Mathematical Aspects of Collective Dynamics:
Kinetic Description and Fractional Diffusion

Compressible fluids, entropy hierarchies, and flocking

Theodore Drivas

Princeton University

We will discuss two types of one-dimensional compressible fluid equations; Navier-Stokes models with local dissipation in which the viscosity depends degenerately on the density and nonlocal models for collective dynamics which exhibit flocking behavior. For the local models, we prove large data global regularity for a class of equations covering viscous shallow water. Another result proves a conjecture of Peter Constantin on singularity formation for a model describing slender axisymmetric fluid jets. For the non-local models, we establish a continuation criterion which says that smooth solutions exist so long as no vacuum states form. The method of proof involves introducing a hierarchy of entropies to control the solution in terms of the minimum density. We also show than any weak solutions which obeying an entropy inequality exhibit flocking. This reports on joint work with P. Constantin, H. Nguyen, F. Pasqualotto and R. Shvydkoy.