  Existence and uniqueness of global in time measure solution for the
multidimensional aggregation equation is analyzed. Such a system can be written
as a continuity equation with a velocity field computed through a
self-consistent interaction potential. In Carrillo et al. (Duke Math J (2011)),
a well-posedness theory based on the geometric approach of gradient flows in
measure metric spaces has been developed for mildly singular potentials at the
origin under the basic assumption of being lambda-convex. We propose here an
alternative method using classical tools from PDEs. We show the existence of a
characteristic flow based on Filippov's theory of discontinuous dynamical
systems such that the weak measure solution is the pushforward measure with
this flow. Uniqueness is obtained thanks to a contraction argument in transport
distances using the lambda-convexity of the potential. Moreover, we show the
equivalence of this solution with the gradient flow solution. Finally, we show
the convergence of a numerical scheme for general measure solutions in this
framework allowing for the simulation of solutions for initial smooth densities
after their first blow-up time in Lp-norms.
