Conformal Bach flow

In this article we introduce conformal Bach flow and establish its well-posedness on closed manifolds. We also obtain its backward uniqueness. To give an attempt to study the long-time behavior of conformal Bach flow, assuming that the curvature and the pressure function are bounded, global and local Shi's type $L^2$-estimate of derivatives of curvatures are derived. Furthermore using the $L^2$-estimate and based on an idea from \cite{St13} we show Shi's pointwise-estimate of derivatives of curvatures without assuming Sobolev constant bound.