  We consider 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 represents 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. As a first step, in this paper we study in detail the case of three
curves in the plane concurring at a single triple junction and with the other
ends fixed. We show some results about the existence, uniqueness and, in
particular, the global regularity of the flow, following the line of analysis
carried on in the last years for the evolution by mean curvature of smooth
curves and hypersurfaces.
