Gradient flows and a generalized Wasserstein distance in the space of Cartesian currents
Kampschulte, Malte; Melcher, Christof Erich (Thesis advisor); Westdickenberg, Michael (Thesis advisor); Jerrard, Robert L. (Thesis advisor)
Aachen (2017, 2018)
Dissertation / PhD Thesis
Dissertation, RWTH Aachen University, 2017
The aim of this thesis is to explore gradient flows in the spaces of currents in general and Cartesian currents (As introduced by Giaquinta, Modica and Sou\v cek) as an important special case. In both cases the focus is on currents without a boundary, however results can be easily extended to the general case. This thesis is split into two parts. The first part is devoted to finding a right choice of a metric for gradient flows, which turns out to be a generalized version of the Wasserstein-distance. The main idea here is to use an analogon of the Benamou-Brenier formulations of optimal transport. To achieve this, first of all, we define a proper notion of transport for currents, in the form of a Lie-derivative. Instead of the conservation of mass, this will lead to a conservation of multiplicity. We then proceed to define two families length-functionals for curves of currents and the resulting generalized Wasserstein-distances, one for the general case and one specifically for Cartesian currents, both parametrized by the exponent $p$. In the case of $p=1$, we show that this generalized distance is in fact equivalent to the flat-metric for currents. For this we use the helpful concept that a curve of $k$-currents of finite $1$-length has a well defined trace in the form of a $k+1$-current. We also show that for Cartesian currents, the newly defined distance results in a form of geometric $L^p$-distance. We then end this part with a discussion of the problem of lower semicontinuity and the correct way to extend previous results to the case of currents with boundary.The second part then details a possible approach for gradient flows of currents using the minimizing movement scheme. Here we show that the powerful compactness properties of currents not only guarantee convergence of the scheme under very mild coercivity and semicontinuity assumptions, but also result in a limit in form of a space-time current. We also discuss various possible improvements to the scheme, as well as connections with the generalized Wasserstein distance. Finally we remark on some possible applications.