Global Weak Solutions to the Cauchy Problem ofNonlinear Balance Laws

• Main Authors: Yuan Chang, 張淵
• Other Authors: John M. Hong
• Format: Others
• Language: en_US
• Published: 2008
• Online Access: http://ndltd.ncl.edu.tw/handle/79227012584072387846

博士 === 國立中央大學 === 數學研究所 === 96 === 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.