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 <\/p>\n\n\n\n
$$\\frac{d}{dt} T_t + \\mathcal{L}_{b_t} T_t = 0 $$<\/p>\n\n\n\n
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. <\/p>\n\n\n\n
This is joint work with Paolo Bonicatto and Giacomo Del Nin.<\/p>\n\n\n\n
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The transport of singular structures, such as vortex lines\/sheets in fluids, topological singularities in magnetism, or dislocation lines in plastic…<\/p>\n