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 $t$.



Yang Mills gradient/heat flow on 4-torus
Donaldson and Kronheimer wrote their book by 1990. There were some further developments about long time behaviour of Yang-Mills flow on four manifolds by, among others, Struwe and collaborators. You may try starting with [Schlatter's dissertation](http://www.ams.org/mathscinet-getitem?mr=1443269). 


Crawling through MathSciNet reference links may get you "somewhere", but it is my impression that higher dimensional Yang-Mills heat flow is still somewhat of an open problem. Is there any reason why you expect $\mathbb{T}^4$ would be better behaved than, say, the unit ball?


