本篇報告主要是由三位教授：J. COLLIANDER, J. HOLMER and N. TZIRAKIS.發表於2008年、題目為 extit{Low Regularity Global Well-Posedness for the Zakharov and Klein-Gordon-Schrodinger Systems}的論文為主體。在這裡， 我們仔細地探討關於一維的扎哈洛夫系統下低正規整體適定性，然後寫下本篇報告。
這篇論文主要是探討由三位教授所證明的定理1.1，即是論文的定理1.3。為了證明此定理，本研究討論一維的薛丁格方程和一維的波動方程的解公式和算子的定義方式。 此外，本研究探討齊次估計、非齊次估計、非線性估計的證明。為了證明定理1.3，本研究利用迭代法、固定點定理和收縮求出方程式有局部解。由質量守恆定律得知，因薛丁格方程解的部分隨著時間變大，解也不會變大。因此，本研究利用前面所得到的估計對波動方程的解做一些控制，然後得到隨著時間變大波動方程解的部分也會被控制住的結果，得到全域解，最後，本論文證明了定理1.3的結果。
在詳讀他們論文的過程中，我們更正一些論文內的打字錯誤和試著用更明瞭的方式來改寫論文內一些模糊或易讓人誤解的式子。

We exploit the details about the 1d Zakharov system in the paper: Low Regularity Global Well-Posedness for the Zakharov and Klein-Gordon-Schrodinger Systems,
by authors: J. COLLIANDER, J. Holmer, and N.TZIRAKIS, in 2008, and then write the report.
This thesis mainly investigates the Theorem 1.1 which was proven by the three professors, and it is the Theorem 1.3 of this thesis. In order to prove this theorem, this study discussed the solution formula of the 1d Schrodinger equation, the solution formula of the 1d wave equation and define some operator forms. In addition, this research explored the proofs of the Group estimates, Duhamel estimates, Multilinear estimates. To prove Theorem 1.3, this study uses Iteration scheme, the fixed point theorem and contraction to obtain local well-posedness in equations. From the law of conservation of mass, the solution part of Schrödinger equation increases over time, and the solutions will not become larger. Therefore, this study controls the solution of the wave equation by using the above obtained estimation, and further obtains the part of the wave equation solution which will be controlled to obtain global well-posedness. Finally, the thesis proves the results of theorem 1.3.
We study their work in details. We correct some types in the paper. Some formula in some parts of the paper are given in
a way of vague or misleading, we rewrite them in a clear way.

References
[BC] J. Bourgain, J. Colliander, On Well-Posedness of the Zakharov System, Internat.
Math. Res. Notices, no.11, 515-546 (1996).

[CHT] J. Colliander, J. Holmer, N. Tzirakes, Low Regularity Global Well-Posedness for the Zakharov
and Klein-Gordon-Schrooo¨dinger Systems, American Mathematical Society, Transactions. 360,
4619-4638 (2008).

[E] Lawrence C. Evans, Partial Differential Equations, American Mathematical Society, Providence,
R.I., (1998).

[GTV] J. Ginibre, Y. Tsutsumi, and G. Velo, On the Cauchy Problem for the Zakharov System, J.
Funct. Anal. 151, no. 2, 384-436 (1997).

[HH] H. Dym, H. P. McKean, Fourier Series and Integrals, A.P, New York, (1972).

[NMPZ] S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, V. E. Zakharov, Theory of Soli- tons: The
Inverse Scattering Method, Springer-Verlag (1984).

[P] H. Pecher, Global Well-Posedness below Energy Space for the One-dimensional Za- kharov System,
Internat. Math. Res. Notices, no.19, 1027-1056 (2001).

[WZ] Richard. L. Wheeden, A. Zygmund, Measure and Integral: An Introduction to Real Analysis,
Marcel Dekker Inc, New York, (1977).

[Z] V. E. Zakharov, What is Integrability?, Springer-Verlag (1991).

[ZLF] V. E. Zakharov, V. S. L’vov, G. Falkovich, Kolmogorov Spectra of Turbulence I: Wave
Turbulence, Springer-Verlag (1992).

[ZV] Vladimir Zakharov, The Paradise for Clouds, Ancient Purple Translations (2009).