CAMS Colloquium: Sticky Particle Methods for the 1D Euler Alignment System

Monday, August 30 at 3:30pm to 4:30pm

This is a past event.

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.

Event Type

Lecture / Talk / Workshop

Campus

University Park Campus

Department
Mathematics
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