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A kinetic flocking model with diffusion. (English) Zbl 1213.35395

The paper is concerned with a kinetic flocking model in the presence of diffusion. The model consists of a nonlinear Fokker-Planck equation which describes the collective behaviour of an ensemble of organisms, animals or devices, and represents the continuous kinetic version of the Cucker-Smale flocking augmented with diffusion. The drift term and the diffusion coefficient depend non-locally on the macroscopic momentum and density respectively. The model conserves both the total mass and the total momentum. The agents are forced to adapt their velocities according to a rule which implies a final mean velocity equal to the initial one. Because of the present lack of a Lyapunov functional, it is necessary to turn to the perturbation theory of equilibrium. The classical energy method together with suitable smallness assumptions produce some uniform a priori estimates in high order Sobolev spaces, which in association with the local existence and the continuum argument yield the global existence. As regards the time-decay rate, a recently developed energy-spectrum method is applied.

MSC:

35Q84 Fokker-Planck equations
82B05 Classical equilibrium statistical mechanics (general)
35Q82 PDEs in connection with statistical mechanics
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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