On the chain rule formulas for divergences and applications to conservation laws

TL;DR

This paper establishes a chain rule formula for divergence of composite functions involving divergence-measure fields and BV functions, and applies it to prove uniqueness in scalar conservation laws with discontinuous flux.

Contribution

It introduces a nonautonomous chain rule formula for divergence of composite functions involving divergence-measure fields and BV functions, with applications to conservation laws.

Findings

Proved a chain rule formula for divergence of composite functions.

Established a uniqueness result for scalar conservation laws with discontinuous flux.

Extended the mathematical framework for divergence and conservation law analysis.

Abstract

In this paper we prove a nonautonomous chain rule formula for the distributional divergence of the composite function $\boldsymbol{v}(x)=\boldsymbol{B}(x,u(x))$, where $\boldsymbol{B}(\cdot,t)$ is a divergence--measure vector field and $u$ is a function of bounded variation. As an application, we prove a uniqueness result for scalar conservation laws with discontinuous flux.