Published Paper

Inserted: 23 dec 2025
Last Updated: 23 dec 2025

Journal: J. Funct. Anal.
Volume: 286
Year: 2024
Doi: https://dx.doi.org/10.1016/j.jfa.2024.110315

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.