In this paper, we define a family of functionals generalizing the Yang–Mills–Higgs functionals on a closed Riemannian manifold. Then we prove the short-time existence of the corresponding gradient flow by a gauge-fixing technique. The lack of a maximum principle for the higher order operator brings us a lot of inconvenience during the estimates for the Higgs field. We observe that the $L^2$ -bound of the Higgs field is enough for energy estimates in four dimensions and we show that, provided the order of derivatives appearing in the higher order Yang–Mills–Higgs functionals is strictly greater than one, solutions to the gradient flow do not hit any finite-time singularities. As for the Yang–Mills–Higgs k-functional with Higgs self-interaction, we show that, provided $\dim (M)<2(k+1)$ , for every smooth initial data the associated gradient flow admits long-time existence. The proof depends on local $L^2$ -derivative estimates, energy estimates and blow-up analysis.

在本文中，我们定义了一个泛函族，该泛函泛化了闭黎曼流形上的 Yang-Mills-Higgs 泛函。然后我们通过规范固定技术证明了相应梯度流的短时存在。高阶算子没有极大值原理给我们在估计希格斯场时带来了很多不便。我们观察到 Higgs 场的 $L^2$ -bound 足以在四个维度上进行能量估计，并且我们表明，如果出现在高阶 Yang-Mills-Higgs 泛函中的导数的阶严格大于一，梯度流的解不会遇到任何有限时间奇点。至于具有 Higgs 自交互作用的 Yang-Mills-Higgs k泛函，我们证明，如果 $\dim (M)<2(k+1)$ ，对于每个平滑的初始数据，相关的梯度流允许长期存在。证明取决于局部 $L^2$ -导数估计、能量估计和爆破分析。