Dynamics and Analysis of Alignment Models of Collective Behavior

1. Preface 2 2. Emergent phenomena of collective behavior. 4 3. Agent-based alignment systems 6 3.1. Types of communication and collective outcomes 6 3.2. Momentum, Energy, and Maximum Principle 7 3.3. Connectivity and spectral method 8 3.4. Alignment with heavy tail communication 10 3.5. Stability 12 3.6. Singular kernels and the issue of collisions 14 3.7. Degenerate communication. Corrector Method 15 3.8. Multi-flocks. Clusters. Multi-species 19 3.9. Notes and References 22 4. Forced systems 24 4.1. 3Zones model. Small crowd flocking 24 4.2. External confinement. Hypocoercivity 26 4.3. 2Zone model: attraction + alignment 29 4.4. Dynamics under self-propulsion and Rayleigh friction 34 4.5. Notes and References 36 5. Kinetic models 37 5.1. BBGKY hierarchy: formal derivation 37 5.2. Weak formulation and basic principles of kinetic dynamics 38 5.3. Kinetic maximum principle and flocking 40 5.4. Stability. Kantorovich-Rubinstein metric. Contractivity 41 5.5. Mean-field limit 43 5.6. Notes and References 45 6. Macroscopic description. Hydrodynamic limit. 46 6.1. Multi-scale model and its justification 47 6.2. Hydrodynamic limit. Kinetic relative entropy 50 6.3. Notes and References 55 7. Euler Alignment System 57 7.1. Basic properties. Energy law 57 7.2. Hydrodynamic flocking and stability 58 7.3. Spectral method. Hydrodynamic connectivity 60 7.4. Topological models. Adaptive diffusion 62 7.5. Notes and References 67 8. Local well-posedness and continuation criteria 68 8.1. Smooth models 68 8.2. Singular models 72

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