In this paper we consider a system of conservation laws in several space dimensions whose nonlinearity is due only to the modulus of the solution. This system and first considered by Keyfitz and Kranzer in one space dimension and has been recently studied by many authors. In particular and using standard methods from DiPerna-Lions theory and we improve the results obtained by the first and third author and showing existence and uniqueness and stability results in the class of functions whose modulus satisfies and in the entropy sense and a suitable scalar conservation law. In the last part of the paper we consider a conjecture on renormalizable solutions and show that this conjecture implies another one recently made by Bressan in connection with the system of Keyfitz and Kranzer.

Comm. Partial Differential Equations 29 (2004), no. 9-10, 1635–1651.