# Flocking hydrodynamics with external potentials

## TL;DR

This paper analyzes the long-term behavior of a hydrodynamic model for collective agent dynamics influenced by external potentials, revealing conditions for flocking, convergence, and smooth solutions in various potential and kernel settings.

## Contribution

It introduces new results on flocking behavior under convex external potentials, including quadratic cases with thin-tail kernels, and establishes global smooth solutions with stability conditions.

## Key findings

- Unconditional flocking with exponential convergence in quadratic potentials.
- Flocking behavior persists under certain stability conditions for convex potentials.
- Existence of global smooth solutions for low-dimensional cases with critical initial data.

## Abstract

We study the large-time behavior of hydrodynamic model which describes the collective behavior of continuum of agents, driven by pairwise alignment interactions with additional external potential forcing. The external force tends to compete with alignment which makes the large time behavior very different from the original Cucker-Smale (CS) alignment model, and far more interesting. Here we focus on uniformly convex potentials. In the particular case of \emph{quadratic} potentials, we are able to treat a large class of admissible interaction kernels, $\phi(r) \gtrsim (1+r^2)^{-\beta}$ with `thin' tails $\beta \leq 1$ --- thinner than the usual `fat-tail' kernels encountered in CS flocking $\beta\leq\frac{1}{2}$: we discover unconditional flocking with exponential convergence of velocities \emph{and} positions towards a Dirac mass traveling as harmonic oscillator. For general convex potentials, we impose a stability condition, requiring large enough alignment kernel to avoid crowd scattering. We then prove, by hypocoercivity arguments, that both the velocities \emph{and} positions of smooth solution must flock. We also prove the existence of global smooth solutions for one and two space dimensions, subject to critical thresholds in initial configuration space. It is interesting to observe that global smoothness can be guaranteed for sub-critical initial data, independently of the apriori knowledge of large time flocking behavior.