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