The Yang-Mills heat flow and the caloric gauge
The Yang-Mills heat flow and the caloric gauge
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DOI：
10.24033/ast.1179
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发表时间：
2017-09
期刊：
影响因子：
--
通讯作者：
Sung-Jin Oh;D. Tataru
中科院分区：
文献类型：
--
作者：
Sung-Jin Oh;D. Tataru
这是四篇论文序列的第一部分，它建立了阈值猜想和孤子泡与~（4 + 1）维Minkowski时空中能量临界双曲型杨-米尔斯方程的散射二分法本文的主要研究对象是四维欧氏空间上的另一个偏微分方程，即能量临界杨-米尔斯热流。我们的第一个目标是建立这个系统在$\dot{H}^{1}$中的全局存在性和渐近收敛到平坦联络的尖锐准则，包括二分法定理（即，或者上述性质成立或者谐波杨-米尔斯连接起泡）和阈值定理（即，如果初始能量小于基态能量的两倍，则上述性质成立）。我们的第二个目标是使用杨-米尔斯热流，以确定热量规范，这将发挥重要作用，在分析双曲杨-米尔斯方程在随后的文件。
This is the first part of the four-paper sequence, which establishes the Threshold Conjecture and the Soliton Bubbling vs.~Scattering Dichotomy for the energy critical hyperbolic Yang--Mills equation in the (4 + 1)-dimensional Minkowski space-time. The primary subject of this paper, however, is another PDE, namely the energy critical Yang--Mills heat flow on the 4-dimensional Euclidean space. Our first goal is to establish sharp criteria for global existence and asymptotic convergence to a flat connection for this system in $\dot{H}^{1}$, including the Dichotomy Theorem (i.e., either the above properties hold or a harmonic Yang--Mills connection bubbles off) and the Threshold Theorem (i.e., if the initial energy is less than twice that of the ground state, then the above properties hold). Our second goal is to use the Yang--Mills heat flow in order to define the caloric gauge, which will play a major role in the analysis of the hyperbolic Yang--Mills equation in the subsequent papers.