On the role of sharp chains in the transport theorem

TL;DR

This paper develops a generalized transport theorem for irregular domains using Federer's geometric measure theory, representing domains as flat chains and analyzing their evolution via currents and cochains.

Contribution

It introduces a novel framework for the transport theorem involving sharp chains and currents, extending classical results to irregular, convecting domains.

Findings

Domains modeled as flat chains of finite mass.

Continuity of domain evolution in flat norm.

Representation of properties via r-forms.

Abstract

A generalized transport theorem for convecting irregular domains is presented in the setting of Federer's geometric measure theory. A prototypical $r$-dimensional domain is viewed as a flat $r$-chain of finite mass in an open set of an $n$-dimensional Euclidean space. The evolution of such a generalized domain in time is assumed to be in accordance to a bi-Lipschitz type map. The induced curve is shown to be continuous with respect to the flat norm and differential with respect to the sharp norm on currents in $\mathbb{R}^{n}$. A time dependent property is naturally assigned to the evolving region via the action of an $r$-cochain on the current associated with the domain. Applying a representation theorem for cochains the properties are shown to be locally represented by an $r$-form. Using these notions a generalized transport theorem is presented.