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As a counterpart of the classical Yamabe problem, a fractional Yamabe flowhas been introduced by Jin and Xiong (2014) on the sphere. Here we pursue itsstudy in the context of general compact smooth manifolds with positivefractional curvature. First, we prove that the flow is locally well posed inthe weak sense on any compact manifold. If the manifold is locally conformallyflat with positive Yamabe invariant, we also prove that the flow is smooth andconverges to a constant scalar curvature metric. We provide different proofsusing extension properties introduced by Chang and Gonz\'alez (2011) for theconformally covariant fractional order operators.

Weak and smooth solutions for a fractional Yamabe flow: the case of general compact and locally conformally flat manifolds