在本篇論文中我們是採取跟Bourgain不同的方法去對非線性項做估計。我們在第一到三章是一開始是利用fourier transform先把解直接比出來，再利用非線性項有null form structure去估計非線性項並且利用charge 有守恆律，由此可以得知DKG方程式有唯一Global 的解。
從第四章到第十一章我們是採取 Klainerman和 Machedon 的想法並且改進齊次波方程的Bilinear estimate。
最後在第十二章到第十四章我們去考慮另一個方程式Davey-Stewartson system on a cycle。

The purpose of this work is to state a new proof of null form estimate, see [B], by proving a solution representation of DKG equations which is more simple. In this way, we give an interpretation of the null form structure depicted within the nonlinear term ψγ5ψ, and it is different form that in [B]. The nonlinear term has the null form structure, see [KM].
The approach that we adopt in this work is as follow : In chapter one, we derive the conservation law of charge,

∫∣x∣2dx = constant (0.1)

which can be applied to derive the global solution existence for the DKG equations. Second, the solution representation has derived in fourier transform, then we write down the direct solution representation. Third, we will use it to estimate the nonlinear form ψγ5ψ, and the derivations of some necessary estimates become straight forward in the chapter two. Finally, we can prove the local and global existence results of DKG equations with ψ0∊L2, φ0∊H1, and φ1∊L2, which are called charge class solutions in chapter three. We use the idea of Klainerman and Machedon [KM1] and then the proof of the bilinear space-time estimates for homogeneous wave equation in [KM2] has been modified from chapter four to chapter eleven. We consider the Cauchy problem of the following Davey-Stewartson system on a cycle from the chapter twelve to the chapter fourteen. Now, so I am interested in DKG. I hope I can do more deeply research.

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