| Abstract: |
| This talk presents a new perspective on the one-dimensional pressureless Euler-alignment system, which can be seen as a hydrodynamic limit of the Cucker--Smale model for collective behaviour.
While previous analyses of the Cucker--Smale model typically allowed particle trajectories to cross, recent works by Leslie and Tan suggest that such solutions are not suitable for hydrodynamic limits, and instead advocate for sticky particle dynamics.
Building on this perspective, we characterise the sticky particle Cucker--Smale dynamics as an $L^2$-gradient flow of a convex functional, providing a new variational framework for the system.
This approach not only recovers the unique entropy solution of the associated scalar balance law in Leslie and Tan's framework, but also clarifies how (non-)monotonicity properties of the model's so-called \textit{natural velocities} determine cluster formation.
Our results place the Euler-alignment system within a rigorous gradient flow framework in one dimension, already established for the pressureless Euler and Euler--Poisson systems. |
|