Dynamics of one-fold symmetric patches for the aggregation equation and collapse to singular measure

We are concerned with the dynamics of one fold symmetric patches for the two-dimensional aggregation equation associated to the Newtonian potential. We reformulate a suitable graph model and prove a local well-posedness result in sub-critical and critical spaces. The global existence is obtained only for small initial data using a weak damping property hidden in the velocity terms. This allows to analyze the concentration phenomenon of the aggregation patches near the blow up time. In particular, we prove that the patch collapses to a collection of disjoint segments and we provide a description of the singular measure through a careful study of the asymptotic behavior of the graph.
Comment: 59 pages