On the Cucker-Smale- Fokker-Planck Model under Random Environment

Abstract

In this dissertation, we mainly focus on a kinetic Cucker--Smale--Fokker--Planck (CS-FP) type equation with a degenerate diffusion coefficient. The CS-FP equation is described in a differential equation for a probability distribution function $f$ of the infinitely many Cucker--Smale flocking particles in a random environment. We will present a priori estimates for proving the global existence of classical solutions to the CS-FP equation. The global existence of classical solutions under a given sufficiently smooth initial datum will be obtained by applying sobolev embedding theorem to the a priori estimates and iterating the solutions of uniformly parabolic equations which approximates the CS-FP equation. We also present the Cucker-Smale-Kuramoto model which describes flocking and synchronization coupled phenomena. Sufficient conditions for the asymptotic flocking and synchronization will be derived with the Lyapunov functional approach. We provide the numerical compuations for a special case to suggest the future works on clustering.