Stochastic and Multiscale Modeling and Computation Seminar by Trevor Leslie - More Basics of the Euler Alignment System: 1D Classical Solutions and the Heavy-Tail Condition for Flocking

Speaker: Trevor Leslie, Illinois Institute of Technology

Title: More Basics of the Euler Alignment System: 1D Classical Solutions and the Heavy-tail Condition for Flocking

Abstract:

Last time, starting from a kinetic version of the Cucker--Smale
ODE's, we derived the Eulerian and Lagrangian formulations of the
pressureless Euler Alignment system.  This time, we show how both of
these formulations simplify drastically in 1 spatial dimension, due to
the appearance of additional conserved quantities.  These quantities
furthermore lead naturally to an essentially sharp criterion for the
existence of classical solutions [Carrillo-Choi-Tadmor-Tan 2016,
Leslie 2020].  Next, we describe the now well-known "heavy-tail"
criterion which guarantees (in any spatial dimension) exponentially
fast convergence to a flock.  If time allows, we will give the proof
of this statement, which boils down to finding an appropriate Lyapunov
function.  The argument we present is essentially due to [Ha-Liu 2009]
but has been optimized by other authors.

 

Stochastic and Multiscale Modeling and Computation Seminar