  In this note, we propose a new relative entropy combination of the methods
developed by P.--E. Jabin and Z.~Wang [Inventiones (2018)] and by S. Serfaty
[Proc. Int. Cong. of Math, (2018) and references therein] to treat more general
kernels in mean field limit theory. This new relative entropy may be understood
as introducing appropriate weights in the relative entropy developed by P.-E.
Jabin and Z. Wang (in the spirit of what has been recently developed by
D.~Bresch and P.--E. Jabin [Annals of Maths (2018)]) to cancel the more
singular terms involving the divergence of the flow. As an example, a full
rigorous derivation (with quantitative estimates) of the Patlak-Keller-Segel
model in some subcritical regimes is obtained. Our new relative entropy allows
to treat singular potentials which combine large smooth part, small attractive
singular part and large repulsive singular part.
