On the Cucker-Smale- Fokker-Planck Model under Random Environment
확률 환경 하에서의 쿠커-스메일-포커-플랑크 모델에 대하여

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dc.description학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2016. 8. 하승열.-
dc.description.abstractIn 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.
dc.description.tableofcontentsChapter 1 Introduction 1

Chapter 2 Preliminaries 5
2.1 The Cucker-Smale Model 5
2.1.1 The Vicsek Model 5
2.1.2 The Cucker-Smale Model 6
2.1.3 The Kinetic Cucker-Smale Model 7
2.2 The Cucker-Smale-Fokker-Planck Equation 9

Chapter 3 The Cucker-Smale Model with White Noise 16
3.1 The Additive Noise Case 16
3.2 The Multiplicative Noise Case 19

Chapter 4 Wellposedness of the CS-FP Equation 23
4.1 Estimates of Classical Solutions 23
4.1.1 A priori Estimates 25
4.2 A Local Existence Result 31
4.2.1 Extention of Local Existence 35

Chapter 5 The Cucker-Smale-Kuramoto Model 37
5.1 The Cucker-Smale-Kuramoto Models 38
5.2 Frameworks 40
5.3 Estimates in the CSK model 42
5.4 Estimates in the CSK model with Hebbian Coupling 46
5.5 Numerical Simulations 56
5.5.1 Natural Frequency 56
5.5.2 Intensity of Interaction 60

Chapter 6 Conclusion 65

Bibliography 66

Abstract (in Korean) 71
dc.format.extent3492630 bytes-
dc.publisher서울대학교 대학원-
dc.titleOn the Cucker-Smale- Fokker-Planck Model under Random Environment-
dc.title.alternative확률 환경 하에서의 쿠커-스메일-포커-플랑크 모델에 대하여-
dc.contributor.affiliation자연과학대학 수리과학부-
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College of Natural Sciences (자연과학대학)Dept. of Mathematical Sciences (수리과학부)Theses (Ph.D. / Sc.D._수리과학부)
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