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.

中文翻译：

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关于具有一大类奇异核的平均场限制和定量估计：在 Patlak-Keller-Segel 模型中的应用

在本说明中，我们提出了 P.--E 开发的方法的新相对熵组合。Jabin 和 Z.~Wang [发明 (2018)] 和 S. Serfaty [Proc. 国际。丛。数学，（2018 年）和其中的参考文献] 在平均场极限理论中处理更一般的内核。这个新的相对熵可以理解为在 P.-E 开发的相对熵中引入适当的权重。Jabin 和 Z. Wang（本着 D.~Bresch 和 P.--E. Jabin [Annals of Maths (2018)] 最近开发的精神）取消了涉及流散度的更奇异的术语。例如，获得了在某些亚临界状态下 Patlak-Keller-Segel 模型的完全严格推导（带有定量估计）。我们新的相对熵允许处理结合大平滑部分的奇异势，