Aggregation Equation and Collapse to Singular Measure

Abstract

We are concerned with the dynamics of onefold symmetric patches for the two-dimensional aggregation equation associated with the Newtonian potential. We reformulate a suitable graph model and prove a local well-posedness result in subcritical 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 blowup 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.