  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.
