Transport of currents and geometric Rademacher-type theorems

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 and usually boundaryless $k$-currents $t \mapsto T_t$ in an ambient space $\mathbb {R}^d$ (or a subset thereof). Here, $b_t = b(t,\cdot )$ is the driving vector field and $\mathcal {L}_{b_t} T_t$ is the Lie derivative of $T_t$ with respect to $b_t$ (introduced in this work via duality). 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 thus 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 of a heavily singular evolution, which is nevertheless Lipschitz-regular in time.