In "The Yang Mills flow in four dimensions", M. Struwe proves that this flow converges, up to bubbling phenomena. And he has conjectured that this explosion in finite time should happen as proven for the harmonic heat flow by Chang, Ding and Ye. Looking on MathSciNet, I have found no proof or disproof of this fact. Is this question still open?

On the other hand, Donaldson proves that if the underlying bundle is stable there is a global existence. Has this hypothesis been weakened?

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    $\begingroup$ Link to Struwe's paper: download.springer.com/static/pdf/421/… $\endgroup$ – Noah Schweber Jun 12 '14 at 18:48
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    $\begingroup$ I assume you are only interested in Dimension 4. But for the benefit of the casual reader: Råde (1992) proved global existence in dimensions 2 and 3 for the Yang Mills heat flow over compact manifolds. And in dimensions 5 and higher Naito (1994) showed finite time blow-up. As of 2002 dimension 4 is still open. $\endgroup$ – Willie Wong Jun 13 '14 at 8:44
  • $\begingroup$ In terms of global existence, the results I am aware of are all under symmetry assumptions. There's originally the work of Schlatter, Struwe, and Tahvildar-Zadeh (1998) which proved global existence for $SU(2)$ over $\mathbb{R}^4$ under an equivariance assumption, and this has been generalised a bit by Hong and Tian (2004). $\endgroup$ – Willie Wong Jun 13 '14 at 9:04

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