On Existence and Admissibility of Singular Solutions for Systems of Conservation Laws

Abstract

A weak notion of solution for systems of conservation laws in one dimension is put forward. In the framework introduced here, it can be shown that the Cauchy problem for any $$n\times n$$ n × n system of conservation laws has a solution. The solution concept is an extension of the notion of singular $$\delta $$ δ -shocks which have been used to provide solutions for Riemann problems in various systems, for example in cases where strict hyperbolicity or the genuine-nonlinearity condition are not satisfied, or in cases where initial conditions have large variation. We also introduce admissibility conditions which eliminate a wide range of unreasonable solutions. Finally, we provide an example from the shallow water system which justifies introduction of $$\delta $$ δ -distributions as a part of solutions to systems of conservation laws.