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arXiv 0903.3189math.AP
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二维Dirac-Klein-Gordon方程组的整体解

Global solutions for the Dirac-Klein-Gordon system in two space dimensions

Axel Gruenrock, Hartmut Pecher

摘要:

对于H ^{1/2} \times H ^{-1/2}中的L^2 Schoedinger数据和波动数据,二维经典Dirac-Klein-Gordon系统的Cauchy问题是整体适定的.在光滑数据的情况下,存在全局光滑(经典)解。证明使用基于Besov空间的Bourgain类型的函数空间-以前由Colliander,Kenig和Staffilani应用于广义Benjamin-Ono方程,也由Bejenaru,Herr,Holmer和Tataru应用于2D Zakharov系统-和由d'Ancona,Foschi和Selberg检测到的系统的零结构,以及由于Selberg的改进的双线性Schahartz估计。整体存在性证明使用了Colliander,Holmer和Tzirakis关于1D Zakharov系统的思想。

英文摘要:

The Cauchy problem for the classical Dirac-Klein-Gordon system in two space dimensions is globally well-posed for L^2 Schoedinger data and wave data in H^{1/2} \times H^{-1/2}. In the case of smooth data there exists a global smooth (classical) solution. The proof uses function spaces of Bourgain type based on Besov spaces - previously applied by Colliander, Kenig and Staffilani for generalized Benjamin-Ono equations and also by Bejenaru, Herr, Holmer and Tataru for the 2D Zakharov system - and the null structure of the system detected by d'Ancona, Foschi and Selberg, and a refined bilinear Strichartz estimate due to Selberg. The global existence proof uses an idea of Colliander, Holmer and Tzirakis for the 1D Zakharov system.

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