*preprint*

**Inserted:** 24 oct 2022

**Year:** 2022

**Abstract:**

The transport of many kinds of singular structures in a medium, such as
vortex points*lines*sheets in fluids, dislocation loops in crystalline plastic
solids, or topological singularities in magnetism, can be expressed in terms of
the geometric (Lie) transport equation \[
\frac{\mathrm{d}}{\mathrm{d} t} T_t + \mathcal{L}_{b_t} T_t = 0 \] for a
time-indexed family of integral or normal $k$-currents $t \mapsto T_t$ in
$\mathbb{R}^d$. Here, $b_t$ is the driving vector field and $\mathcal{L}_{b_t}
T_t$ is the Lie derivative of $T_t$ with respect to $b_t$. Written in
coordinates for different values of $k$, this PDE encompasses the classical
transport equation ($k = d$), the continuity equation ($k = 0$), as well as the
equations for the transport of dislocation lines in crystals ($k = 1$) and
membranes in liquids ($k =d-1$). The top-dimensional and bottom-dimensional
cases have received a great deal of attention in connection with the
DiPerna--Lions and Ambrosio theories of Regular Lagrangian Flows. On the other
hand, very little is rigorously known at present in the
intermediate-dimensional cases. This work develops the theory of the geometric
transport equation for arbitrary $k$ and in the case of boundaryless currents
$T_t$, covering in particular existence and uniqueness of solutions, structure
theorems, rectifiability, and a number of Rademacher-type differentiability
results. The latter yield, given an absolutely continuous (in time) path $t
\mapsto T_t$, the existence almost everywhere of a ''geometric derivative'',
namely a driving vector field $b_t$. This subtle question turns out to be
intimately related to the critical set of the evolution, a new notion
introduced in this work, which is closely related to Sard's theorem and
concerns singularities that are ''smeared out in time''. Our differentiability
results are sharp, which we demonstrate through an explicit example.