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Topological and Purely Topological Alignment Dynamics

Trevor M. Leslie, Jan Peszek

TL;DR

The paper studies the Euler Alignment system with topological interaction protocols, focusing on both regular and purely topological kernels. It develops a reformulation in mass coordinates that decouples the velocity equation from a scalar conservation law, enabling rigorous analysis in 1D, including a threshold condition $e_0\ge0$ that guarantees global classical solutions for regular kernels. For singular, purely topological kernels, it establishes well-posedness via mass-distributional and $v$-strong solutions, employing regional fractional Laplacians $\Lambda_{\mathsf{D}}^s$, $\Lambda_{\mathsf{N}}^s$ and corresponding strong solutions of a fractional heat equation. The work proves stability and exponential velocity alignment in both admissible and $v$-strong regimes and proves equivalence between the original Euler system and its reformulation, linking continuum descriptions to mass-transport frameworks. Overall, the results illuminate how purely topological interactions can drive global well-posedness and flocking, while providing a rigorous bridge to mean-field and particle perspectives.

Abstract

We study the Euler Alignment system of collective behavior, equipped with `topological' interaction protocols, which were introduced to the mathematical literature by Shvydkoy and Tadmor. Interactions subject to these protocols may depend on both the Euclidean distance between agents and on the mass distribution between them -- the `topological' component. When the interaction protocol is regular, we prove sufficient conditions for the existence of global-in-time classical solutions, related to the initial nonnegativity of a conserved quantity of the system. The remainder of our results explore the case where the interactions are `purely' topological and the interactions do not depend on the Euclidean distance. We show that in this case, the system decouples into an autonomous velocity equation in mass coordinates together with a scalar conservation law with time-dependent flux determined by the velocity. We analyze the long-time behavior for the dynamics associated to both regular and singular protocols.

Topological and Purely Topological Alignment Dynamics

TL;DR

The paper studies the Euler Alignment system with topological interaction protocols, focusing on both regular and purely topological kernels. It develops a reformulation in mass coordinates that decouples the velocity equation from a scalar conservation law, enabling rigorous analysis in 1D, including a threshold condition that guarantees global classical solutions for regular kernels. For singular, purely topological kernels, it establishes well-posedness via mass-distributional and -strong solutions, employing regional fractional Laplacians , and corresponding strong solutions of a fractional heat equation. The work proves stability and exponential velocity alignment in both admissible and -strong regimes and proves equivalence between the original Euler system and its reformulation, linking continuum descriptions to mass-transport frameworks. Overall, the results illuminate how purely topological interactions can drive global well-posedness and flocking, while providing a rigorous bridge to mean-field and particle perspectives.

Abstract

We study the Euler Alignment system of collective behavior, equipped with `topological' interaction protocols, which were introduced to the mathematical literature by Shvydkoy and Tadmor. Interactions subject to these protocols may depend on both the Euclidean distance between agents and on the mass distribution between them -- the `topological' component. When the interaction protocol is regular, we prove sufficient conditions for the existence of global-in-time classical solutions, related to the initial nonnegativity of a conserved quantity of the system. The remainder of our results explore the case where the interactions are `purely' topological and the interactions do not depend on the Euclidean distance. We show that in this case, the system decouples into an autonomous velocity equation in mass coordinates together with a scalar conservation law with time-dependent flux determined by the velocity. We analyze the long-time behavior for the dynamics associated to both regular and singular protocols.
Paper Structure (21 sections, 10 theorems, 80 equations)

This paper contains 21 sections, 10 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. The Euler Alignment System
  3. Flocking Dynamics and the Role of Topological Distance
  4. Well-posedness and the Role of Dimension 1
  5. Purely Topological Protocols and Reformulation of the System
  6. Outline of the Paper and Statement of Results
  7. A Threshold Condition for Global Existence of Classical Solutions
  8. Types of Solutions
  9. Well-posedness and Focus on the $v$-equation
  10. Stability, Uniqueness and Long-Time Behavior
  11. Global-in-Time Classical Solutions for Regular Topological Protocols
  12. Weak/Mass-Distributional, $v$-Strong, and Admissible Solutions
  13. Solutions of the $(v,M)$ System
  14. Solutions of the 1D Euler Alignment System
  15. Equivalence for Admissible Solutions
  16. ...and 6 more sections

Key Result

Theorem 1.1

Consider the Euler Alignment system e:EA in one spatial dimension, with communication protocol of the form e:topokernel, where $\varphi = \varphi(\!\, \mathrm{d},z)$ is Lipschitz and nonincreasing in $\, \mathrm{d}$, as well as even and bounded in $z$. Assume $\rho_0, u_0\in C^1(\mathbb{R})$ and def If $\psi_0$ is nondecreasing, then there exists a unique global-in-time classical solution associat

Theorems & Definitions (39)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 2.1
  • Remark 2.2
  • Definition 3.1
  • Definition 3.2
  • ...and 29 more