Existence of weak solutions to the elastic string equations in three dimensions

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. We examine the equations modeling an elastic string of infinite length in three-dimensional space, restricted to possess non-simple eigenvalues of constant multiplicity. We show that there exists a weak solution of the nonstrictly hyperbolic conservation law when the total variation of the initial data is sufficiently small. The proof technique is similar to Glimm's classical existence for hyperbolic conservation laws, but necessarily departs from Glimm's proof by not requiring strict hyperbolicity.