Rigorous Estimates on Balance Laws in Bounded Domains

The initial-boundary value problem for a general balance law in a bounded domain is proved to be well posed. Indeed, we show the existence of an entropy solution, its uniqueness and its Lipschitz continuity as a function of time, of the initial datum and of the boundary datum. The proof follows the general lines in Bardos, Leroux and N\'ed\'elec (1979), striving to provide a rigorous treatment and detailed references.