A transport theorem for nonconvecting open sets on an embedded manifold

TL;DR

This paper develops a transport theorem for nonconvecting domains on embedded manifolds, allowing irregular boundaries and using geometric measure theory to handle complexities.

Contribution

It introduces a novel transport theorem for nonconvecting domains on manifolds, extending classical results to irregular boundaries and generalized boundary evolution.

Findings

Established a transport theorem for nonconvecting domains on manifolds.

Utilized geometric measure theory to handle Lipschitz boundaries.

Proved the theorem by considering the domain as a fixed set in higher dimension.

Abstract

Most transport theorems---that is, a formula for the rate of change of an integral in which both the integrand and domain of integration depend on time---involve domains that evolve according to a flow map. Such domains are said to be convecting. Here a transport theorem for nonconvecting domains evolving on an embedded manifold is established. While the domain is not convecting, it is assumed that the boundary of the domain does evolve according to a flow map is some generalized sense. The proof relies on considering the evolving set as a fixed set in one higher dimension and then using the divergence theorem. The domains considered can be irregular in the sense that their boundaries need only be Lipschitz. Tools from geometric measure theory are used to deal with this irregularity.