Existence of Solutions of the Cauchy Problem

Abstract

In this chapter we study the generalization of the front-tracking algorithm to systems of conservation laws, and how this generalization generates a convergent sequence of approximate weak solutions. We shall then proceed to show that the limit is a weak solution. Thus we shall study the initial value problem 6.1(formula presented) where (formula presented) and u 0 is a function in $$L^{1}(\mathbb{R})$$. In doing this, we are in the setting of Lax’s theorem (Theorem 5.17); we have a system of strictly hyperbolic conservation laws, where each characteristic field is either genuinely nonlinear or linearly degenerate, and the initial data are close to a constant. This restriction is necessary, since the Riemann problem may fail to have a solution for initial states far apart, which is analogous to the appearance of a ‘‘vacuum’’ in the solution of the shallow-water equations. The convergence part of the argument follows the traditional method of proving compactness in the context of conservation laws, namely, via Kolmogorov’s compactness theorem or Helly’s theorem. Again, the basic ingredient in front tracking is the solution of Riemann problems, or in this case, the approximate solution of Riemann problems. Therefore, we start by defining these approximations.