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研究生： 張惟傑
Jhang, Wei-Jie
論文名稱： 量子札哈羅夫系統的半古典極限
On the Semi-classical Limit of Quantum Zakharov System
指導教授： 方永富
Fang, Yung-Fu
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系應用數學碩博士班
Department of Mathematics
論文出版年： 2016
畢業學年度： 104
語文別： 英文
論文頁數： 45
中文關鍵詞： 量子札可哈羅夫系統 、整體適定性 、古典極限
外文關鍵詞： Quantum Zakharov system, Global well-posedness, Classical limit
相關次數： 點閱：109  下載：1 
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    • 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。

    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.

    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

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    [9] Halsey Royden, Patrick Fitzpatrick,Real Analysis 4th,(2010)

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