Mathematical Aspects of Collective Dynamics:
Kinetic Description and Fractional Diffusion

Regularity and Long-Time Behavior for Hydrodynamic Flocking Models

Trevor Leslie

University of Wisconsin-Madison

The focus of this talk is two-fold: First, we will consider wellposedness theory for low regularity solutions of the Fractional Euler Alignment (FEA) system on the 1D torus. Next, we will discuss recent joint work with Roman Shvydkoy; the latter is concerned with models closely related to the FEA model, but with possibly local kernels (including the 'topological kernels' recently considered by Shvydkoy and Tadmor). We will show that for large times, the deviation from a uniform flock can be controlled by an auxiliary quantity that depends only on the initial data.