Transport, Fluids and Mixing

Moving currents: on the Lie transport equation and a Rademacher type theorem

speaker: Paolo Bonicatto (University of Warwick)

abstract: In the classical theory, given a vector field $b \colon 0,T \times \mathbb R^d \to \mathbb R^d$, one usually studies the transportcontinuity equation drifted by $b$ looking for solutions in the class of functions (with certain integrability) or at most in the class of measures. In this seminar I will talk about recent efforts, motivated by the modeling of defects in plastic materials, aimed at extending the previous theory to the case when the unknown are instead k-currents in $\mathbb R^d$, i.e. generalised $k$-dimensional surfaces. The resulting equation involves the Lie derivative $L_b$ of currents in direction $b$ and reads $\partial_t T_t + L_b T_t = 0$. It is easily seen that the continuity equation corresponds to the case of 0-currents, while the transport equation to the case of d-currents. I will explain the main challenges this problem presents and some recent results based on an ongoing project with Giacomo Del Nin and Filip Rindler (University of Warwick).

timetable:
Wed 26 Jan, 15:20 - 15:50,
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