Geometric singularities for solutions of single conservation laws

Abstract: "In this paper we describe the geometric framework for the study of generation and propagation of shock waves in R[superscript n] appearing in weak solutions of scalar conservation laws. We first define the notion of geometric solutions for scalar conservation laws in the framework of one-parameter unfoldings of immersions. The geometric solutions are, in general, multi-valued and they are constructed by the method of characteristics. We use singularity theory techniques to classify the generic types of multi-valuedness of the geometric solutions. Such a classification is used to construct the unique entropy solution of the scalar conservation law by selecting the proper single-valued discontinuous branch of the geometric solution satisfying the entropy condition across the discontinuity."