System of kinematical conservation laws (KCL) a review article

TL;DR

This review article discusses the theory of kinematical conservation laws (KCL), which describe the evolution of propagating surfaces with singularities, focusing on identifying conserved densities and fluxes for non-expert readers.

Contribution

It provides a comprehensive review of KCL theory, emphasizing the identification of conserved variables and fluxes, and presents the material in an accessible manner for non-specialists.

Findings

Identification of conserved variable density in KCL

Determination of flux functions for KCL

Explanation of singularities like kinks in propagating surfaces

Abstract

In a wide range of physical phenomena, we find propagating surfaces {\Omega}t which need mathematical treatment. In this article, we review the theory of the system of kinematical conservation laws (KCL), which govern the evolution of these surfaces and have been developed by the second author and his collaborators. KCL are the most general equations in conservation form, governing the evolution of {\Omega}t with physically realistic singularities. A very special type of singularity is a kink, which is a point on {\Omega}t when {\Omega}t is a curve in R2 andisacurveon{\Omega}t when{\Omega}t isasurfaceinR3. Acrossakinkthenormalnto{\Omega}t and the normal velocity m on {\Omega}t are discontinuous. The main aim of this article is to identify density of the conserved variable and the flux for the KCL which we did not do earlier. The presentation of this article is like that in a popular article, which which aims at non-experts in the field.