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Formation of multiple flocks in a simple nonlocal aggregation model

Carrie Clark

Abstract

We consider a family of interaction energies given by kernels having a ''well-barrier'' shape, and investigate how these kernels drive the formation of multiple flocks within a larger population. We use the dichotomy case of the concentration compactness principle to obtain an energy minimizer, even though minimizing sequences are expected to break apart into flocks which may move far away from one another.

Formation of multiple flocks in a simple nonlocal aggregation model

Abstract

We consider a family of interaction energies given by kernels having a ''well-barrier'' shape, and investigate how these kernels drive the formation of multiple flocks within a larger population. We use the dichotomy case of the concentration compactness principle to obtain an energy minimizer, even though minimizing sequences are expected to break apart into flocks which may move far away from one another.

Paper Structure

This paper contains 12 sections, 16 theorems, 92 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Statement of Main Results
  3. Preliminaries on shape of minimizers
  4. Balls are minimal for small mass
  5. Minimizers have compact support
  6. Subadditivity
  7. Proof of Main Results
  8. Separation Lemma
  9. Proof of Theorem for Compactly supported kernels
  10. Proof of Theorem \ref{['thm:inf is many balls']}
  11. Linear Growth
  12. Size of droplets

Key Result

Theorem 1.1

Let $K$ be a bounded kernel which satisfies $\mathbf{(K1)-(K5)}$. Further, assume $K$ is strictly increasing on $[0,a]$. Then for any $m>0,$ there exists some $k\in \mathbb{N}$ and radii $r_1, \dots, r_k >0$ such that

Figures (1)

  • Figure 1: Sketch of a "well-barrier" type kernel. The parameters $d$ and $w$ represent the depth and width of the attractive well, and the parameters $h$ and $W$ represent the height and width of the repulsive barrier.

Theorems & Definitions (31)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3: $L^1$ local minimizers have compact support
  • proof
  • Lemma 2.4: Subadditivity of the minimal energy
  • proof
  • Lemma 3.1
  • ...and 21 more