  This paper collects results concerning global rates and large time
asymptotics of a fractional fast diffusion on the Euclidean space, which is
deeply related with a family of fractional Gagliardo-Nirenberg-Sobolev
inequalities. Generically, self-similar solutions are not optimal for the
Gagliardo-Nirenberg-Sobolev inequalities, in strong contrast with usual
standard fast diffusion equations based on non-fractional operators. Various
aspects of the stability of the self-similar solutions and of the entropy
methods like carr{\'e} du champ and R{\'e}nyi entropy powers methods are
investigated and raise a number of open problems.
