论文标题
$ su（2）$ yang-mills热流$ \ mathbb {r}^4 $的无限时间冒泡
Infinite time bubbling for the $SU(2)$ Yang-Mills heat flow on $\mathbb{R}^4$
论文作者
论文摘要
我们研究了bundle $ \ mathbb {r}^4 \ times su（2）$上的Yang-Mills热流的长时间行为。 Waldron \ cite {waldron2019}证明了封闭$ 4- $歧管流量的全球存在和平滑度，在无限时间内使行为的问题打开了。我们展示了两种长期冒泡的类型：首先，我们构建一个初始数据和一个全球定义的解决方案，该解决方案在$ \ Mathbb r^4 $中的给定点中{\ sl buld-up-up}。其次，我们证明了{\ sl sl bubble-tower}解决方案的存在，也可以在无限的时间内。这回答了Yang-Mills连接在临界尺寸$ 4 $中的热流的基本动力学特性，并特别表明通常不能期望这种梯度流会收敛到Yang-Mills连接。我们强调，我们不假定第一个结果任何对称假设。而泡沫塔的存在的第二个结果是$（4）$ - 均等级别，但仍然是新的。
We investigate the long time behaviour of the Yang-Mills heat flow on the bundle $\mathbb{R}^4\times SU(2)$. Waldron \cite{Waldron2019} proved global existence and smoothness of the flow on closed $4-$manifolds, leaving open the issue of the behaviour in infinite time. We exhibit two types of long-time bubbling: first we construct an initial data and a globally defined solution which {\sl blows-up} in infinite time at a given point in $\mathbb R^4$. Second, we prove the existence of {\sl bubble-tower} solutions, also in infinite time. This answers the basic dynamical properties of the heat flow of Yang-Mills connection in the critical dimension $4$ and shows in particular that in general one cannot expect that this gradient flow converges to a Yang-Mills connection. We emphasize that we do not assume for the first result any symmetry assumption; whereas the second result on the existence of the bubble-tower is in the $SO(4)$-equivariant class, but nevertheless new.
