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論文名稱(中文) 札卡洛夫的區域非良態解及其能量收斂
論文名稱(英文) LOCAL ILL-POSEDNESS AND THE ENERGY CONVERGENCE OF 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) 方冠智
研究生(英文) Kuan-Chih Fang
學號 L16031077
學位類別 碩士
語文別 英文
論文頁數 31頁
口試委員 指導教授-方永富
口試委員-史習偉
口試委員-吳宗芳
中文關鍵字 札卡洛夫系統  薛丁格  Strichartz估計 
英文關鍵字 Zakharov system  Schrodinger  Strichartz estimate 
學科別分類
中文摘要 在這篇碩士論文中,我們考慮一維古典札卡洛夫系統。我們探討他的區域非良態解,同時也探討三維度古典札卡洛夫系統的能量收斂性質。我們查詢了Ginibre, Tsutsumi, Velo的工作並研讀了Holmer的論文並整理細節。而能量收斂到薛丁格方程的部分我們讀了Masmoudi and Nakanishi的論文。我們亦整理了他們的細節並呈現在碩士論文中。
英文摘要 In this thesis, we consider the classical Zakharov system. We study the illposedness of the system in 1D and the adiabatic limit at the energy level of the system in 3D. We investigate the work of Ginibre, Tsutsumi, Velo [7] and of Holmer [8] for the ill-posedness problem of the system.
We elaborate their works in details. For the energy convergence of the Zakharov system to the cubic nonlinear Schrodinger equation, we study the work of Masmoudi and Nakanishi [10]. We also elaborate their work in details.
論文目次 1.Introduction p1
2.Notations and Tools p4
3.Local Wellposedness p10
4.Norm Inflation p11
5.Soliton Solution p15
6.Phase Decoherence p16
7.Data-to-Solution Map Not C^2 p23
8.Zakharov to Cubic Nonlinear Schrodinger p25
9.References p31
參考文獻 [1] Robert A. Adams John J. F. Fournier, Sobolev Spaces, second edition
[2] J. Bourgain and J. Colliander, On wellposedness of the Zakharov system, Internat. Math. Res.
Notices (1996), no. 11, 515546. MR MR1405972(97h:35206)
[3] Christ-Colliander-Tao, Ill-posedness for nonlinear Schrodinger and wave equations, arxiv.org
preprint math.AP/0311048 (2003).
[4] Lawrence C. Evans, Partial Di erential Equations, American Mathematical Society, 2010
[5] Yung-Fu Fang, Hsi-Wei Shih, and Kuan-Hsiang Wang, Local Well-Posedness for the Quantum Za-
kharov System in One Spacial Dimension. To appear in Journal of Hyperbolic Di erential Equations.
[6] A. Eduardo Gatto1, Product Rule and Chain Rule Estimates for Fractional Derivatives on Spaces
that Satisfy the Doubling Condition, Journal of Functional Analysis 188, 2737 (2002)
[7] J. Ginibre, Y. Tsutsumi, and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct.
Anal. 151 (1997), no. 2, 384436. MR MR1491547(2000c:35220)
[8] Justin Holmer, Local ill-posedness of the 1D Zakharov system , Electron. J. Di erential Equations
(2007), No. 24, 22. MR 2299578 (2007k:35465).
[9] Guo Bo-Lin, Gan Zai-Hui, Zhang Jing-Jun, Zakharov Equation And The Soliton Solution, Beijingsciencepress,
2011
[10] Nader Masmoudi,Kenji Nakanishi, Energy convergence for singular limits of Zakharov type sys-
tems, Invent. math. 172, 535583 (2008) DOI: 10.1007/s00222-008-0110-5
[11] Robert McOwen, Partial Di erential Equations: Methods and Applications, Pearson Education,
2008
[12] Halsey Royden, Patrick Fitzpatrick, Real Anlysis, Fourth Editio
[13] Tosinobu Muramatsu, On Imbedding Theorems for Besov Spaces of Functions De ned in General
Regions, Publ. RIMS, Kyoto Univ.7 (1971/72), 261-285
31
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