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

Abstract

We study the formation of singularities for the Euler-alignment system with influence function ψ = k α | x | 1 + α \psi =\frac {k_\alpha }{|x|^{1+\alpha }} in 1D. As in [Commun. Math. Sci. 17 (2019), pp. 1779–1794] the problem is reduced to the analysis of a nonlocal 1D equation. We show the existence of singularities in finite time for any α \alpha in the range 0 > α > 2 0>\alpha >2 in both the real line and the periodic case and with just a point of vacuum.