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系統識別號 U0026-0607201616024000
論文名稱(中文) 量子札哈羅夫系統的半古典極限
論文名稱(英文) On the Semi-classical Limit of Quantum Zakharov System
校院名稱 成功大學
系所名稱(中) 數學系應用數學碩博士班
系所名稱(英) Department of Mathematics
學年度 104
學期 2
出版年 105
研究生(中文) 張惟傑
研究生(英文) Wei-Jie Jhang
學號 L16034067
學位類別 碩士
語文別 英文
論文頁數 45頁
口試委員 指導教授-方永富
口試委員-郭鴻文
口試委員-吳宗芳
中文關鍵字 量子札可哈羅夫系統  整體適定性  古典極限 
英文關鍵字 Quantum Zakharov system  Global well-posedness  Classical limit 
學科別分類
中文摘要 在這篇論文中,我們主要是讀Yanfeng Guo, Jingjun Zhang,和 Boling Guo所寫的“量子札可哈洛夫系統的整體適定性及其古典極限” 這篇論文,並且研究這個工作的細節。
首先,作者們證明量子札可哈洛夫系統在良好的初始值下,對於所有的時間,會有唯一的解落在好的空間,並且這個解在初始值上是連續的。為了證明這個結論,我們使用了非線性薛丁格方程和非線性波方程的公式解,並使用標準收縮法來局部解的存在與唯一性。然後我們用解均勻有屆的估計來拓展到整體適定性。此外,當我們把初始條件放寬,我們仍可得到同樣的結論。
第二,在適當的初始值下,我們可以使用能量估計來得到系統的古典極限,換句話說,當量子參數趨近於零,量子札可哈洛夫系統會收斂到古典系統。此外,當我們把初始條件放寬,我們仍可得到同樣的結論。
英文摘要 In this thesis, we mainly study the paper with title" Global well-posedness and the classical limit of the solution for the quantum Zakharov system" authored by Yanfeng Guo, Jingjun Zhang, and Boling Guo, and elaborate the details of this work.
First the authors prove that the quantum Zakharov system with nice initial data admits a unique solution staying in a nice space for all time , and the solution depends continuously on initial data. In order to prove this result, we use the solution formula of nonlinear Schrodinger equation and nonlinear wave equation, and make use of the standard contraction method to get the existence and uniqueness of local solution. And then we use the uniform bound estimate of the solution to extend it to global well-posedness. Also the same result hold if the condition of initial data is relaxed.
Second, with proper initial data, we can use the energy estimate to get the classical limit for the system, that is, the quantum Zakharov system converges to the classical system as the the quantum parameter tends to zero. Also the initial data can be relaxed for gettin the same result.
論文目次 Content



Chapter 1 Introduction ................................................................................ 1

1.1 Background ............................................................................................................... 1

1.2 Quantum Zakharov System and Zakharov System ................................................. 1

1.3 Global well-posedness .............................................................................................. 2

1.4 Classical limit ............................................................................................................ 2

Chapter 2 Notations and Preknowledge ..................................................... 3

2.1 Notations and Preknowledge .......................................................................... 3

Chapter 3 Energy estimate of the Solution .................................................. 5

3.1 Uniform Estimate for Theorem 1.1 ....................................................................... 5

3.2 Uniform Estimate Independ on h for Theorem 1.2 ................................................ 9

3.3 Uniform Estimate for Theorem 1.3......................................................................... 13

3.4 Uniform Estimate Independ on h for Theorem 1.4 ................................................ 16

Chapter 4 Global Well-posedness of the Solution ..................................... 19

4.1 The Uniqueness of the Weak Solution ................................................................... 19

4.2 Proof of Theorem 1.1.............................................................................................. 21

4.3 Proof of Theorem 1.2 with Relaxing Initial Data .................................................... 30

Chapter 5 The Limit Behavior of the Quantum Zakharov System............... 39

5.1 Proof of Theorem 1.3 ............................................................................................ 39

5.2 Proof of Theorem 1.4 with Relaxing Initial Data .................................................. 42




References.................................................................................................... 45
參考文獻 [1] Lawrence C.Evens., Partial Di erential Equation 1st, (1997)
[2] Yung-Fu Fang, Chi-Kun Lin, Jun-Ichi Segata, The Fourth Order Nonlinear Schrdinger Limit for
Quantum Zakharov System,Gauthier-Villars, (2016)
[3] Ginibre, J., Tsutsumi, Y., Velo, G., On the Cauchy problem for the Zakharov system., J. Funct. Anal.,
151, 384{436 (1997).
[4] Boling Guo,Yanfeng Guo and Jingjun Zhang, Global well-posedness and the classical limit of the
solution for the quantum Zakharov system., Z. Angew. Math. Phys., 64, 53{68 (2013).
[5] Haas, F., Shukla, P.K., Quantum and classical dynamics of Langmuir wave packets., Phys. Rev. E,
79, 066402 (2009).
[6] Jacques-Louis Lions,Quelques methodes de resolution des problemes aux limites non lineaires,Gauthier-
Villars, (1969)
[7] Majda, A.J., Bertozzi, A.L., Vorticity and Incompressible Flow.,Cambridge University Press, Cambridge,
(2002).
[8] Robert McOwen,Partial Di erential Equations:Method and Applications, Pearson Education, (2008)
[9] Halsey Royden, Patrick Fitzpatrick,Real Analysis 4th,(2010)
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