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Malte Kampschulte - Gradient flows in the framework of (Cartesian) Currents

Date: 09/12/2014 14:00
Place: Seminární místnost KNM
When considering variational problems and PDEs where the function
only takes values on a fixed manifold, for example magnetization as
a unit-vector field, one often encounters topological singularities
which are nontrivial to handle using standard methods. The framework
of Cartesian Currents as introduced by Giaquinta, Modica and Soucek
is well suited to describe those singularities in static variational
problems. However it would be desirable to use this powerful tool to
also describe the singularity formation in dynamic problems, e.g.
the bubbling in the harmonic map heat flow or vortex annihilation in
micromagnetics. One step in this direction is looking at gradient
flows of Currents. For this, a surprisingly general all-time
existence result can be derived using minimizing movements.
Additionally I will present a generalization of the Wasserstein
distance which seems to be the right a candidate for a possible
metric on (Cartesian) Currents, being sensitive to topological
changes, but still resulting in the correct gradient flow.