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

TL;DR

This paper studies the evolution and collapse of symmetric patches in a 2D aggregation model, showing how they form singular measures and describing their asymptotic behavior near blow-up.

Contribution

It introduces a graph-based reformulation and proves local well-posedness, along with global existence for small data, and characterizes the singular measure resulting from collapse.

Findings

Patch collapses to disjoint segments near blow-up

Global existence for small initial data

Asymptotic analysis of the singular measure

Abstract

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.