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In this note, we propose a new relative entropy combination of the methodsdeveloped 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 generalkernels in mean field limit theory. This new relative entropy may be understoodas introducing appropriate weights in the relative entropy developed by P.-E.Jabin and Z. Wang (in the spirit of what has been recently developed byD.~Bresch and P.--E. Jabin [Annals of Maths (2018)]) to cancel the moresingular terms involving the divergence of the flow. As an example, a fullrigorous derivation (with quantitative estimates) of the Patlak-Keller-Segelmodel in some subcritical regimes is obtained. Our new relative entropy allowsto treat singular potentials which combine large smooth part, small attractivesingular part and large repulsive singular part.

On Mean Field Limit and Quantitative Estimates with a Large Class of Singular Kernels: Application to the Patlak-Keller-Segel Model