I'm going to talk about recent work on aggregation models in which the velocity field is a prescribed nonlocal function of the density. Such models arise in population dynamics in which the species undergoes social aggregation based on local sensing mechanisms. I will talk about aspects of the models that lead to rotation, clumping, and dispersal. Then I will focus on the aggregation problem and the role of the regularity of the interaction kernel in determining whether solutions blow up in finite time. Solutions of the equations are shown to behave like a multidimensional, nonlocal analogue of the derivative of Burgers' equation. Numerical solution of finite time singularities involves anomalous “second kind” scaling with weak blowup in an Lp space. However after the intial weak blowup instantaneous mass concentration occurs, with a unique continuation like the entropy solution of Burgers' equation. Information is lost after the blowup due to characteristics impinging on the mass concentration in finite time.
The talk will overview joint work with a number of colleagues including Chad Topaz, Mark Lewis, Thomas Laurent, Jeremy Brandman, Jose Carrillo, Yanghong Huang, and Jesus Rosado.