Scalar Balance Law with Singular
Source Term														 10
2 The Classical Solutions of the Perturbed Riemann Problem 				 11
2.1 The Global Classical Solutions of the Perturbed Riemann Problem
(2.1)-(2.4) by the Characteristic Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.1 xε Issued from ¡0L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 13
2.1.2 xε Issued from ¡0 ε . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .. . 17
2.1.3 xε Issued from ¡0R . . . . . . . . . . . . . . .. . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . 20
2.2 Behavior of the Perturbed Riemann Solutions (2.45) as ε → 0 .. . . . . . . . . 21
2.3 The Classical Solutions of the Perturbed Riemann Problem with Dis-
continuous Initial Data by the Characteristic Method . . . . . . . . . . . . . . . . . . . . . . 23
2.3.1 xε Issued from ¡L 0ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23
2.3.2 xε Issued from ¡R 0ε . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . .. . . . . . . . . . . 24
3 The Shock Waves of the Riemann and Perturbed Riemann Problems		 29
3.1 The ShockWaves of the Perturbed Riemann Problem (2.1),(2.3),(2.4) and
(3.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29
3.2 Behavior of the Solutions of Perturbed Riemann Problem (2.1),(2.3),(2.4)
and (3.1) as ε → 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4 Stability of the Perturbed Riemann Solutions and the Generalized Entropy
Solutions                          							     34
4.1 An Example Demonstrating the Non-Stability of Perturbed Riemann
Solutions with respect to the General Profile of ε(x)  . . . . . . . . . . . . . . . . . . . . . . 34
4.2 Stability of the Perturbed Riemann Solutions with respect to Some Class of
{aε(x)} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 38
4.3 Uniqueness of the Generalized Entropy Solution of Lax-Type to the Riemann Problem (1.1)-(1.3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 45
II Globally Lipschitz Continuous Solutions to a Class of
Quasilinear Wave Equations                                        47
5 The Riemann and Perturbed Riemann Problems                         48
5.1 The Approximate Solution of the Perturbed Riemann Problem (5.6) . . . . . . . 50
5.2 The Approximate Weak Solution of the Riemann Problem (1.16) and (5.1) . . 55
6 The Generalized Glimm’s Scheme and Its Stability 						 60
6.1 The Generalized Glimm’s Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.2 Stability of the Generalized Glimm’s Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 62
7 Weak Convergence of the Residuals     				                 66
7.1 The Estimations of the Residuals of (uε θ",△x,Uεθ",△x) . . . . . . . . . . . . 66
7.2 Existence of a Weak Solution for the Cauchy Problem(1.16) . . . . . . . . . . . . . .71
Bibliography                                           			 74

                                

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