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

Paolo Bonicatto; Giacomo Del Nin; Filip Rindler

paper

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

arXiv:2303.03218

Abstract

This work establishes the existence and uniqueness of solutions to the initial-value problem for the geometric transport equation $\frac{d}{d t} T_{t} + L_{b} T_{t} = 0$ 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.

16 pages