Existence of weak solutions to a class of nonstrictly hyperbolic conservation laws with non-interacting waves

Anthony J. Kearsley and Andrea M. Reiff

Vol. 205 (2002), No. 1, 153–170

DOI: 10.2140/pjm.2002.205.153

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.

Milestones

Received: 2 February 2000

Revised: 15 November 2001

Published: 1 July 2002