Long time existence and bounded scalar curvature in the Ricci-harmonic flow

In this paper we study the long time existence of the Ricci-harmonic flow in terms of scalar curvature and Weyl tensor which extends Cao's result [6] in the Ricci flow. In dimension four, we also study the integral bound of the “Riemann curvature” for the Ricci-harmonic flow generalizing a recently result of Simon [38].