Abstract

Many applied problems resulting in hyperbolic conservation laws are nonstrictly hyperbolic. As of yet, there is no comprehensive theory to describe the solutions of these systems. In this paper, a proof of existence is given for a class of nonstrictly hyperbolic conservation laws using a proof technique first applied by Glimm to systems of strictly hyperbolic conservation laws. We show that Glimm’s scheme can be used to construct a subsequence converging to a weak solution. This paper necessarily departs from previous work in showing the existence of a convergent subsequence. A novel functional, shown to be equivalent to the total variation norm, is defined according to wave interactions. These interactions can be bounded without any assumptions of strict hyperbolicity.

Authors