Abstract:
We propose a hydrodynamic Cucker-Smale type flocking model with external potential forcing, which describes the collective behavior of continuum of agents under alignment interaction and external forcing. The external potential force tends to compete with the Cucker-Smale interaction, which makes the long time behavior very different from the original Cucker-Smale model and far more interesting. We discover that the harmonic potential is a good prototype of the external potential, for which we prove that smooth solution must flock at exponential rate and converge to a traveling Dirac delta function, using hypocoercivity. When the Cucker-Smale interaction is strong enough, we prove the same conclusion for general strictly convex potentials that can be bounded above and below by harmonic potentials. For possibly non-convex potentials that can be bounded above and below by harmonic potentials, we prove thresholds for the existence of global smooth solutions for one or two space dimensions. It is interesting to see that for both one and two space dimensions, global smoothness of solution can be guaranteed for a class of initial data, even without the knowledge of long time flocking behavior. |