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

Arnaiz, Victor; Castro, Ángel

doi:10.1090/tran/8228

Singularity formation for the fractional Euler-alignment system in 1D
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by Victor Arnaiz and Ángel Castro PDF

Trans. Amer. Math. Soc. 374 (2021), 487-514 Request permission

Abstract:

We study the formation of singularities for the Euler-alignment system with influence function $\psi =\frac {k_\alpha }{|x|^{1+\alpha }}$ 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 $\alpha$ in the range $0<\alpha <2$ in both the real line and the periodic case and with just a point of vacuum.

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