| We consider the following system of conservation laws with source terms: Ut+FU x=a′GU Ux,0 =U0x or at=0u t+fa,ux=a ′ga,u a,ux,0 =a0x ,u0x In the part I of the thesis we will study systems in the strictly hyperbolic case. For the systems without source terms, the modern theory of Riemann problem was originally put forward by Peter Lax, and the first definition and analysis of the Cauchy problem was given after this by James Glimm. Glimm's analysis is based on the work of Lax in that it employs Lax's solution of the Riemann problem in constructing approximate solutions for Cauchy problem. For our system, we introduce a reformulation of the source term a ′G that enables us to extend the method introduced by Lax to construct general solutions of the Riemann problem. This enables us to correspondly extend Glimm's method to prove the existence of the solution of the Cauchy problem when general source term a′G are present under the assumption that the variable a is a Lipschitz-continuous function of x. Our generalized solution of the Riemann problem is “weaker than weak” in the sense that it is weaker than a distribution solution. Thus we prove that a weak solution of the Cauchy problem is the limit of a sequence of Glimm scheme approximate solution that are based on “weaker than weak” solutions of the Riemann problem. Since our approximate scheme employs only a pointwise approximate of a(x), it follows that a′G( U) does not converge strongly. However, we show that it converges weakly, by oscillation, when a is Lipschitz-continuous. This is enough to demonstrate convergence of the residual. We interpret this result as extending Glimm's method to inhomogenous systems.; In the second part of the thesis, we will focus on the scalar nonlinear balance law of the nonstrictly hyperbolic case which has been studied by Isaacson and Temple. By adding an extra equation, at=0, we have a 2 x 2 resonant system of conservation laws. Because of the coinciding of the eigenvalues for the system in some domain, the resonant situation happens. And the oscillation of the approximate solution constructed by Glimm scheme causes the blow-up of the total variation, which puts the solution beyond the range of standard compactness argument developed by Oleinik and Glimm. Here we give a sharp estimate for the maximum possible total variation of the solution at time t, in terms of a new norm defined on the initial data. This norm is infinite precisely when the total variation in the approximate solution is unbounded. This provides the proof that the total variation of the conserved quantities is uniformly bounded in time for a resonant nonlinear system. Our global bound on the total variation in the presence of nonlinear resonant is a purely nonlinear affect, because, for the linearized equation, the total... |