Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws

TL;DR

This paper develops a rigorous mathematical framework for divergence-measure fields, defining normal traces on sets of finite perimeter, and applies it to balance laws and entropy solutions in hyperbolic conservation laws.

Contribution

It introduces a new approximation theorem for sets of finite perimeter to define normal traces of divergence-measure fields, extending the Gauss-Green theorem to these sets.

Findings

Normal trace of divergence-measure fields can be defined via approximation by smooth boundary sets.

Gauss-Green theorem holds for divergence-measure fields on sets of finite perimeter.

Framework enables derivation of balance laws with measure-valued sources and recovery of entropy fluxes.

Abstract

We analyze a class of weakly differentiable vector fields (\FF \colon \rn \to \rn) with the property that (\FF\in L^{\infty}) and (\div \FF) is a Radon measure. The primary focus of our investigation is to introduce a suitable notion of the normal trace of any divergence-measure field $\FF$ over the boundary of an arbitrary set of finite perimeter, which ensures the validity of the Gauss-Green theorem. To achieve this, we establish a fundamental approximation theorem which states that, given a Radon measure $\mu$ that is absolutely continuous with respect to $\mathcal{H}^{N-1}$ on $\rn$, any set of finite perimeter can be approximated by a family of sets with smooth boundary essentially from the measure-theoretic interior of the set with respect to the measure $\|\mu\|$. With this approximation theorem, we derive the normal trace of $\FF$ on the boundary of any set of finite perimeter, (E), as the limit of the normal traces of $\FF$ on the boundaries of the approximate sets with smooth boundary, so that the Gauss-Green theorem for $\FF$ holds on (E). With these results, we analyze the Cauchy fluxes that are bounded by a Radon measure over any oriented surface (i.e. an $(N-1)$-dimensional surface that is a part of the boundary of a set of finite perimeter) and thereby develop a general mathematical formulation of the physical principle of balance law through the Cauchy flux. Finally, we apply this framework to the derivation of systems of balance laws with measure-valued source terms from the formulation of balance law. This framework also allows the recovery of Cauchy entropy fluxes through the Lax entropy inequality for entropy solutions of hyperbolic conservation laws.