Monday, August 30 at 3:30pm to 4:30pm
Kaprielian Hall (KAP), 414
3620 South Vermont Avenue, Los Angeles, CA 90089
Trevor Leslie, USC
Abstract: The Euler Alignment system is a hydrodynamic version of the celebrated Cucker--Smale ODE's of collective behavior. It can have a hyperbolic or parabolic character, depending on the specified nonlocal interaction protocol; this talk concerns the hyperbolic case in 1D. It is well-established that solutions may lose regularity in finite time, but it has been unknown until recently how to continue to evolve the dynamics after a blowup. After brief orientation on the special structure of these equations, I will describe a recent joint work with Changhui Tan (University of South Carolina), where we developed a theory of weak solutions. Inspired by Brenier and Grenier's work on the pressureless Euler equations, we show that the dynamics of our system are captured by a nonlocal scalar balance law. We generate the unique entropy solution of a discretization of this balance law by introducing the 'sticky particle Cucker--Smale' system to track the shock locations. Our approximation scheme for the density converges in the Wasserstein metric; it does so with a quantifiable rate as long as the initial velocity is at least Hölder continuous.