On $2\times2$ Conservation Laws at a Junction

Abstract

This paper deals with $2\times 2$ conservation laws at a junction. For the Cauchy problem, existence, uniqueness, and Lipschitz continuous dependence of the solution from the initial data as well as from the conditions at the junction are proved. The present construction comprehends the case of the p-system used to describe gas flow in networks and hereby unifies different approaches present in the literature. Furthermore, different models for water networks are considered.