本篇報告主要是由三位教授：J.COLLIANDER, M.GRILLAKIS and N.TZIRAKIS.發表於2009年、題目為Tensor Products and Correlation Estimates with Application to Nonlinear Schrodinger Equations的論文為主體， 我們仔細地探討裡面的細節，然後寫下本篇報告。

首先，我們討論在所有維度的相關估計。接著，我們分別討論相互作用的Morawetz不等式在維度大於等於3和2維的情況。然後，我們分別探討相互作用的Morawetz估計在1+2維的新證明。在2維的估計符合Strichartz的Bourgain的雙線性細化的非線性對角線模擬。在2維的情況下，作者們跟隨Lin and Strauss最初的方法，但是利用到解的張量積；而在1維的情況下，作者們利用Gauss-Weierstrass求和的方法作用在守恆律。最後，作者們利用非線性的薛丁格方程在2維的估計和推導出一個直接證明在每一個L^2超臨界非線性的Nakanishi's H^1 散射結果。

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 all the details in the paper: Tensor Products and Correlation Estimates with Application to Nonlinear Schrödinger Equations, by authors: J.COLLIANDER, M.GRILLAKIS and N.TZIRAKIS, in 2009, and then write the report.
First, we discuss the correlation estimates in all dimensions. Second, we discuss interaction Morawetz Inequality in dimension n ≥ 3 and in two dimensions. Third, we investigate the proof of new interaction Morawetz-type (correlation) estimates in one and two dimensions in details. In dimension 2 the estimate corresponds to the nonlinear diagonal analogue of Bourgain's bilinear refinement of Strichartz. In the case of dimension 2, the authors follow the original approach of Lin and Strauss but applied to tensor products of solutions. In one dimension, the authors use the Gauss-Weierstrass summability method acting on the conservation laws. Finally, the authors then apply the two-dimensional estimate to nonlinear Schrödinger equations and derive a direct proof of Nakanishi's H^1 scattering result for every L^2-supercritical nonlinearity.
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.

[A] Andrew Browder. Mathematical analysis :an introduction Springer,New York(1996).
[AF] Robert A. Adams and John J.F. Fournier., Sobolev spaces, Academic Press, Amsterdam(2003).
[B] Bourgain J. Re_nements of Strichartz' ineqality and applications to 2D-NLS with critical nonlinearity. Internat. Math. Res. Notices 1998, no. 5,253-283.
[BD] Barbara D. MacCluer. Elementary functional analysis. Springer,New York (2009).
[BJ] Bourgain, J. Global solutions of nonlinear Schrodinger equations. American Mathematical Society Colloquium Publications, 46. American Mathematical Society, Providence,R.I.,(1999)
[BSU] Y.M. Berezansky, Z.G. Sheftel, G.F. Us ; translated from the Russian by Peter V. Malyshev. Functional analysis Basel ;Birkhauser Verlag,(1996).
[CKST] Colliander, J.; Keel, M.; Sta_lani, G.; Takaoka, H.; Tao, T. Global existence and scattering for rough solutions of a nonlinear Schrodinger equation on R3. Comm. Pure Appl. Math.57(2004), no. 8, 987-1014.
[F] F. Max Stein, An introduction to vector analysis, Harper and Row,New York(1963).
[FG] Fang, Y. F.; Grillakis, M. G. On the global existence of rough solutions of the cubic defocusing Schrodinger equation in R2+1. J. Hyperbolic Di_er. Equ.4 (2007), 233-257.
[FIS] Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence. Linear algebra N.J.:Pearson Education, Upper Saddle River,(2003).
[HH] H. Dym, H. P. McKean. Fourier series and integrals. A.P, New York,(1972)
[JM] Jerrold E. Marsden, Michael J. Ho_man. Elementary classical analysis.
W.H.Freeman and company, New York,(1993).
[JMN] J.Colliander, M.Grillakis, N.Tzirakis. Tensor Products and Correlation Estimates with Application to Nonlinear Schrodinger Equations Comm. Pure Appl.Math.57(2009), Vol. LXII, 0920-0968.
[KN] V.P. Khavin, N.K. Nikol'skij (eds.). Commutative harmonic analysis. Berlin
;Springer-Verlag,New York(1991).
[KT] Keel,M.; Tao, T. Endpoint Strichartz estimates. Amer. J. Math. 120(1998), 955-980.
[M] Morawetz, C.S. Time decay for the nonlinear Klein-Gordon equations. Proc. Roy. Soc. Ser. A306(1968), 291-296.62
[MR] Maxwell Rosenlicht. Introduction to analysis Scott, Foresman,(1968)
[R]Raymond A. Serway.Physics for scientists and engineers,Philadelphia :Saunders College Pub.,c1986.
[VZ] Visan, M.; Zhang, X. Global well-posedness and scattering for a class of nonlinear Schrodinger equations below the energy space. arXiv: 0606611v1, 2006.