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Soliton Model On long time dynamic and singularity formation of NLS MASSACHTS ITTUTE by OF TECHNOLOGY Chenjie Fan AUG 0 12017 B.S., Peking University (2012) LIBRARIES Submitted to the Department of Mathematics ARCHIVES in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2017 @ Massachusetts Institute of Technology 2017. All rights reserved. Signature redacted Author ............................................ Department of Mathematics May 3rd, 2017 Certified by. Signature redacted ... Gigliola Staffilani LAbby Rockefeller Mauze Professor Thesis Supervisor Accepted by... Signature redacted .................. William Minicozzi Chairman, Department Committee on Graduate Theses 2 On long time dynamic and singularity formation of NLS by Chenjie Fan Submitted to the Department of Mathematics on May 3rd, 2017, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract In this thesis, we investigate the long time behavior of focusing mass critical nonlinear Schr6dinger equation (NLS). We will focus on the singularity formation and long time asymptotics. To be specific, there are two parts in the thesis. In the first part, we give a construction of log-log blow up solutions which blow up at m prescribed points simultaneously. In the second part, we show weak convergence to ground state for certain radial blow up solutions to NLS at well chosen time sequence. We also include a lecture note on concentration compactness. Concentration compactness is one of the main tool we use in the second part of the thesis. Thesis Supervisor: Gigliola Staffilani Title: Abby Rockefeller Mauze Professor 3 I Acknowledgments I am very fortunate to work with my advisor, Gigliola Staffilani. I am very grateful to her consistent support and encouragement. I want to take this chance to thank her generous sharing of mathematical insights, her patience and her kindness. I have learned a lot from my advisor, not only disperisve PDE, but also how to become a better person. I also want to thank Larry Guth and David Jerison for being in my thesis com- mittee. I definitely benefit a lot from discussion with them during my five years in MIT. The material in the thesis benefits a lot from discussion and comments of Patrick Gerard, Carlos Kenig, Yvan Martel , Pierre RaphaBd, Svetlana Roudenko, Vedran Sohinger and Boyu Zhang. I want to take this chance to express my thanks. I have the chance to collaborate with Alice Guionnet, Peter Kleinhenz, Yuqi Song, Andi Wang, Hong Wang, Bobby Wilson, and also with my advisor. That's very nice experience, thank you all. During my graduate life, I have the chance to discuss mathematics with various mathematicians, and their generous sharing of mathematical ideas and insights are really appreciated. I want to show my thanks here. I was also very fortunate to spend one semester in MSRI, for the program: New Challenges in PDE: Deterministic Dynamics and Randomness in High and Infinite Dimensional Systems, in fall 2015-2016. It is a very precious experience for me. The program not only improved my understanding of my own field, but also helped me see the world in different fields, broadening my view of math. I am very proud and fortunate to be a graduate student in MIT math department. The MIT community is known to be nice and supportive, and help me go through graduate life here, which is known to be not so easy. MIT is a wonderful place for me to do research and study. I want to thank my friends here, though it is impossible to list them all, I want to name a few: Rui Chen, Qiang Guang, Hai-Hao Lu, Xin Sun, Teng Fei, Guo-Zhen 5 Wang, Hong Wang, Wen-Zhe Wei, Wei-Jun Xu, Ben Yang, Yi Zeng, Rui-Xun Zhang, Rui-Xiang Zhang, Xu-Wen Zhu. Thanks for the friendship. And, to all my friends, I want to say thank you. I want to in particular thank Rui Chen, Hong Wang, Wei-Jun Xu, and Ben Yang. There were very hard moments in my graduate life, and their support help me walk through that. During the preparation of the material, I was partially supported by NSF Grant DMS 1069225, DMS 1362509 and DMS 1462401. Last but not the least, I want to thank my parents, for their unconditional and endless love. 6 Contents 1 Introduction 11 1.1 O verview .......... ........................ 11 1.2 Log Log blow up solutions with m blow up points ...... ..... 19 1.3 Weak convergence to ground state . ................... 22 1.4 Structure of the Thesis ...... .................... 26 2 On m points log-log blow up solutions 27 2.1 Introduction ................................ 27 2.1.1 N otation ....... .. 30 2.1.2 A quick review of Merle and Rapha81's work and heuristics for the localization of log-log blow up ....... ......... 31 2.1.3 Strategy and structure of the paper ............... 42 2.2 Description of the initial data and dynamic/modification of system: one soliton m odel ............................ 43 2.2.1 Description of the initial data .............. .... 43 2.2.2 Modification of the system ..... ............ ... 44 2.2.3 Description of the dynamic ... ............ ..... 45 2.3 Description of the initial data and dynamic: multi-soliton model .. 46 2.3.1 Description of the initial data .................. 46 2.3.2 Modification of the system ................ .... 48 2.3.3 Description of the dynamic .......... .......... 51 2.3.4 Further remarks on Lemma 2.3.2 ... ............. 54 2.4 Proof of Lemma 2.2.4: One Soliton Model ............ ... 55 7 2.4.1 Setting up ..... ......... .......... .... 55 2.4.2 An overview of the proof ...... .......... .... 56 2.4.3 Rough control ..... .................... 58 2.4.4 Propagation of regularity .......... ......... 63 2.5 Proof of Lemma 2.3.2: Multi Solitons Model ............ 72 2.5.1 Outline of the Proof ...................... 73 2.5.2 Recovering the Lyapounouv functional under bootstrap hypoth- esis ........... .............. ....... 80 2.5.3 Propagation of regularity under bootstrap hypothesis .... 84 2.5.4 Proof of bootstrap estimate except (2.3.57), (2.3.54),(2.3.55) . 88 2.6 Proof of Main Theorem ......................... 92 2.6.1 Preparation of data ....................... 94 2.6.2 Log-log blow up and almost sharp blow up dynamic ..... 96 2.6.3 A quick discussion of blow up at the same time ...... 97 2.6.4 Prescription of blow up points ........ ......... 99 3 On weak convergence to the ground state 105 3.1 Introduction ........ ..................... .. 105 3.2 P relim inary ..................... .......... 107 3.2.1 Local well posedness (LWP) and stability . .......... 107 3.2.2 Scattering below the mass of the ground state ........ 110 3.2.3 Concentration compactness ............ ...... 110 3.2.4 Variational characterization of the ground state . ...... 114 3.3 The dynamic of non-positive energy solution ............. 115 3.4 An overview for the proof for Theorem 1.3.1, Theoreml.3.2 ..... 117 3.4.1 Step 1: First extraction of profile ............... 117 3.4.2 Step 2: Second extraction of profile .............. 117 3.4.3 Step 3: Fast cascade case .................... 118 3.4.4 Step 4: Quasisoliton case .................... 119 3.4.5 Step 5: Approximation argument and conclusion of the proof . 120 8 3.5 Proof of Corollary 3.3.3 ..................... ..... 120 3.6 Proof of Lemma 3.4.1 ............. .............. 122 3.7 Proof for Subsection 3.4.4 ........................ 125 3.7.1 A quick review of Dodson's work [191 . 125 3.7.2 Proof of Lemma 3.4.11 ...................... 131 3.7.3 Proof of Lemma 3.4.9 .......... ............. 132 3.7.4 Proof of Lemma 3.4.8 ....................... 133 3.7.5 Proof of Lemma 3.7.4 ....................... 136 3.8 Proof of Theorem 1.3.1 .......................... 137 3.9 Proof of Theorem 1.3.4 ............. ............ 138 A A few technical lemma-ta 141 A.1 The local well posedness of the modified system ............ 141 A.2 Proof of Lemma 2.6.7 ...................... ..... 142 A.3 Proof of Lemma 2.6.9 ............ ............... 143 B Lecture notes on concentration compactness 147 B .1 Introduction .............. .............. 147 B .2 N otation .......... ........... ......... 148 B.3 A preliminary model . ........... ........... 149 B.4 A working example . ....... ....... ........ 154 B.5 One more working example in dispersive PDE ......... 157 B.5.1 Basic setting of profile decomposition in dispersive PDE 157 B.5.2 On defocusing energy critical NLS, (B.5.1) ....... 164 9 10 Chapter 1 Introduction 1.1 Overview Our thesis is about the long time dynamic of nonlinear Schr6dinger equation, which is a typical example of nonlinear dispersive PDE. Nonlinear dispersive PDEs have been an active research field in recent years. While it is impossible to give a full survey of the field here, we want to highlight certain developments and research directions in recent years, which are closely related to the topic of our thesis. To be more specific ,we mainly restrict ourselves in mass critical nonlinear Schrddinger equation, (1.1.1). The main results of our thesis are also about (focusing) mass critical nonlinear Schr6dinger equation. Iut+ Au = pu 14/du, u(O) = uo. (1.1.1) y = -1. Here d is the dimension, and u is a complex valued function on I x Rd, where I is a time interval. The equation is called focusing when p = -1 and defocusing when p = 1. We say equation (1.1.1) is a dispersive equation because the linear problem asso- 11 ciated to (1.1.1), iut+ Au = 0, U(0) = no. Is dispersive. Let etA be the propagator of (1.1.2), one has the well known dispersive estimate, e uOIILoo -td/2luolILl- (1.1-3) On the other hand, unlike the heat equation, (1.1.2) is time reversible and it also conserves mass, i.e.
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