A Kinetic Flocking Model with Diffusion

We study the stability of the equilibrium states and the rate of convergence of solutions towards them for the continuous kinetic version of the Cucker-Smale flocking in presence of diffusion whose strength depends on the density. This kinetic equation describes the collective behavior of an ensemble of organisms, animals or devices which are forced to adapt their velocities according to a certain rule implying a final configuration in which the ensemble flies at the mean velocity of the initial configuration. Our analysis takes advantage both from the fact that the global equilibrium is a Maxwellian distribution function, and, on the contrary to what happens in the Cucker-Smale model (IEEE Trans Autom Control 52:852–862, 2007), the interaction potential is an integrable function. Precise conditions which guarantee polynomial rates of convergence towards the global equilibrium are found.