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。

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.

[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