We investigate the long time behaviour of the Yang-Mills heat flow on the bundle \(\mathbb {R}^4\times SU(2)\) . Waldron (see Waldron, A.: Long-time existence for Yang-Mills flow. Invent. Math. 4, 217, 1069–1147 (2019)) 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 blows-up in infinite time at a given point in \(\mathbb {R}^4\) . Second, we prove the existence of 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.
