Existence and uniqueness for the transport of currents by Lipschitz vector fields

TL;DR

This paper proves the existence and uniqueness of solutions to the geometric transport equation for currents driven by Lipschitz vector fields, utilizing the concept of decomposability bundles, and offers a new proof for measure-based continuity equations.

Contribution

It introduces a novel approach to establish existence and uniqueness for currents under Lipschitz conditions, leveraging decomposability bundles, and provides a new proof for measure-based continuity equations.

Findings

Proves existence and uniqueness of solutions for geometric transport equations.

Utilizes decomposability bundle concept for the analysis.

Provides a new proof for the uniqueness of the continuity equation for signed measures.

Abstract

This work establishes the existence and uniqueness of solutions to the initial-value problem for the geometric transport equation $$ \frac{\mathrm{d}}{\mathrm{d} t}T_t+\mathcal{L}_b T_t=0 $$ in the class of $k$-dimensional integral or normal currents $T_t$ ($t$ being the time variable) under the natural assumption of Lipschitz regularity of the driving vector field $b$. Our argument relies crucially on the notion of decomposability bundle introduced recently by Alberti and Marchese. In the particular case of $0$-currents, this also yields a new proof of the uniqueness for the continuity equation in the class of signed measures.