Transport-Reaction Models Benedikt Geiger Born August 5 1998 in Gelnhausen

2025-05-06 0 0 2.41MB 121 页 10玖币
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Transport-Reaction Models
Benedikt Geiger
Born August 5, 1998 in Gelnhausen
August 16, 2021
Master’s Thesis Mathematics
Advisor: Dr. Christina Lienstromberg
Second Advisor: Prof. Dr. Juan J. L. Vel´azquez
Institut f¨
ur Angewandte Mathematik
Mathematisch-Naturwissenschaftliche Fakult¨
at der
Rheinischen Friedrich-Wilhelms-Universit¨
at Bonn
arXiv:2210.00416v1 [math.AP] 2 Oct 2022
At first, I would like to express my gratitude to my advisors Dr. Christina Lienstromberg and Prof.
Dr. Juan J. L. Vel´azquez. They always took time to provide guidance and feedback, which led to
many fruitful discussions.
I would also like to thank my parents and my sister. The intense work on my thesis during
the last months would not have been possible without their continuous support.
Contents
1. Introduction 4
1.1. Function Spaces and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2. The Transport Semigroup 7
3. The Transport-Reaction Semigroup 16
4. Spectral Analysis of the Transport-Reaction Semigroup 27
4.1. Spectral Analysis of the Transport Semigroup . . . . . . . . . . . . . . . . . . . . . . 28
4.2. Spectral Analysis of the Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3. Weak Spectral Mapping Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5. Arbitrary Side Lengths and Neumann Boundary Conditions 48
5.1. ArbitrarySideLengths................................... 48
5.2. Symmetric Models with Neumann Boundary Conditions . . . . . . . . . . . . . . . . 49
6. Applications: Long-Time Behavior of Hyperbolic Models 54
6.1. AGoldstein-KacModel .................................. 54
6.2. ModelswithKilling .................................... 59
6.2.1. Positive and Mass Conserving Models with Killing . . . . . . . . . . . . . . . 61
6.3. A Reaction Random Walk System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
7. Pattern Formation 68
7.1. Reaction-Diffusion Equations and Turing Patterns . . . . . . . . . . . . . . . . . . . 69
7.1.1. ReactionNetworks................................. 69
7.1.2. The Full Model and Diffusion-Driven Instabilities . . . . . . . . . . . . . . . . 70
7.2. Transport-Driven Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7.2.1. Transport-Driven Instabilities for N=2..................... 85
7.3. Supplementary Figures and Simulations . . . . . . . . . . . . . . . . . . . . . . . . . 86
7.3.1. Figures of The Spectrum for N=2........................ 87
7.3.2. Figures of The Spectrum for N=3........................ 90
7.3.3. A Simulation for N=3 .............................. 93
A. Postponed Proofs 100
B. C0-Semigroups, Generators and Abstract Cauchy Problems 106
C. Perturbation and Approximation of C0-Semigroups 111
D. Spectral Theory and Long-Term Behavior of C0-Semigroups 113
Bibliography 116
2
List of Figures
1. Domain Generator (W1,p: An Illustration of the Idea . . . . . . . . . . . . . . . . . 14
2. Typical Spectrum for N= 2.Example1......................... 87
3. Typical Spectrum for N= 2.Example2......................... 88
4. Hyperbolic Instabilities for N= 2.Example1...................... 88
5. Hyperbolic Instabilities for N= 2.Example2...................... 89
6. Eventually Constant Real Parts for N= 2.AnExample................ 89
7. Spectrum for N=3.AnExample ............................ 90
8. Hyperbolic Instabilities for N=3.AnExample..................... 91
9. Turing Patterns for N=3.Example1.......................... 91
10. Turing Patterns for N=3.Example2.......................... 92
11. Turing Patterns for N=3.Example3.......................... 92
12. Turing Patterns for N=3.Example4.......................... 93
13. A Simulation of Turing Patterns for N=3. ....................... 94
14. A Simulation of the Time Evolution of Turing Patterns . . . . . . . . . . . . . . . . 95
15. A Simulation of Hyperbolic Instabilities for N=3.................... 96
16. A Simulation of the Time Evolution of Hyperbolic Instabilities . . . . . . . . . . . . 97
3
摘要：

Transport-ReactionModelsBenediktGeigerBornAugust5,1998inGelnhausenAugust16,2021Master'sThesisMathematicsAdvisor:Dr.ChristinaLienstrombergSecondAdvisor:Prof.Dr.JuanJ.L.VelazquezInstitutfurAngewandteMathematikMathematisch-NaturwissenschaftlicheFakultatderRheinischenFriedrich-Wilhelms-UniversitatBo...

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