Mathematical Aspects of Collective Dynamics:
Kinetic Description and Fractional Diffusion

On Euler Alignment system with weakly singular interactions

Changhui Tan

University of South Carolina

The Euler Alignment system is a macroscopic model for flocking dynamics. The global behavior varies with different alignment interactions. Bounded alignment operator behaves like a non-local damping, while strongly singular alignment operator behaves like a nonlinear dissipation. I will review recent results regarding the Euler Alignment system in 1D and 2D. I will then focus on the third regime: weakly singular interactions. I will show a sharp critical threshold condition on initial data that distinguishes global regularity and finite time blowup. The condition is different from the one for bounded interactions, due to the unique feature of the weakly singular alignment.