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 behaviour of conformal Bach flow, assuming that the curvature and the pressure function are bounded, global and local Shi's type L2-estimate of derivatives of curvatures are derived. To make the talk more accessible, we will spend some time to survey on high-order parabolic curvature flow.
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