论文标题
Yang-Mills的量规不变i-hethod
The gauge-invariant I-method for Yang-Mills
论文作者
论文摘要
我们证明了$ h^σ$的$ 3D $ yang-mills方程的全局良好性，$ h^σ$ for $σ> \ frac {5} {6} $。 与相关的方程式不同，杨米尔斯不受其傅立叶和全球版本中几乎保护法（i-hethod）的方法的直接调整。我们提出了一种修改的能量，该能量： 1）是衡量不变且易于本地化的 2）提供局部仪表，以控制局部Sobolev规范（通过uhlenbeck-type诱导分数的规律性） 3）与相关系统的经典i-method能量相比，时间略微更平滑。 空间平滑是通过Yang-Mills的热流而不是乘数$ i $实现的。 由于时间条件及其有限的传播速度，局部量规选择与最近的初始数据扩展结果兼容。因此，可以将平滑的能量差异分为本地碎片（适当扩展）的界限，可以求和。在揭示了三联积分内的无效结构后，可以使用已知方法估算这些结构。 在附录中，我们展示了Maxwell-Klein-Gordon不变的修改能量如何将以前的结果扩展到规律性$σ> \ frac {5} {6} $。
We prove global well-posedness of the $ 3d $ Yang-Mills equation in the temporal gauge in $ H^σ $ for $ σ> \frac{5}{6} $. Unlike related equations, Yang-Mills is not directly amenable to the method of almost conservation laws (I-method) in its Fourier and global version. We propose a modified energy which: 1) Is gauge-invariant and easy to localize 2) Provides local gauges which give control of local Sobolev norms (through an Uhlenbeck-type lemma for fractional regularities) 3) Is slightly smoother in time compared to the classical I-method energy for related systems. The spatial smoothing is realized via the Yang-Mills heat flow instead of the multiplier $I$. Due to the temporal condition and its finite speed of propagation, the local gauge selection is compatible with recent initial data extension results. Therefore, smoothened energy differences can be partitioned into local pieces whose (appropriately extended) bounds can be square summed. After revealing the null structure within the trilinear integrals, these can be estimated using known methods. In an appendix we show how an invariant modified energy for Maxwell-Klein-Gordon can extend previous results to regularities $ σ> \frac{5}{6} $.
