The classic Donaldson-Kronheimer book (Geometry of 4-manifolds) uses the Yang Mills gradient flow (sometimes called heat flow) on $M$ all over the place,

$\frac{d A}{dt} = -\frac{\delta YM(A)}{\delta A}$

where $YM(A)$ is the Yang Mills 'action' the integral of the curvature square,

$YM(A) = \int d^4x Tr F_{\mu\nu} F_{\mu\nu} > 0$

The setting is quite general, either $M$ is a general 4-manifold or Kahler manifold and so all theorems, existence, uniqueness, etc, are quite general.

I'm wondering if there are further results somewhere for the specific case of $M = T^4$ the 4-torus. For example, is it known how the long time asymptotics look like in this case? Theorems about possible blow-ups? I'd think asymptotically the gradient flow drives the connection towards a critical point but is it known how it is approached, exponentially or polynomially in $t$?

Actually for $M=T^4$ I suspect $t^2 YM(A(t))$ goes to a constant for $t\to\infty$ as long as the initial condition for the flow is in a sufficiently small neighbourhood of the absolute minimum of $YM(A)$ but I can't prove it. What is certainly true is that $YM(A(t))$ is a decreasing function of $t$.