热规范中的双曲杨-米尔斯方程。能量分散解的局部适定性与控制
The hyperbolic Yang--Mills equation in the caloric gauge. Local well-posedness and control of energy dispersed solutions
摘要：
这是四篇论文中的第二部分，它建立了阈值猜想和孤子泡与~ $(4+1)$维时空中双曲型杨-米尔斯方程的散射二分法。本文给出了双曲型杨-米尔斯方程的关键规范依赖分析。我们认为拓扑平凡的解决方案中的热量规范，这是在第一篇论文arXiv：1709.08599使用杨-米尔斯热流定义。在这个规范中，我们建立了一个强形式的局部适定性，其中存在的时间是有界的能量浓度标度从下面。此外，我们还表明，只要能量色散较小，溶液的规则性和色散行为就会持续存在。我们还观察到，固定时间的规律性（但不分散）的热量计的属性可以转移到时间的规范没有任何损失，证明作为一个结果小数据的整体适定性的时间规范。本文的结果在随后的arXiv：1709.08604，arXiv：1709.08606中得到应用，证明了平凡拓扑类中热规范的锐阈值定理和任意拓扑类中的二分法定理.
英文摘要：
This is the second part in a four-paper sequence, which establishes the Threshold Conjecture and the Soliton Bubbling vs.~Scattering Dichotomy for the hyperbolic Yang--Mills equation in the $(4+1)$-dimensional space-time. This paper provides the key gauge-dependent analysis of the hyperbolic Yang--Mills equation. We consider topologically trivial solutions in the caloric gauge, which was defined in the first paper arXiv:1709.08599 using the Yang--Mills heat flow. In this gauge, we establish a strong form of local well-posedness, where the time of existence is bounded from below by the energy concentration scale. Moreover, we show that regularity and dispersive behavior of the solution persists as long as energy dispersion is small. We also observe that fixed-time regularity (but not dispersive) properties in the caloric gauge may be transferred to the temporal gauge without any loss, proving as a consequence small data global well-posedness in the temporal gauge. The results in this paper are used in the subsequent papers arXiv:1709.08604, arXiv:1709.08606 to prove the sharp Threshold Theorem in caloric gauge in the trivial topological class, and the dichotomy theorem in arbitrary topological classes.
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