cherubino

Transport of currents – Filip Rindler (University of Warwick)

Venue

Aula Seminari

Abstract

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

ddtTt+LbtTt=0

for a time-indexed family of integral or normal k-currents tTt in the ambient space Rd. Here, Lbt denotes the Lie derivative with respect to the vector field bt, 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=d1). 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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