Gradient estimates for positive solutions of the heat equation under geometric flow

We establish first-and second-order gradient estimates for positive solutions of the heat equations under general geometric flows.Our results generalize the recent work of S. Liu, who established similar results for the Ricci flow.Both results can also be considered as the generalization of P. Li, S. T. Yau, and J. Li's gradient estimates under geometric flow setting.We also give an application to the mean curvature flow.