Let \(X\) be a compact Kähler manifold and let \(E\) be a holomorphic vector bundle over \(X\). The author considers a self-dual Yang-Mills Higgs functional defined on; unitary conections \(A\), and smooth sections \(\varphi\) of the bunde \(E\). The functional is of the same type considered in the now classical works of A. Jaffe, C. Taubes and S. Bradlow. For the real parameter \(\lambda > 0\) in the functional the associated vortex equations are studied. The author gives the global existence of smooth solutions to heat flow for a self-dual Yang-Mills-Higgs field on \(E\). Assuming the \(\lambda\)-stability of \((E,\varphi)\), the author shows the existence of the Hermitian Yang-Mills-Higgs metric on the holomorphic bundle \(E\) by study the limiting behaviour of the gauge flow. The paper is a good contribution to understanding of the fine tools involved in the study of the Yang-Mills-Higgs functional.