This is a report on joint work with Alex Waldron.

The Yang-Mills functional is the most studied functional on the space of connections on a vector bundle over an oriented Riemannian manifold. Its negative gradient flow leads to a semi-parabolic PDE known as the Yang-Mills flow.

I will introduce this flow and talk about its properties in the context of manifolds with special holonomy, particularly in Kahler, $G_2$, and $\operatorname{Spin}(7)$-manifolds. I intend to explain a blow-up criteria and talk about relationships with certain minimal “submanifolds” known as calibrated.