Existence of BV solutions for $2\times2$ hyperbolic balance laws for $L^\infty$ initial data

TL;DR

This paper proves the existence of bounded variation solutions for a class of 2x2 hyperbolic balance laws with nonlinear flux and variable forces, under small initial data, and explores their qualitative behavior.

Contribution

It establishes the existence of BV solutions for 2x2 hyperbolic balance laws with nonlinear flux and variable forces, extending previous results to more general settings.

Findings

Existence of BV solutions for the system under small initial data.

Analysis of qualitative behavior of entropy solutions.

Applicability to systems with time and space dependent forces.

Abstract

We prove the existence of BV solutions for $2\times 2$ system of hyperbolic balance laws in one space dimension. The flux is assumed to have two genuinely nonlinear characteristic fields. We consider a general force which may possibly depend on time and space variable as well. To prove the existence, we assume the initial data to be small in $L^\infty$. Furthermore, we also study qualitative behavior for entropy solutions to hyperbolic system of balance laws.