Existence and uniqueness for the transport of currents by Lipschitz vector fields (Q6119943)

In this article the existence and uniqueness of the solution of the initial value problem to the geometric transport equation \[ \frac{d}{dt}T_t +\mathcal{L}_bT_t=0,\text{ for }t\in(0,1) \] $T_0$ = $\overline{T}$ is investigated. Assuming b is a globally bounded Lipschitz vector field $b:\mathbb{R}^d\rightarrow\mathbb{R}^d$ with flow ${\Phi}_t$ and $\overline{T}\in N_k(\mathbb{R}^d)$ one proves the existence of a unique solution given from the pushforward of the initial current under the flow, $T_t=(\Phi_t)_{\ast}\overline{T}$. The second section of the paper is devoted to the introduction of some notations and basics from linear and multilinear algebra, theory of currents, decomposability bundle and Lipschitz fields. The existence of solutions to the stated initial value problem is proved in the third section and the uniqueness in the fourth section of the article. A simplified proof of uniqueness for the transport of 0-currents is presented in the fifth section.