Analysis and applications of evolutionary PDEs

Andrea Bertozzi, UCLA Particle laden flow applications from the Gulf oil spill to spiral concentrators This talk is about the challenges in modeling particle laden flow in viscous liquids and some of their applications. Particle settling in liquid is a complex process in which long range interactions are important due to the incompressibility of the fluid. Hinderance in particle settling is well-known as are other multi-particle effects such as shear-induced migration. We look at the basic physics of particle laden flow on an incline and show that this problem has a natural bifurcation between settling-dominated dynamics and dynamics dominated by the shear flow. In both cases the presence of particles greatly changes the flow structure from that of a single clear fluid. We also present some recent work connecting this physics to the modeling of particle separation in a spiral separator—used in the mining industry for separating different densities and sizes of particles in a slurry. Mathematical models include both systems of conservation laws (involving both classical and singular shock solutions) and flows with surface tension. Andrew Bernoff, Harvey Mudd A Primer of Swarm Equilibria We study equilibrium configurations of swarming biological organisms subject to exogenous and pairwise endogenous forces. Equilibrium solutions are extrema of an energy functional, and satisfy a Fredholm integral equation. In one spatial dimension, for a variety of exogenous forces, endogenous forces, and domain configurations, we find exact analytical expressions for the equilibria. The equilibria typically are compactly supported and may contain δ-concentrations or jump discontinuities at the edge of the support. In two dimensions we show that the Morse Potential and other