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.
