Extended Divergence-Measure Fields, the Gauss-Green Formula, and Cauchy Fluxes

TL;DR

This paper extends the Gauss-Green formula and Cauchy flux concepts to measure-valued fields, providing a generalized framework for balance laws in continuum physics that encompasses discontinuous and measure-based fields.

Contribution

It establishes the Gauss-Green formula for extended divergence-measure fields and generalizes the Cauchy flux, connecting measure-theoretic fields with physical balance laws.

Findings

Gauss-Green formula proven for extended divergence-measure fields.

Normal trace represented as limit over approximating sets.

Balance law extended to measure-based production and flux.

Abstract

We establish the Gauss-Green formula for extended divergence-measure fields (i.e., vector-valued measures whose distributional divergences are Radon measures) over open sets. We prove that, for almost every open set, the normal trace is a measure supported on the boundary of the set. Moreover, for any open set, we provide a representation of the normal trace of the field over the boundary of the open set as the limit of measure-valued normal traces over the boundaries of approximating sets. Furthermore, using this theory, we extend the balance law from classical continuum physics to a general framework in which the production on any open set is measured with a Radon measure and the associated Cauchy flux is bounded by a Radon measure concentrated on the boundary of the set. We prove that there exists an extended divergence-measure field such that the Cauchy flux can be recovered through the field, locally on almost every open set and globally on every open set. Our results generalize the classical Cauchy's Theorem (that is only valid for continuous vector fields) and extend the previous formulations of the Cauchy flux (that generate vector fields within $L^{p}$). Thereby, we establish the equivalence between entropy solutions of the multidimensional nonlinear partial differential equations of divergence form and of the mathematical formulation of physical balance laws via the Cauchy flux through the constitutive relations in the axiomatic foundation of Continuum Physics.