It will be shown how the critical mass classical Keller-Segel system and
the critical displacement convex fast-diffusion equation in two
dimensions are related. On one hand, the critical fast diffusion
entropy functional helps to show global existence around equilibrium
states of the critical mass Keller-Segel system. On the other hand, the
critical fast diffusion flow allows to show functional inequalities such
as the Logarithmic HLS inequality in simple terms who is essential in the
behavior of the subcritical mass Keller-Segel system. HLS inequalities can
also be recovered in several dimensions using this procedure. It is
crucial the relation to the GNS inequalities obtained by DelPino and
Dolbeault. This talk corresponds to two works in preparation together
with E. Carlen and A. Blanchet, and with E. Carlen and M. Loss.