Inserted: 7 apr 2004
Last Updated: 4 jun 2013
Journal: Progress in Nonlin. Diff. Eqs. and Appl.
This survey describes our project to study the motion by curvature of a network of smooth curves with multiple junctions in the plane, that is, the geometric gradient flow associated to the Length functional.
Such a flow can represent the evolution of a two--dimensional multiphase system where the energy is simply the sum of the lengths of the interfaces, in particular it is a possible model for the growth of grain boundaries.
Moreover, the motion of these networks of curves is the simplest example of curvature flow for sets which are ``essentially'' non regular.
In this paper, we introduce the problem and we present some results and open problems about existence, uniqueness and, in particular, the global regularity of the flow.