Coarsening in the network flow

A network is a 1-dimensional connected set in the plane (or in the torus) composed of a finite number of curves that meet at their endpoints in junctions. The network flow has been studied in mathematical materials science to model the growths of polycrystals. In the last years several results for the network flow has been obtained, both for weak definitions, and for strong solutions of PDE. A motivation to study this flow is try to mathematically formalize a “coarsening-type” behavior of the flow, that is clear from numerical simulations. In particular, one expects that "generically" the flow with a highly complicated initial datum (for instance with hundreds of grains) would converge to a critical point of the length functional with a much simpler geometric/topological structure. During the seminar I will present supporting arguments to a “coarsening-type” behavior of the flow and I will describe the mathematical tools developed to describe the evolution as accurately as possible.