Rigorous Estimates on Balance Laws in Bounded Domains

TL;DR

This paper establishes the well-posedness of initial-boundary value problems for general balance laws in bounded domains, including existence, uniqueness, and regularity of entropy solutions.

Contribution

It provides a rigorous and detailed proof of well-posedness for balance laws with boundary conditions, extending previous foundational work.

Findings

Existence of entropy solutions for balance laws in bounded domains

Uniqueness of these solutions under given boundary conditions

Lipschitz continuity of solutions over time

Abstract

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.