임의성이 있는 쿠커-스메일 모형에 대한 정량적 해석에 관하여

학위논문(박사)--서울대학교 대학원 :자연과학대학 수리과학부,2020. 2. 하승열.In this thesis, we introduce random elements into the Cucker-Smale(C-S) model and provide quantitative analyses for those uncertainties. In real applications of the Cucker-Smale dynamics, we can expect that the C-S model contains some intrinsic uncertainties in itself and misses some extrinsic factors that might affect the dynamics of particles. Thus, to provide a better description for the dynamics of a C-S ensemble, one needs to incorporate such uncertain factors to the model and evaluate their effects on the dynamics or stability of the C-S system. To fulfill this, we first consider the macroscopic version of the Cucker-Smale model. Namely, we introduce random inputs from communication weights and initial data into the hydrodynamic Cucker-Smale (HCS) model to yield the random HCS model. Furthermore, we address extrinsic uncertainties in the microscopic and mesoscopic level, respectively. For a microscopic model, we introduce a randomly switching network structure to the Cucker-Smale model and investigate sufficient conditions for the emergence of flocking. As a mesoscopic model, we consider the kinetic Cucker-Smale equation perturbed by multiplicative white noise and study the well-posedness and asymptotic dynamics of solutions.본 학위 논문에서는, 쿠커-스메일 모형에 임의적 요소를 도입하여 그러한 불확실성에 대한 정량적 해석을 제시한다. 쿠커-스메일 모형의 다이나믹스를 실제로 응용함에 있어 우리는 쿠커-스메일 모형 자체가 몇몇 내적 불확실성을 포함하고 있으며 입자들의 다이나믹스에 영항을 줄 수 있는 몇 가지 외부적 요인을 놓치고 있음을 예상할 수 있다. 그러므로 쿠커-스메일 총체의 다이나믹스를 더 잘 서술하기 위해, 이러한 불확실성이 있는 요소를 모형에 도입하여 그것들이 쿠커-스메일 계의 다이나믹스나 안정성에 주는 영향을 평가할 필요가 있다. 이를 달성하기 위해, 우리는 우선 쿠커-스메일 모형의 거시적인 형태를 고려한다. 즉, 우리는 통신 가중치 함수와 초기값에서 오는 임의적 입력치를 유체역학 쿠커-스메일 모형에 포함시켜 임의적 유체역학 쿠커-스메일 모형을 유도한다. 더 나아가 미시적 그리고 중간보기적 단계에서 외적 불확실성에 대해 다룬다. 미시적 모형에 대해서, 쿠커-스메일 모형에 임의로 변하는 네트워크 구조를 도입하여 플로킹의 창발에 대한 충분조건을 알아본다. 중간보기적 단계의 모형으로서, 우리는 곱셈 백색 소음으로 동요된 쿠커-스메일 운동방정식을 고려하고 해의 존재성 및 유일성과 점근적 다이나믹스를 공부한다.1 Introduction 1 2 Preliminaries 9 2.1 Notation 9 2.2 Previous results 10 3 A local sensitivity analysis for the hydrodynamic Cucker-Smale model with random inputs 15 3.1 Pathwise well-posedness of z-variations 16 3.1.1 First-order z-variations 18 3.1.2 Higher-order z-variations 26 3.2 The local sensitivity analysis for stability estimates 32 3.2.1 Higher-order L2-stability 32 3.2.2 L2-stability estimates for z-variations 37 3.3 A local sensitivity analysis for flocking estimate 41 4 On the stochastic flocking of the Cucker-Smale flock with randomly switching topologies 48 4.1 Preliminaries 49 4.1.1 Pathwise dissipative structure 49 4.1.2 A directed graph 52 4.1.3 A scrambling matrix 53 4.1.4 A state transition matrix 54 4.1.5 Previous results 55 4.2 A description of main result 57 4.2.1 Standing assumptions 57 4.2.2 Main result 58 4.3 Emergent behavior of the randomly switching system 61 4.3.1 A matrix formulation 61 4.3.2 Pathwise flocking under a priori assumptions 62 4.3.3 Emergence of stochastic flocking 70 5 Collective stochastic dynamics of the Cucker-Smale ensemble under uncertain communication 74 5.1 Preliminaries 75 5.1.1 Derivation of the SPDE 75 5.1.2 Presentation of main results 78 5.1.3 Elementary lemmas 80 5.2 A priori estimates for classical solutions 82 5.2.1 Quantitative estimates for classical solutions 86 5.3 Global well-posedness and asymptotic dynamics of strong solutions 92 5.3.1 Construction of approximate solutions 94 5.3.2 Estimates on approximate solutions 95 5.3.3 Proof of Theorem 5.1.3 103 6 Conclusion and future works 110 Appendix A Detailed proof of Chapter 3 112 A.1 Proof of Lemma 3.1.2 112 A.2 Proof of Lemma 3.1.5 115 A.3 Proof of Lemma 3.2.4 119 A.4 Proof of Theorem 3.3.2 121 Appendix B Detailed proof of Chapter 5 124 B.1 A proof of Theorem 5.2.1 124 B.2 A proof of Proposition 5.3.3 129 Bibliography 133Docto

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