Singularity formation for the fractional Euler-Alignment system in 1D

TL;DR

This paper investigates the conditions under which singularities form in the one-dimensional fractional Euler-Alignment system with a specific influence function, demonstrating finite-time singularity formation for a range of parameters.

Contribution

It establishes the finite-time singularity formation for the 1D fractional Euler-Alignment system across all relevant influence function parameters, extending previous results.

Findings

Finite-time singularities occur for all 0<α<2 in 1D system.

Singularity formation occurs on both the real line and periodic domains.

The analysis reduces the problem to a nonlocal 1D equation.

Abstract

We study the formation of singularities for the Euler-Alignment system with influence function $\psi=\frac{k_\alpha}{|x|^\alpha}$ in 1D. As in [20] the problem is reduced to the analysis of a nonlocal 1D equation. We show the existence of singularities in finite time for any $\alpha$ in the range $0<\alpha<2$ in both the real line and the periodic case.