论文标题
关于Yang-Mills-higgs方程的全球动力学
On the global dynamics of Yang-Mills-Higgs equations
论文作者
论文摘要
我们研究了Yang-Mills-Higgs方程的解决方案，介绍了在$ \ Mathbb {R}^3 $的半径$ r $中给出的数据的最大开发。数据的能量可以是无限的，并且解决方案在$ r-t $中以$ t \ rightarrow r $ $ $ $ $ $ $。作为应用程序，我们将未来的Yang-mills-higgs字段的衰减估计值或Minkowski Space $ \ Mathbb {r}^{1+3} $在加权能量空间中绑定的，具有权重$ | x | x |^{1+ε} $。此外，对于麦克斯韦 - 克莱因·戈登系统的阿贝尔案例，我们将Lindblad和Sterbenz的小数据结果扩展到一般的大数据（在相同的假设下，但没有任何较小的假设）。该证明是独立的，它基于Eardley和Moncrief的框架，以及由Klainerman和Rodnianski构建的几何Kirchhoff-Sobolev参数。新成分是通过适合初始数据的向后光锥进行的一类加权能量估计。
We study solutions to the Yang-Mills-Higgs equations on the maximal Cauchy development of the data given on a ball of radius $R$ in $\mathbb{R}^3$. The energy of the data could be infinite and the solution grows at most inverse polynomially in $R-t$ as $t\rightarrow R$. As applications, we derive pointwise decay estimates for Yang-Mills-Higgs fields in the future of a hyperboloid or in the Minkowski space $\mathbb{R}^{1+3}$ for data bounded in the weighted energy space with weights $|x|^{1+ε}$. Moreover, for the abelian case of Maxwell-Klein-Gordon system, we extend the small data result of Lindblad and Sterbenz to general large data (under same assumptions but without any smallness). The proof is gauge independent and it is based on the framework of Eardley and Moncrief together with the geometric Kirchhoff-Sobolev parametrix constructed by Klainerman and Rodnianski. The new ingredient is a class of weighted energy estimates through backward light cones adapted to the initial data.
