2022/08/29 by Sire, Yannick, Wei, Juncheng, Zheng, Youquan
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2208.13875
We investigate the long time behaviour of the Yang-Mills heat flow on the bundle ℝ4× SU(2). Waldron \citeWaldron2019 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 R4. 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.