Skeleton for the one-dimensional aggregation equation

TL;DR

This paper investigates the behavior of solutions to the one-dimensional aggregation equation, focusing on the limit measure and support set (skeleton) at blow-up time for initial densities that are characteristic functions.

Contribution

It introduces a detailed analysis of the limit measure and the skeleton set at blow-up time for initial data as characteristic functions, providing new insights into the solution's structure.

Findings

Characterization of the limit measure at blow-up time

Description of the support set (skeleton) for initial characteristic functions

Analysis of the evolution for open or compact initial sets

Abstract

For the aggregation equation in $\mathbb{R}$, we consider the evolution of an initial density corresponding to the characteristic function of some set $\Omega_0$. We study the limit measure at the blow up time 1 for $\Omega_0$ open or compact and we inspect the limit set (skeleton) where this measure is supported.