Flocking Hydrodynamics with External Potentials

We study the large-time behavior of a 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 the 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 quadratic potentials, we are able to treat a large class of admissible interaction kernels, ϕ ( r ) ≳ ( 1 + r 2 ) - β with ‘thin’ tails β ⩽ 1 —thinner than the usual ‘fat-tail’ kernels encountered in CS flocking β ⩽ 1 2 ; we discover unconditional flocking with exponential convergence of velocities and positions towards a Dirac mass traveling as harmonic oscillator. For general convex potentials, we impose a stability condition, requiring a large enough alignment kernel to avoid crowd scattering. We then prove, by hypocoercivity arguments, that both the velocities and positions of a 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.