Transport of currents – Filip Rindler (University of Warwick)


Aula Seminari


The transport of singular structures, such as vortex lines/sheets in fluids, topological singularities in magnetism, or dislocation lines in plastic solids, can all be seen as fundamentally governed by the geometric (Lie) transport equation

$$\frac{d}{dt} T_t + \mathcal{L}_{b_t} T_t = 0 $$

for a time-indexed family of integral or normal $k$-currents $t \mapsto T_t$ in the ambient space $\mathbb{R}^d$. Here, $\mathcal{L}_{b_t}$ denotes the Lie derivative with respect to the vector field $b_t$, defined by duality. Written in coordinates, this PDE encompasses the classical transport equation ($k = d$), the continuity equation ($k = 0$), the equation for the transport of lines ($k = 1$), and the advection of membranes ($k = d-1$). This talk will report on recent progress on the analysis of this equation for arbitrary $k$, covering in particular existence and uniqueness of solutions, structure theorems, rectifiability, and Rademacher-type differentiability results.

This is joint work with Paolo Bonicatto and Giacomo Del Nin.

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