A chain rule formula in BV and applications to conservation laws

TL;DR

This paper establishes a new chain rule formula for BV functions and applies it to handle discontinuous flux in one-dimensional conservation laws, advancing the mathematical understanding of such systems.

Contribution

It introduces a novel chain rule formula for BV functions and applies it to conservation laws with discontinuous flux, providing an intrinsic analytical approach.

Findings

New chain rule formula for BV functions derived.

Application to conservation laws with discontinuous flux.

Enhanced mathematical tools for analyzing conservation laws.

Abstract

In this paper we prove a new chain rule formula for the distributional derivative of the composite function $v(x)=B(x,u(x))$, where $u:]a,b[\to\R^d$ has bounded variation, $B(x,\cdot)$ is continuously differentiable and $B(\cdot,u)$ has bounded variation. We propose an application of this formula in order to deal in an intrinsic way with the discontinuous flux appearing in conservation laws in one space variable.