Infinite time bubbling for the $SU(2)$ Yang-Mills heat flow on $\mathbb{R}^4$

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.