Definitions of solutions to the IBVP for multiD scalar balance laws

TL;DR

This paper compares four different definitions of solutions for multi-dimensional scalar balance laws in initial-boundary value problems, demonstrating their equivalence and clarifying boundary condition implementations.

Contribution

It generalizes and compares four solution definitions for multiD scalar balance laws, proving their equivalence and elucidating boundary condition treatments.

Findings

All four definitions are equivalent.

Clarified how boundary conditions are satisfied in each definition.

Provided a unified framework for solution concepts.

Abstract

We consider four definitions of solution to the initial-boundary value problem for a scalar balance laws in several space dimensions. These definitions are generalised to the same most general framework and then compared. The first aim of this paper is to detail differences and analogies among them. We focus then on the ways the boundary conditions are fulfilled according to each definition, providing also connections among these various modes. The main result is the proof of the equivalence among the presented definitions of solution.