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arXiv math/0611719math.AP
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一维Dirac-Klein-Gordon系统在荷范数下的整体适定性

Global well-posedness below the charge norm for the Dirac-Klein-Gordon system in one space dimension

Sigmund Selberg

摘要:

我们证明了全局适定性低于收费标准(即,Dirac旋量的$L^2$范数)的Dirac-Klein-Gordon方程组(DKG)在一维空间中。自适应的方法,由于布尔增益,我们分裂的高频率部分的初始数据的旋量,并利用非线性平滑效应来控制的演变的高频部分。为了证明非线性平滑,我们依赖于DKG系统的零结构,以及Bourgain-Klainerman-Machedon空间中的双线性估计。

英文摘要:

We prove global well-posedness below the charge norm (i.e., the $L^2$ norm of the Dirac spinor) for the Dirac-Klein-Gordon system of equations (DKG) in one space dimension. Adapting a method due to Bourgain, we split off the high frequency part of the initial data for the spinor, and exploit nonlinear smoothing effects to control the evolution of the high frequency part. To prove the nonlinear smoothing we rely on the null structure of the DKG system, and bilinear estimates in Bourgain-Klainerman-Machedon spaces.

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