  This paper concerns periodic solutions for a 1D-model with nonlocal velocity
given by the periodic Hilbert transform. There is a rich literature showing
that this model presents singular behavior of solutions via numerics and
mathematical approaches. For instance, they can blow up by forming
mass-concentration. We develop a global well-posedness theory for periodic
measure initial data that allows, in particular, to analyze how the model
evolves from those singularities. Our results are based on periodic mass
transport theory and the abstract gradient flow theory in metric spaces
developed by Ambrosio et al. [2]. A viscous version of the model is also
analyzed and inviscid limit properties are obtained.
