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On the fluid coupling and hydrodynamic limit of the kinetic thermomechanical Cucker-Smale equation
| DC Field | Value | Language |
|---|---|---|
| dc.contributor.advisor | 하승열 | - |
| dc.contributor.author | 김정호 | - |
| dc.date.accessioned | 2019-10-21T03:38:00Z | - |
| dc.date.available | 2019-10-21T03:38:00Z | - |
| dc.date.issued | 2019-08 | - |
| dc.identifier.other | 000000156400 | - |
| dc.identifier.uri | https://hdl.handle.net/10371/162415 | - |
| dc.identifier.uri | http://dcollection.snu.ac.kr/common/orgView/000000156400 | ko_KR |
| dc.description | 학위논문(박사)--서울대학교 대학원 :자연과학대학 수리과학부,2019. 8. 하승열. | - |
| dc.description.abstract | 본 학위 논문에서는, 열역학적 쿠커-스메일 모델을 유도하고, 해당 모델과 관련된 여러 가지 주제들에 대해 다루고자 한다. 가장 먼저, 다중 온도 혼합 기체 모델에서부터 무질서도 원칙 및 갈릴레이 불변 법칙을 이용하여 완전한 열역학적 쿠커-스메일 모델을 도출하였다. 그다음, 도출된 국소 열역학적 쿠커-스메일 모델 및 기존의 쿠커-스메일 모델의 기본적인 성질들과 점근적 행동들에 대한 기존의 결과를 복습한다. 국소 모델의 경우, 입자의 개수가 많아지면, 각 입자의 자취를 모두 추적하는 것은 거의 불가능하며, 이를 해결 하기 위해 열역학적 쿠커-스메일 모델의 운동방정식 및 유체방정식을 도출하였다. 이와 관련 하여, 국소 모델에서 운동방정식을 도출하는 평균장 극한과 운동방정식, 유체방정식에서의 점근적 성질들을 복습한다. 실제 현상에서는, 입자들의 움직임은 닫힌계가 아닌, 주변의 매질들과 상호작용하는 열린계이다. 따라서, 열역학적 쿠커-스메일 운동 방정식과 유체 모델과의 결합을 연구하는 것은 이론적인 측면 및 응용 측면 모두에서 흥미로운 주제이다. 본 논문에서는 열역학적 쿠커-스메일 운동 방정식과 비압축성/압축성 나비어-스토크스 방정식의 결합모델을 제시하고 해의 존재성 및 긴 시간 현상을 탐구하였다. 또한, 운동방정식에서 유체방정식으로의 극한은 전통적으로 매우 흥미로운 문제 중 하나였다. 본 논문에서는, 이와 관련 하여, 열역학적 쿠커-스메일 운동방정식에서부터 쿠커-스메일 유체방정식으로의 극한에 대한 결과를 제시하였다. | - |
| dc.description.abstract | In this thesis, we introduce the thermomechanical Cucker-Smale (TCS) model and study various topics regarding it. We provide the full description of deriving TCS model from the multi-temperature gas mixtures, using entropy
principle and Galilean invariance principle. We review some previous works regarding the microscopic (particle) Cucker-Smale (C-S) model and the TCS model, including several basic properties and asymptotic behaviors. Next, we formally derive the mesoscopic (kinetic) and macroscopic (hydrodynamic) models for the TCS model, using BBGKY hierarchy and taking velocity moments. We also briefly review the mean- eld limit procedure from micro- to mesoscopic scale and some results about kinetic/hydrodynamic models. In reality, the dynamics of particle ensemble does not form a closed system and interact with surrounding environments. Thus, it is interesting to study the TCS system coupled with fluid. We propose the kinetic TCS equation coupled with incompressible/compressible fluids and study the well-posedness and large-time behavior of them. On the other hand, the hydrodynamic limit from kinetic to the hydrodynamic model has been one of the historically interesting problems. In this regards, we studied the hydrodynamic limit from the variation of the kinetic TCS equation towards hydrodynamic C-S equations. | - |
| dc.description.tableofcontents | Abstract
1. Introduction 2. The Thermomechanical Cucker-Smale model 2.1 Derivation of the TCS model 2.2 A brief review on C-S model 2.3 Emergence of flocking behavior of the TCS model 2.3.1 Flocking behavior of (1.0.2) 2.3.2 Flocking behavior of (1.0.3) 3. Derivation of kinetic and hydrodynamic TCS equations 3.1 The kinetic TCS equation 3.2 The hydrodynamic TCS equations 3.3 Emergence of flocking of the kinetic and hydrodynamic TCS equations 4. The kinetic TCS equation coupled with incompressible fluid 4.1 Auxiliary lemmas 4.2 Presentation of main results 4.2.1 An existence of global weak solutions 4.2.2 An existence and uniqueness of global strong solution 4.2.3 Asymptotic flocking estimate 4.3 A global existence of weak solutions 4.3.1 A regularized system and its solvability 4.3.2 Proof of Theorem 4.2.1 4.4 A global existence and uniqueness of strong solutions 4.4.1 Construction of approximated solution 4.4.2 Compactness of approximated solutions 4.4.3 Proof of Theorem 4.2.2 4.5 Asymptotic flocking estimate 4.5.1 Time-decay of the Lyapunov functional 4.5.2 Proof of Theorem 4.2.3 5. The kinetic TCS equation coupled with compressible fluid 5.1 Preliminaries and main results 5.1.1 Monotonicity of the temperature support 5.1.2 Conservation laws and energy dissipation 5.1.3 Bogovskii type estimate 5.1.4 Main results 5.2 Asymptotic emergence of flocking 5.2.1 Proof of Theorem 5.1.1 5.3 Global existence and uniqueness of strong solutions 5.3.1 Approximate solutions 5.3.2 Convergence of approximated solutions 5.3.3 Proof of Theorem 5.1.2 6 Hydrodynamic limit of kinetic TCS equation 6.1 Preliminaries 6.1.1 Presureless Euler equations with nonlocal alignments 6.1.2 Relative entropy method 6.1.3 A priori estimates 6.2 Description of main results 6.2.1 Weak solutions and flocking dynamics 6.2.2 Hydrodynamic limit 6.3 Existence of weak solutions and flocking dynamics 6.3.1 A global existence of a weak solution 6.3.2 Proof of Theorem 6.2.2 6.4 From the kinetic TCS equation to hydrodynamic equations 6.4.1 Structural hypotheses 6.4.2 The first part of Theorem 6.2.3 6.4.3 The second part of Theorem 6.2.3 7. Conclusion and future works Appendix A. Detailed proof of Chapter 4 A.1 Solvability of the kinetic TCS equation A.2 Proof of Lemma 4.3.1 A.2.1 Estimate for $\ | - |
| dc.description.tableofcontents | f^n(t)-f^{n-1}(t)\ | - |
| dc.description.tableofcontents | _{L^\infty}$
A.2.2 Estimate for $\ | - |
| dc.description.tableofcontents | \mathbf{z}^n(t)-\mathbf{z}^{n-1}(t)\ | - |
| dc.description.tableofcontents | _{L^\infty}$
Appendix B. Detailed proof of Chapter 4 B.1 Proof of Lemma 5.2.7 B.2 Proof of Lemma 5.3.3 B.2.1 $L^2$-estimate for $f^{n+1}$ B.2.2 $H^s$-estimate for $f^{n+1}$ B.3 Proof of Lemma 5.3.4 Bibliography Abstract (in Korean) Acknowledgement (in Korean) | - |
| dc.language.iso | eng | - |
| dc.publisher | 서울대학교 대학원 | - |
| dc.subject | Flocking | - |
| dc.subject | hydrodynamic limit | - |
| dc.subject | large-time behavior | - |
| dc.subject | Navier-Stokes equations | - |
| dc.subject | thermomechanical Cucker-Smale model | - |
| dc.subject.ddc | 510 | - |
| dc.title | On the fluid coupling and hydrodynamic limit of the kinetic thermomechanical Cucker-Smale equation | - |
| dc.type | Thesis | - |
| dc.type | Dissertation | - |
| dc.contributor.department | 자연과학대학 수리과학부 | - |
| dc.description.degree | Doctor | - |
| dc.date.awarded | 2019-08 | - |
| dc.identifier.uci | I804:11032-000000156400 | - |
| dc.identifier.holdings | 000000000040▲000000000041▲000000156400▲ | - |
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