We establish new local regularity results for the harmonic map and Yang–Mills heat flows on Riemannian manifolds of dimension greater than 2 and 4, respectively, obtaining criteria for the smooth local extensibility of these flows. As a corollary, we obtain new characterisations of singularity formation and use this to obtain a local estimate on the Hausdorff measure of the singular sets of these flows at the first singular time. Finally, we show that smooth blow-ups at rapidly forming singularities of these flows are necessarily nontrivial and admit a positive lower bound on their heat ball energies. These results crucially depend on some local monotonicity formulæ for these flows recently established by Ecker (Calc Var Partial Differ Equ 23(1):67–81, 2005) and the Afuni (Calc Var 555(1):1–14, 2016; Adv Calc Var 12(2):135–156, 2019).

我们分别建立了分别在尺寸大于2和4的黎曼流形上的谐谱图和Yang-Mills热流的新局部规则性结果，从而获得了这些流的平滑局部可扩展性的标准。作为推论，我们获得了奇异点形成的新特征，并用它来获得在第一个奇异时刻对这些流的奇异集的Hausdorff测度的局部估计。最后，我们表明，在这些流的快速形成的奇异点处的平稳爆炸必定是不平凡的，并允许它们的热球能量具有正的下限。这些结果主要取决于一些局部单调性公式，这些公式由Ecker（Calc Var Partial Differ Equ 23（1）：67–81，2005）和Afuni（Calc Var 555（1）：1-14，2016; Adv Calc Var 12（2）：135–156，2019）。