Maximum and minimum generalized entropy solutions to the Cauchy problem for a first-order quasilinear equation

The existence of the maximum and minimum generalized entropy solutions of the Cauchy problem for a first-order quasilinear equation is proved in the general case of a flux vector that is merely continuous, when the uniqueness property of a generalized entropy solution does not necessarily hold. Some useful applications are presented. In particular, the uniqueness of the generalized entropy solution is established for input data that are periodic with respect to $n-1$ linearly independent space vectors ($n$ is the number of space variables).