  We study the pressureless Euler equations with nonlocal alignment
interactions, which arises as a macroscopic representation of complex
biological systems modeling animal flocks. For such Euler-Alignment system with
bounded interactions, a critical threshold phenomenon is proved by Tadmor-Tan
in 2014, where global regularity depends on initial data. With strongly
singular interactions, global regularity is obtained by Do-Kiselev-Ryzhik-Tan
in 2018, for all initial data. We consider the remaining case when the
interaction is weakly singular. We show a critical threshold, similar to the
system with bounded interaction. However, different global behaviors may happen
for critical initial data, which reveals the unique structure of the weakly
singular alignment operator.

Analysis of PDEs