Local curvature estimates for the Ricci-harmonic flow

In this paper we give an explicit bound of ${\Delta }_{g(t)}u(t)$ and the local curvature estimates for the Ricci-harmonic flow under the condition that the Ricci curvature is bounded along the flow. In the second part these local curvature estimates are extended to a class of generalized Ricci flow, introduced by the author, whose stable points give Ricci-flat metrics on a complete manifold, and which is very close to the $(K,N)$-super Ricci flow recently defined by Xiangdong Li and Songzi Li. Next we propose a conjecture for Einstein's scalar field equations motivated by a result in the first part and the bounded $L^2$-curvature conjecture recently solved by Klainerman, Rodnianski and Szeftel. In the last two parts of this paper, we discuss two notions of "Riemann curvature tensor" in the sense of Wylie-Yeroshkin, respectively, and Li, whose "Ricci curvature" both give the standard Bakey-Émery Ricci curvature, and the forward and backward uniqueness for the Ricci-harmonic flow.