The Higher Order Yang–Mills–Higgs Flow Over Riemannian Manifold

In this paper, we introduce a novel higher-order Yang–Mills–Higgs functional $$\mathcal {YMH}_k(\nabla ,u)$$, which is closely associated with a generalized Higgs-like potential W. First, by employing de Turck’s trick, we establish the local existence of the negative gradient flow, thus refining the ellipticity argument presented in [Car. Var. PDE 58 (2019), 100]. We establish the long-time existence of the flow under conditions on the order of derivatives appearing in the functional and on the degree of the potential W. This result refines and extends the one given in [J. Aust. Math. Soc. 113 (2022), 257-287]. At last, we discuss the physical applications of our higher-order Yang–Mills–Higgs theory and its long-time flow, clarifying the rationale for the higher-order terms and the polynomial potential, rooted in specific mathematical models.